Advances in Imaging and Electron Physics, Volume 94

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 94 EDITOR-IN-CHIEF PETER W. HAWKES CEMEULaboratoire d’Optique Electr...

Author: Peter W. Hawkes

12 downloads 595 Views 16MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 94

EDITOR-IN-CHIEF

PETER W. HAWKES CEMEULaboratoire d’Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

ASSOCIATE EDITORS

BENJAMIN KAZAN Xerox Corporation

Palo Alto Research Center Palo Alto, California

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics EDITED BY PETER W. HAWKES CEMES/Laboratoire d’Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

VOLUME 94

u ACADEMIC PRESS San Diego New York Boston London Sydney Tokyo Toronto

This book is printed on acid-free paper. @ Copyright 0 1995 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NWI 7DX

International Standard Serial Number: 1076-5670 International Standard Book Number: 0-12-014736-X PRINTED IN THE UMTED STATES OF Ah4EFXA 95 96 9 7 9 8 99 0 0 B B 9 8 7 6

5

4

3 2

1

CONTENTS CONTRIBUTORS. PREFACE . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Group Algebras in Signal and Image Processing DENNIS WENZELA N D DAVIDEBERLY 1. Introduction . . . . . . . . . . . . . . . . . . . 11. Convolution by Sections . . . . . . . . . . . . . . 111. Group Algebras . . . . . . . . . . . . . . . . . IV. Direct Product Group Algebras . . . . . . . . . . V. Semidirect Product Group Algebras . . . . . . . . VI. Inverses of Group Algebra Elements . . . . . . . . VII. Matrix Transforms . . . . . . . . . . . . . . . . VIII. Feature Detection Using Group Algebras . . . . . . IX. The GRASP Class Library . . . . . . . . . . . . . X. GRASP Class Library Source Code . . . . . . . . . References. . . . . . . . . . . . . . . . . . . .

. . . . . .

. .

. . . . .

. . .

. . . . . .

ix xi

2 4 8 14 19 22 27 44 55 63 78

Mirror Electron Microscopy REINHOLD GODEHARDT

Introduction . . . . . . . . . . . . . . . . . . . . Principles of Mirror Electron Microscopy . . . . . . . Applications of the Mirror Electron Microscope . . . . Fundamentals of Quantitative Mirror Electron Microscopy Quantitative Contrast Interpretation of One-Dimensional Electrical Distortion Fields . . . . . . . . . . . . . VJ. Quantitative Contrast Interpretation of One-Dimensional Magnetic Distortion Fields . . . . . . . . . . . . . . VII. Quantitative Mirror Electron Microscopy, Example 1: Contrast Simulation of Magnetic Fringing Fields of Recorded Periodic Flux Reversals . . . . . . . . . . 1. 11. 111. IV. V.

V

. . . .

81 83 95 98

.

109

.

114

.

119

vi

CONTENTS

VIII . Quantitative Mirror Electron Microscopy. Example 2: Investigation of Electrical Microfields of Rotational Symmetry and Evaluation of Current Filaments in Metal-Insulator-Semiconductor Samples . . . . . . . . IX. Concluding Remarks . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

124 144 145

Rough Sets JERZYW . GRZYMALA-BUSSE I. Fundamentals of Rough Set Theory . . . . . . . . . . . I1. Rough Set Theory and Evidence Theory . . . . . . . . I11. Some Applications of Rough Sets . . . . . . . . . . . IV. Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

152 171 182 194 194

Theoretical Concepts of Electron Holography FRANKKAHLAND HARALDROSE

I. Introduction . . . . . . . . . . . . . . . . . . . . . I1. The Scattering Amplitude . . . . . . . . . . . . . . . . I11. Image Formation in Transmission Electron Microscopy . . IV. Off-Axis Holography in the Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . . . V. Image Formation in Scanning Transmission Electron Microscopy . . . . . . . . . . . . . . . . . . . . . VI . Off-Axis Scanning Transmission Electron Microscope Holography . . . . . . . . . . . . . . . . . . . . . VII . Conclusion . . . . . . . . . . . . . . . . . . . . . . Appendix: Fresnel Diffraction in Off-Axis Transmission Electron Microscope Holography . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

197 200 203 213 221 232 252 253 256

Quantum Neural Computing SUBHASH C. KAK

I. Introduction . . . . . . . . . . . . . . . . . . . . . I1. Turing Test Revisited . . . . . . . . . . . . . . . . .

260 266

vii

CONTENTS

I11. Neural Network Models . . . . . . . IV. The Binding Problem and Learning . . V. On Indivisible Phenomena . . . . . . VI . On Quantum and Neural Computation VII . Structure and Information . . . . . . VIII . Concluding Remarks . . . . . . . . References . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

272 284 293 298 305 309 310

Signal Description: New Approaches. New Results ALEXANDER M. ZAYEZDNY AND ILANDRUCKMANN I . Introduction . . . . . . . . . . . . . . . . . . . . . I1. Theoretical Basis of the Structural Properties Method . . I11. Generalized Phase Planes . . . . . . . . . . . . . . IV. Applications . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

INDEX

. . . . . . . . . . . . . . . . . . . . . . . . . .

. .

315 327 347 365 394

397

This Page Intentionally Left Blank

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin.

ILAN DRUCKMANN (315), Lasers Group, Nuclear Research Center Negev, Beer Sheeva 84190, Israel DAVIDEBERLY(l),Department of Computer Science and Neurosurgery, University of North Carolina, Chapel Hill, North Carolina 27599 REINHOLD GODEHARDT (81), Department of Materials Science, Martin Luther University Halle Wittenberg, D-06217 Merseburg, Germany JERZY W. GRZYMALA-BUSSE (151), Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, Kansas 66045

FRANK KAHL(197), Institute of Applied Physics, Technical University of Darmstadt, D-64289 Darmstadt, Germany SUBHASH C. KAK(259), Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, Louisiana 70803 HARALD ROSE(197), Institute of Applied Physics, Technical University of Darmstadt, D-64289 Darmstadt, Germany DENNIS WENZEL(l), Phototelesis Corporation, San Antonio, Texas 78230 ALEXANDER M. ZAYEZDNY (315), Department of Electrical Engineering and Computers, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

ix


PREFACE

The six chapters that make up this volume are concerned with themes of current interest in image processing, electron microscopy and holography, signal handling, and set theory. We begin with a study by Dennis Wenzel and David Eberly of the role of group algebras in image processing, which is inspired by the heavy use of convolutional filters in image enhancement. The opening part of this chapter is inevitably rather abstruse but the authors have taken considerable trouble to provide examples of each stage of development to assist the reader. It emerges that there is a much wider class of convolutions than the familiar cyclic ones and that the members of this class may well be valuable for image analysis; once again examples are provided. The chapter ends with a formal description of a C++ class library called Group Algebras for Signal Processing (GRASP). The second contribution, by Reinhold Godehardt, contains a full account, the first such survey for several years, of a type of electron microscope in which the beam is reflected from the surface of the specimen, the mirror electron microscope. Here the beam is incident approximately normal to the specimen plane and hence a specially designed instrument is needed, unlike the reflection modes of the everyday transmission electron microscope, for which only the specimen holder is different. The author first discusses the history and instrumental aspects of these microscopes, which go back to the first decade of the electron microscope; he then presents in detail quantitative mirror electron microscopy of several kinds of objects. A short section is devoted to the principles, after which one-dimensional electric and magnetic distortion fields are examined. Several concrete examples are investigated in great detail, notably metal-insulator-semiconductor structures. The chapter is dedicated to Johannes Heydenreich, who has contributed to our understanding of these relatively unfamiliar instruments, on the occasion of his 65th birthday and we associate our good wishes with those of the author. In the third chapter, by Jerzy W. Grzymala-Busse, we are introduced to the notions of rough sets. I came across these in a volume of conference proceedings and, suspecting that I was not alone in not knowing what they were, I was delighted that the acknowledged expert agreed to prepare an account of them and their practical uses for these Advances. They go back to the work of Z . Pawlak in 1982 and are proving to be invaluable for xi

xii

PREFACE

handling granularity of data, inconsistent data, and other related questions. Jerzy W. Grzymala-Busse first sets out the theory of these sets (which are not to be confused with fuzzy sets) and then discusses several practical examples. I have no doubt that this account of the subject will be welcomed and heavily used. Holography was invented by D. Gabor in the late 1940s in the hope of circumventing the limited resolution of the electron microscope imposed by the aberrations of the objective lens. With the invention of the laser, photons took over, but, thanks to the introduction of the electron biprism by Mbllenstedt and Diiker and to the development of field-emission electron sources, it was revived in electron microscopy in the late 1960s. For many years the most impressive results came from A. Tonomura and colleagues in the Hitachi Research Laboratory in Japan, but they were soon joined by H. Lichte’s team in Tiibingen and another group in Bologna around G. Pozzi. Very impressive reconstructions are now being made from electron holograms, especially since numerical reconstruction (rather than optical reconstruction) has become current. A study of the statistical properties of these techniques is thus urgently needed so that we can understand how far such reconstructions can be expected to go and how reliable they will be. This is one of the main preoccupations of Frank Kahl and Harald Rose, from the Technical University of Darmstadt, who have contributed a careful study of the theoretical aspects of electron holography. Both the conventional and the scanning transmission instruments are considered and the account of the latter contains much new material not available in such detail elsewhere. In order to introduce the fifth chapter, by Subhash C. Kak, I cannot do better than quote the opening sentences: “This chapter reviews the limitations of the standard computing paradigm and sketches the concept of quantum neural computing. . . . A quantum neural computer is a single machine that reorganizes itself, in response to a stimulus, to perform a useful computation.” The author leads us through the Turing test, neural network models, the binding problem and learning, uncertainty and complementarity, quantum and neural computation, and structure and information. I have to say that this is a challenging chapter and not all Kak’s ideas will find acceptance, but I believe that it is important to examine, in a series such as this, the difficulties that arise when the intrinsically imprecise cognitive sciences and the exact sciences are brought into contact or even collision. The volume concludes with a long contribution by Alexander M. Zayezdny and Ilan Druckman of Beer Sheva on new approaches and results in the field of signal description. They are principally concerned with the method based on the structural properties of signals, which are expressed in terms of the signal, its derivatives, and various transforms of it. The first author

PREFACE

...

Xlll

of this chapter has been deeply concerned with this method since its introduction and the presentation here is thus authoritative. I wish to stress once again that the authors have taken care to illustrate their material with concrete examples to help the newcomer to grasp the essentials of the subject. In conclusion, I thank all the authors most warmly for the trouble they have taken with the preparation of their contributions and conclude with a list of forthcoming chapters. We have two special volumes to look forward to in the near future, one of which is wholly given over to an account of map methods for the design of particle accelerators by M. Berz and colleagues. The other is in a sense a sequel to “The Beginnings of Electron Optics” (Supplement 16), entitled “The Growth of Electron Optics”; this contains a collection of essays on the history of the International Federation of Societies of Electron Microscopy (IFSEM), contributed by many of the member societies of that Federation, and gives an extremely vivid picture of the way the subject has developed.

Peter W. Hawkes

FORTHCOMING ARTICLES Nanofabrication Use of the hypermatrix Image processing with signal-dependent noise The Wigner distribution Parallel detection Discontinuities and image restoration Hexagon-based image processing Microscopic imaging with mass-selected secondary ions Modern map methods for particle optics Nanoemission Cadmium selenide field-effect transistors and display

H. Ahmed D. Antzoulatos H. H. Arsenault M. J. Bastiaans P. E. Batson L. Bedini, E. Salerno, and A. Tonazzini S . B. M. Bell M. T. Bernius M. Berz and colleagues (Vol. 97) Vu Thien Binh (Vol. 95) T. P. Brody, A. van Calster, and J. F. Farrell

xiv

PREFACE

ODE methods Electron microscopy in mineralogy and geology Projection methods for image processing Space-time algebra and electron physics Fuzzy morphology The study of dynamic phenomena in solids using field emission Gabor filters and texture analysis Miniaturization in electron optics Liquid metal ion sources The critical-voltage effect Stack filtering Median filters

RF tubes in space Relativistic microwave electronics The quantum flux parametron The de Broglie-Bohm theory Contrast transfer and crystal images Seismic and electrical tomographic imaging

Morphological scale-space operations Algebraic approach to the quantum theory of electron optics Surface relief Spin-polarized SEM Sideband imaging The recursive dyadic Green’s function for ferrite circulators Ernst Ruska, a memoir

J. C. Butcher P. E. Champness P. L. Combettes (Vol. 95) C. Doran and colleagues (Vol. 95) E. R. Dougherty and D. Sinha M. Drechsler J. M. H. Du Buf A. Feinerman R. G. Forbes A. Fox M. Gabbouj N. C. Gallagher and E. Coyle A. S. Gilmour V. L. Granatstein W. Hioe and M. Hosoya P. Holland K. Ishizuka P. D. Jackson, D. M. McCann, and S. L. Shedlock P. Jackway R. Jagannathan and S. Khan J. J. Koenderink and A. J. van Doorn K. Koike W. Krakow C. M. Krowne

L. Lambert and T. Mulvey (Vol. 95)

xv

PREFACE

Regularization Near-field optical imaging Vector transformation SEM image processing Electronic tools in parapsychology The Growth of Electron Microscopy The Gaussian wavelet transform Phase-space treatment of photon beams Image plate Z-contrast in materials science HDTV The wave-particle dualism Scientific work of Reinhold Riidenberg Electron holography X-ray microscopy Accelerator mass spectroscopy Applications of mathematical morphology Set-theoretic methods in image processing Texture analysis Wavelet vector transforms Focus-deflection systems and their applications New developments in ferroelectrics Electron gun optics Very high resolution electron microscopy Morphology on graphs

A. Lannes A. Lewis W. Li N. C . MacDonald R. L. Morris T. Mulvey, ed. (Vol. 96) R. Navarro, A. Taberno, and G. Cristobal G. Nemes T. Oikawa and N. Mori S. J. Pennycook E. Petajan H. Rauch H. G. Rudenberg D. Saldin G. Schmahl J. P. F. Sellschop J. Serra M. I. Sezan H. C. Shen (Vol. 95) E. A. B. da Silva and D. G. Sampson T. Soma J. Toulouse Y.Uchikawa D. van Dyck L. Vincent


ADVANCES IN IMAGING AND ELECTRON PHYSICS. VOL. 94

Group Algebras in Signal and Image Processing DENNIS WENZEL Phototelesis Corporation. San Antonio. Texas

and

DAVID EBERLY Departments of Computer Science and Neurosurgery. University of North Carolina. Chapel Hill. North Carolina

I . Introduction . . . . . . . . . . . . . . . . . . . . I1. Convolution by Sections I11. Group Algebras . . . . . . . . . . . A . FiniteGroupsandIndexNotation . . . . B. Group Algebras and Group Convolution . . C. Matrix Representation of Group Elements . . . . . . . . IV. Direct Product Group Algebras A. Direct Product Groups and Index Notation . B. Tensor Products of Group Algebras . . . . C. Fast Convolution . . . . . . . . . . D. Examples . . . . . . . . . . . . V . Semidirect Product Group Algebras . . . . . A. Semidirect Product Groups and Index Notation B. Semidirect Tensor Products of Group Algebras C. Example . . . . . . . . . . . . VI . InverseofGroup AlgebraElements . . . . . A . Inverses . . . . . . . . . . . . . B. Ideals and Quotient Rings . . . . . . . VII. Matrix Transforms . . . . . . . . . . A . Vector Spaces of Group Algebra Elements . . B. Matrices with the Convolution Property . . C. Examples . . . . . . . . . . . . D . Direct Tensor Product Matrices . . . . . E. Semidirect Tensor Product Matrices . . . . F. Inverses of Matrices with Group Algebra Elements . G . Fast Computation of the Matrix Transform VIII . Feature Detection Using Group Algebras . . . A . Value Functions . . . . . . . . . . . . . . . . . . . B. Feature Detection . . . . . . . . C. Application to Signals . . . . . . . . D . Application to Images 1

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . . . .

. . . . . . . . . . . . . . . . . .

. 1 4 8 8 10 12 14 14 16 17 19 19 20 21 22 22 23 24 27 28 29 32 39 40 42 43 44 45 45 46 46

Copyright 0 1995 by Academic Press. Inc. All rights ofreproduction in any form reserved.

2

DENNIS WENZEL AND DAVID EBERLY

IX. The GRASP Class Library . . A. Groupclasses . . . . . B. Group Algebraclass . . C. Group Algebra Matrices Class . . . . D. Example Usage X. GRASP Class Library Source Code A. gr0up.h . . . . . . . B. group.cpp . . . . . . C. grpa1g.h . . . . . . . D. grpalg.cpp . . . . . . E. gmatrix.h . . . . . . F. gmatrix.cpp . . . . . References . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . .

. . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . . . . .

55 55 58 60 62 63 63 65 68 69 13 74 78

I. INTRODUCTION Filtering is an important operation in digital signal and image processing, used to enhance or deemphasize certain features. Although a variety of filtering methods exist, such as using look-up tables (e.g., thresholding), morphological operators (e.g., dilations and erosions), and neighborhood rankorder statistics (e.g., min, max, or median filtering), the primary filtering method used is convolution. For execution of the convolution operation, the coefficients of the filter function are organized into a template. In a patternmatching application, the template of coefficients represents the pattern to be matched and the correlation operation for measuring the quality of a match is mathematically equivalent to a convolution. In other applications, the coefficients in a convolution template are organized to generate output in which low-spatial frequencies are removed (e.g., Laplacian filter), highspatial frequencies are removed (e.g., Gaussian filter), or spatial discontinuities are identified (e.g., edge operators). A good discussion of these linear techniques can be found in Rosenfeld and Kak (1982). Nonlinear processes which generalize the linear convolution are based on nonlinear diffusion equations and can be found in ter Haar Romeny (1994). Two standard convolutions used are linear convolution and cyclic convolution, both of which are based on polynomial arithmetic. Treating the input data and the filter weights as coefficients of polynomials, linear convolution is computed directly as a product of these polynomials. Cyclic convolution is used as an efficient means of computing the linear convolution. In this case, the input is organized into sections, each section being treated as the coefficients of a polynomial. The output data are generated by selecting the last coefficients of the products of the section polynomials with the filter polynomial. We review this process in Section I1 as motivation for the later ideas in

GROUP ALGEBRAS IN SIGNAL AND IMAGE PROCESSING

3

the chapter. We develop a more general methodology for filtering signals and images based on two ideas. Group algebras. The linear and cyclic convolutions are computed as polynomial products. Group algebras are generalizations of polynomials and polynomial arithmetic based on algebraic groups. The multiplication in a group algebra is called group convolution. Loui (1976) and Willsky (1978) used group algebras based on a finite cyclic group in the context of random Markov processes. In particular, they constructed generalized Fourier transforms which preserve group convolution. A similar application of group algebras to signal processing is described by Eberly and Hartung (1987). Eberly and Wenzel (1991) and Wenzel (1988) extended these results to image processing. Applications of group algebras to interconnection networks is found in Gader (1986) and is formulated using the image algebra framework (Ritter et al., 1990). Vulue functions. The selection of coefficients from the cyclic convolution to generate output for the linear convolution can be extended to other functions, called value functions, to allow for a wider variety of outputs.

Section 111 contains the formal definitions for group algebras. Background material on groups and group convolution is also provided. Group convolution of a signal requires selecting a single base group to work with. For images, group convolution is naturally based on the direct product of groups. Matrix representation of group algebra elements is also discussed. Section IV provides a discussion of direct product group algebras. The ideas extend to convolution of data sets whose dimension is larger than 2. Section V is a discussion of semidirect product group algebras, which are used mainly in studying group algebras based on dihedral groups. Section VI discusses inverting group algebra elements. Two methods for inverting elements are given. In applications where nonzero and noninvertible group algebra elements may be a problem, the section also provides a way to eliminate these elements via ideals and quotient rings. Section VII discusses matrices whose elements are from a group algebra. In particular, we are interested in those matrices which have the convolution property, that is, those matrices for which the transform of a convolution is the product of the individual transforms. We also look at fast algorithms for group convolutions. Section VIII provides examples of the application of group algebras to edge detection in signals and images. The examples illustrate the use of direct product cyclic groups (the standard convolution case), direct product dihedral groups, and direct product quaternions. Section IX provides details of how to implement group algebras in an object-oriented programming language. Source code for a sample implementation is provided in the Appendix.

4


11. CONVOLUTION BY SECTIONS To motivate the ideas of group algebra and value function, we consider first the filtering of a sequence of numbers using the standard convolution by sections. Normally the input sequence is very long, appearing to be infinite, and it is assumed that the numbers in the sequence are sampled at equal intervals (temporally or spatially). A commonly used filter is the nonrecursive finite-impulse-response (FIR) filter. The output sequence from an FIR filter is a linear convolution of the input sequence and the sequence described by the filter tap weights. be the input sequence, where p is large. Let {fj}:=-, represent Let the sequence of filter tap weights, where N p and fj = 0 for j < 0 and j > N - 1. The output sequence is { r k } k p , + t - l . Another form of convolution is the cyclic convolution, which is closely related to the linear convolution. The input sequence {si}&* and the filter sequence {A}:=-,, have the same length. The cyclic convolution of the two sequences is defined by the polynomial product T(X)

= s ( X ) f ( X ) mod(xN- 1) = k=O

( c

(i+j)modN=k

Sifi

)

N-l Xk

=

1

rkXk.

k=O

The output sequence is {rk}[:t. The linear convolution can be computed from cyclic convolutions by sectioning the input sequence. Each short section is individually processed as a cyclic convolution and properly merged with the other sections to produce the linear convolution. Such an approach is advantageous because many fast algorithms exist for computing a short cyclic convolution (Blahut, 1988; Elliott and Rao, 1982). A common technique for performing this convolution by sections is the overlap-save method, which is briefly described below. Assume that the cyclic convolution will be performed on sequences of length N . Let { s ~ } &be~ the input sequence and let {fi}T=o be the


5

filter sequence, where q < N and p >> N . Construct the set of polynomials { ~ ‘ ~ ’ ( xs(’)(x), ), . . .}, where sy) = s ~ + ~ ( for ~ - ~0 )5 i < N and for some j < p , such that all coefficients of s(x) are assigned. Note that there is an overlap of q coefficients. Let r ( x ) = s ( x ) f ( x ) represent the linear convolution and let r(k)(x)= d k ’ ( x ) f ( x )mod(xN- 1 ) represent the kth cyclic convolution segment. Except for the first q coefficients, the coefficients of r(x) are found among the coefficients of r(k)(x),0 I k I j , given by = rij), for q I i < N . Each cyclic convolution except for the last one produces N - q coefficients of the linear convolution, with the remaining q coefficients discarbed. Example. Let f ( x ) = x 2 - 1, which represents a one-dimensional edge detection filter. Let s(x) represent the signal with p = deg(s) large. The linear convolution of f ( x ) with s ( x ) is given by r ( x ) = -so - S I X

+ (so

- s2)x2

+ (sl

- s3)x3

+ ... + ( s p p 2- S P ) X P + 2 .

Using the overlap-save method with cyclic convolutions of length 3, the input sequence is sectioned into blocks of three values, with each block treated as the coefficients for a polynomial d k ) ( x )= s k + s k + l x + S k + 2 X 2 , k = 0, . . ., p - 2. The output sequence values (starting with the x 2 term) are obtained by selecting the x 2 coefficient of each product, I ( ~ ) ( x= ) s(k)(x)f(x)(mod x3 -

+

1)

+ s k + 2 x 2 ) ( - 1 + x2)(modx 3 - 1) = (sk+l - s k ) + + - sk+2)x2, = (sk

sk+lx

(sk+2

sk+l)x

(sk

for k = 0,. . ., p - 2; that is, the output sequence is rk = s k - s k + 2 for k = 0, . . . , p - 2. The linear convolution of the input sequence and the filter sequence as a convolution by sections can be viewed pictorially as shown in Fig. 1, where the filter is superimposed over part of the input sequence, the necessary ...

input sequence multiplication filter sequence

addition output sequence

FIGURE1. Illustration of convolution for signals

6

DENNIS WENZEL A N D DAVID EBERLY

arithmetic is performed, and then the filter is moved to the right, where the process is repeated again. Note that the coefficients of the filter polynomial appear is reverse order. We consider next the filtering of an image { s i , , i 2 } @ ~ i 2 = The o . filter sequence is (fil,j2}j”,L-,;{&1, where Nl Nd - 1 (d = 1, 2). The Output Sequence is {rkl,kz}P:=+ON.Lk;l’~+N2-1. If the input polynomial s(x, y) has degrees p1 = Nl - 1 and p 2 = N2 - 1, then the cyclic convolution is defined by the polynomial product

jd

r(x, y) = s(x, y)f(x, y) mod(xN1- 1) mod(yNz- 1) = C k l e Z C k 2 E Z (C(il+jl)rnodN1=kl C(i,+j2)rnodN2=k2 Si,,i2fj~.j2)Xk1ykz~

The output sequence is {r~,,k2}~:f-j,f$~’. Similar to the case of signal processing, the linear convolution for images can be computed by sectioning the image into Nl x N2 blocks, treating the entries of the block as coefficients of a polynomial, and computing the cyclic convolutions of these polynomials with the filter polynomial.

+ +

Example. Let f(x, y ) = (x2 x l)(y2 - l), which represents a 2-dimensional vertical-edge emphasis filter. Section the image into 3 x 3 blocks and let (kl,k , ) be the upper left coordinates of the input block. The block polynomial is

7


and the cyclic convolution of input with filter is y ) = s ( ~ ~ * ~y z) f)((xx,y)(mod , x3 - l)(mod y 3 - 1)

r(kl,k2)(x,

+ ' k 1 + 2 , k Z + l - Sk1+2.kz + ' k I + l , k , + l - ' k l + l , k Z + S k l , k z + l - ' k 1 , k z + 'k1+2,k,+l

= (Skl,kz+l - 'kl,kZ

+ (Skl+l,kz+l + (Skl+2,k2+1

- Skl+2,k2

+ 'kl+l,kZ+l

- 'kI+l,kz

- 'kl+l,kZ)

- sk1+2.k2)x

+ Skl,kz+l

- sk1,k2) xz

+ Skl+2,kz+2 - ' k I + 2 , k z + l + S k ~ + l . k 2 + 2 - Sk,+2.kz+l)Y + ('kl+l.kZ+2 - ' k l + l , k Z + l + sk1,k,+2 - ' k l, k , + l + Sk1+2,k2+2 - s k ~ + 2 , k z + I ) x Y + ('kI+2,k2+2 - S k l + 2 , k 2 + l + S k l + l , k ~ + 2- S k l + l . k z + l + ' k l , k z + 2 - S k 1 , k z + 1 ) X 2 y + ( S k l , k z - S k l . k 2+ 2 + S k1 + 2 . kz - S k , + 2 , k ~ + 2 + Sk1+l.k2 - ' k 1 + l , k 2 + 2 ) Y 2 2 + ( S k l + l , k z - ' k l + l , k 2 + 2 + ' k 1 , k z - S k 1 , k ~ + 2-k s k1+2.k2 - S k ~ + 2 , k ~ + 2 ) X Y

f (S k , , k z +Z - S k l , k z + l

+ (skl+2,kz

+ 'kl+l,kZ

+ Skl.kz

2

2

Y The output sequence is given by the last coefficients (x2y2 term) of r(k13k2), r k l , k Z = ' k I , k z - 'k l. k 2 + 2 + S k l + l , k z - S k l + l . k z + 2 + 'k1+2,k2 - ' k 1 + 2 . k 2 + 2 ) for k , = 0,. . ., p 1 - 2 and k 2 = 0,. . ., p 2 - 2. Viewing convolution by sections pictorially, the filter and data block are represented in Fig. 2. Superimposing the filter over the block, multiplying entries, and summing yields the output r k l r k 2 . - S k l+ 2 . k 2 + 2

- Skl+l,kz+2

-' k l , k ~ + 2 ) ~ *

Although the goal of convolution by sections is to reconstruct the linear convolution, it is not necessary to do so from an applications point of view. What is important is the construction of the output sequence from the coefficients of the convolution. Selecting the last coefficient of r(l')(x)(in the onedimensional case) or r(klpkz)(x, y ) (in the two-dimensional case) is appropriate for the application, since each coefficient is a finite difference which measures, in come sense, the presence of an edge in the signal. Instead of retaining the last coefficient, it is possible to select any other coefficient, or more generally, to use as output a function of all the coefficients. For signals,

FIGURE2. Illustration of convolution for images.

8


+

one could choose as output rk = J ( S k + , - Sk)’ + (sk+z - Sk+1)’ (sk - s k + l ) 2 . Of course, such a choice of function makes the filtering a nonlinear problem.

111. GROUPALGEBRAS The material in this section assumes that the reader is familiar with the basis concepts of groups, rings, and fields. Some standard algebra texts which cover these topics are Fraleigh (1982), Herstein (19751, and Hillman and Alexanderson (1983). We provide some simple examples of groups, namely, the cyclic, dihedral, and quaternion groups, which are used throughout this article. We introduce the concepts of groups algebras and group convolution, which are based on groups, rings, and fields. A . Finite Groups and Index Notation

Define the index set I N = (0, 1, . . ., N - l}. Let G = { g i : i E I N } be a multiplicative group of order N with identity element g o . Let i , j , k E IN and g i , g j , gk E G. The multiplication of group elements induces an operation of index summation on the index set I N , denoted @, and defined by gigj=gk-i@j=

k.

Note that i Q j = j @ i for all i and j if and only if G is a commutative group. Define the mapping 4: G -,IN by 4 ( g i ) = i for g i E G,i E I N . The mapping 4 is bijective and satisfies 4 ( S i ) 0 4(Sj) = i @ j

= 4 ( S i e j ) = #(SigjX

which implies that 4 is a homomorphism. Thus, 4 is an isomorphism between (G,.) and ( I N , @), say (C, .) g ( I N , @), where . denotes the operation for G and @ denotes the operation for I N . Since ( I N , @) is a group, it has an identity element 0 with the property i @ 0 = i for all i E I N . Also, 0 must be an associative operation, i @ ( j @ k) = ( i Q j ) @ k. Finally, each index i must have an inverse, denoted i-’, such that i @ i-’ = 0 = i-’ @ i. 1. Cyclic g r o u p

Let G = {xi:i E I N } be a multiplicative group of order N, where the group multiplication is defined by xixi

= #+j)modN

for i , j E I N , xi,x i

E

G;


9

then G is isomorphic to the set of integers IN under addition modulo N . The group G is called a cyclic group of order N , where the element x is the group generator. If g i = x i , then index summation for I , corresponding to the cyclic group is

iO j

= (i

+ j ) mod N .

Geometrically, the cyclic group can be associated with rotations in the plane. The group element x corresponds to a rotation by 211,” radians.

2. Dihedral Group The dihedral group of order 2N is gN = { r i d :i E I , , j E IN}, where T, = 1, oN = 1, and ?up= Q , - ~ T . If we define g k = akI2for k even and gk = T Q ( ~ - ” I ~ for k odd, then index summation for I N corresponding to the dihedral group is

iOj = j

+ (-ly’imod

2N.

Geometrically, the dihedral group 9, can be associated with rotations and reflections in the plane. Element Q corresponds to a rotation by 2 z / N radians, and element T corresponds to reflection through the x axis. It is intuitive that a rotation followed by a reflection does not necessarily yield the same effect as a reflection followed by a rotation. In the dihedral groups g Nfor N > 2, it can be verified that T Q # QT, which implies that such groups are noncommutative. Note, however, that 9, is commutative.

0 1

2 3 4 5 6 7

1

0 1 2 3 4 5 6

0 3 2 5 4

I

6

I

I

3 2 0 1 6

6

I

4 5

4

2 3 1

0

5

4 5

5 4

6

I

I

6

6

I

5

4

I

6 0 1 2 3

4

5 3 2 0 1

1 0 3 2

2 3 1 0

10


where index i corresponds to the ith row and index j corresponds to the jth column. Geometrically, 9 can be associated with rotations in a threedimensional space (Shoemake, 1983). The elements i, j, and k correspond to three-dimensional coordinate vectors defining the axis of rotation, and the 1 element corresponds to the angle of rotation. The - i, -j, and - k elements correspond to negated vectors, and the -1 element corresponds to a negated angle of rotation. Multiplication of quaternion group elements corresponds to a three-dimensional cross product. As three-dimensional translations are more easily described in rectangular coordinates than in spherical coordinates, three-dimensional rotations are more easily described by using quaternion coordinates. B. Group Algebras and Group Convolution

The construction of an algebra over a finite group involves the application of group theory and algebraic properties. Although extensive material on the subject of group algebras can be found in Curtis and Reiner (1962) and Serre (1977), we present the structure, operations, and properties of a group algebra needed for later developments. Let IF be a field and G = { g i : i E IN} be a finite multiplicative group of order N with identity element g o and index summation operation 0.Define the group aigebra of G over IF to be the set of formal sums

Let c E IF and let tl, /?E FCG] be given by a=

1 aigi

and

fl =

iefN

1 Bjgj. JEIN

Addition in the group algebra is defined by

and scalar multiplication of a group algebra element is defined by ctl = c

1 aigi) = 1 ctligi.

(iefN

iefN

Multiplication in the group algebra is defined by


11

where 0 is the index summation corresponding to the group G.The multiplication in the group algebra is called group convolution. Note the resemblance of the operation to that of linear and cyclic convolution shown in the last section. Some observations are in order. The center of the group algebra is defined by C(IF[G]) = { a E IF[G]: ap = pa, Vg E IF[G]}. The centralizer of an element a in the group algebra is defined by C,(IF[G]) = E IFCG]: olp = pol). The set {ago: CI E IF} is isomorphic to IF, so IF _c C(IF[G]). Also, if a E C(IF[G]), then C,(IF[G]) = IF[G]. Under the operations of addition and multiplication, the group algebra IF[G] is a ring. Under the operations of addition and scalar multiplication, the group algebra F[G] is isomorphic to the N-dimensional vector space IFN over the field IF. The group convolution induces a multiplication on the vector space IFN. Given elements u, v E IFN, where u = (uo,ul,. .. , u,-~) and v = (uo, u l , .. . , u , - ~ )for ui, uj E IF, and i, j E Z, vector addition is defined by

{a

u

+ v = (uo, ..., + (uo, ..., = + U O , + .. ., + UN-1)

u1,

u1

(u0

u1,

uN-1

01 9

UN-1)

uN--l)?

and scalar multiplication is defined by cu = C(u0, u1, * * .

9

uN-1)

= (cue, cui, * * .

9

CuN-l),

where c E IF. Let e, E IFN be the vector which has all zero components except for the ith component, which is 1. The vectors can be rewritten as u=

uiei

and v =

ielN

1ujej. jEIN

The group convolution imposes a product on u and v, denoted u * v, by

where we refer to as vector convolution. Consequently, the vector convolution of em, en E IFN for m,n E Z , is given by em * e n

(1

= k e I N i bj=k

dmidnj)ek?

where a, is the Kronecker delta. Thus, em* en = emen. Note that if G is a noncommutative group, then in general, u * v # v * u. Example. The cyclic convolution of order N arises by choosing the group G = ( x i : i E I N )to be isomorphic to the cyclic group Z,, where the index summation operation is addition of integers modulo N, and the field IF = IR, for the group algebra IF[G].

12


Example. The linear convolution arises by choosing G = { x i :i E I } to be a multiplicative group of infinite order which is isomorphic to the set of integers I = Z,where the index summation operation is simply the addition of integers: xixj = xi+j for i, j E I . Typically, the sequences used in the linear convolution, { u i } E - , and {uj}j"-,, have the property that ui = uj = 0 for i < 0, j < 0, i > p , and j > q (for some large positive integers p and q). For such sequences, say {ui}f=oand { U ~ } J = ~the , linear convolution is given by

i=O

j=O

C. Matrix Representation of Group Elements

We present a way to represent group elements by matrices which is useful for converting matrices whose elements are from a group algebra into equivalent (but larger) matrices whose elements are in the given field. Let IF" be an n-dimensional vector space over the field IF and let GL(n, IF) be the group of isomorphisms which map IF" onto itself. An element M E IF" is, by definition, a linear mapping and with the set {eo, e l , . . .,e n - l } of basis vectors for the vector space IF", 9: IF" + IF" is defined by a square matrix [ m i j ] for order n, where the coefficients mij, for i, j E I,, are elements in the field IF. Since Y is an isomorphism, it must have an inverse map 3 - l which is also linear and which implies that the determinant of 9, denoted det(Y), is not zero. The group G L ( n , IF) is thus identified with the group of invertible square matrices of order n. Let G = { g i :i E I,} be a multiplicative group of order N with identity element go and index summation operation 0 . The matrix representation of the group G in the vector space IF" is given by the homomorphism +: G + GL(n, IF). An element g i E G, i E I , is represented as $ ( g i ) such that $ ( g i e j ) = $(gigj) = $ ( g i ) $ ( g j ) for gi, gj E G and i, j E I,. This formula also implies that $(go) = 3"and $(g[') = $-'(gi), where 9"is the n x n identity matrix which is a column ordering of the basis vectors e,, e l , . . ., for the vector space IF". Note that the dimension n of the vector space IF" does not necessarily equal the order N of the group G. Let IF[G] be a group algebra over the field IF and the group G. Let IF[GL(n, IF)] also be a group algebra, or a ring of the square matrices of order n. Through a linear extension of the $ homomorphism, the group algebra IF[G] is homomorphic to IF[GL(n, IF)]. Let a = l i P I Naigi E IFCG]. It has a matrix representation


13

Example. Let 9,= (1, z, o, zc,.. . , cN-',z o N - l ) be the dihedral group of order 2 N . Since the elements in 9, can be identified with operations on vectors in the plane IF2, let S and T be the matrices which represent the elements a and z, respectively. The matrices are given by cos(g) *(a) = s =

);(

-sin(;) and $(z)= T

);(

sin

=

cos

Consequently, + ( z P c 4= ) TPSq. Example. Let Z, = { 1,(, C2, . . ., CN-'} be the cyclic group of order N . Similar to the dihedral group, the elements in Z, can also be identified with operations on vectors in the plane IF2. The matrix C which represents the element C is

$(C)

cos

G)

sin

($)

=c =

(5) ($)

- sin

cos

Consequently, t,6(Cq) = Cq.

+

Example. Let 22 = { 1, +i, +j, +k} be the quaternion group of order 8. Elements in the quaternion group can be represented by 4 x 4 matrices where the group identity element 1 is represented by X4,the 4 x 4 identity matrix. The group element - 1 is represented by the matrix -Y4. Thus, $(I) = Y4 and $(- 1) = -Y4. Define the matrix representations for i and j by 0 1 0 0 1 0 $(i)=[

-'

0 0 0

-1

and $ ( j ) = [

0 0 1

-8

::

:I.

0 - 1 0 0

Since ij = k in the quaterion group and since $ is a homomorphism, $(k) = + ( i j ) = $(i)$( j ) and the matrix representation for the group element k is forced to be 0 0

*(k) = - 1 0

0 0

14


The matrix representation for the remaining group elements are determined by - $ ( i ) = $ ( - i ) = W j ) , - $ ( A = $ ( - j ) = $(ik), and - $ ( k ) = $ ( - k ) =

$W.

IV. DIRECT PRODUCT GROUPALGEBRAS

For the applications of group algebras to signal processing, we will choose a single group G and a field IF and work with the group algebra IF[G]. For processing of images and higher-dimensional data sets, one could also choose a single group for the group algebra. However, it is more natural to choose a group for each dimension of the data and work with the algebraic equivalent of a Cartesian product, called a direct product group. By working with group algebras based on direct products, fast algorithms can be produced for group algebra matrix transforms. These algorithms generalize the idea of fast Fourier transforms. The background for direct product group algebras is presented in this section. Fast convolution is discussed here, but the fast algorithms for matrix transforms will be discussed in a later section. A . Direct Product Groups and Index Notation

Let G and H be two finite multiplicative groups of order M and N , respectively, and let I , and Z, be their respective index sets. Define the direct product of groups G and H by G @H

=

{gi @ hj:gi E G, hj E H }

with the following definition for multiplication. For gi,, gi, E G and h,,, hj2 E H, define (gi, @ hj,)(gi,@hj,) = (gi, eiz @ hjl gj2)'

where 0 is the index summation for ,Z and 6 is the index summation for I,. If it is clear from context which index summation is being used, then the tilde will be omitted. The set G @ H with the operation defined above is a multiplicative group with identity go @ ho, where go is the identity of G and h, is the identity of H . The inverse of gi @ hj is given by (gi 6 hj)-' = g;' 0 h,Tl. The multiplication of group elements in G @ H induces an index summation on the Cartesian product of the index sets,Z x ,Z by (gi, @ hjl)(giz@ hjz) = (gi, eiz @ hjl ej2)*(i1,j1) O (i2,j2)= (il o i 2 9 j 1 Oj,)

, j , ) E ,Z x Z,. The Cartesian product I, x Z, with index sumfor ( i l , j 1 ) (i,, mation @ defined above will be denoted by ,Z @ Z,.

15


The direct product group G 0 H has order M N and is isomorphic to a group F = {h:k E I M N )via isomorphism $: G 6H -+ F, which itself induces an isomorphism 8: I, 0 IN + ( I M N ,0) by $(gi Q hi) =foci, j ) . If (bG(gi) = i, &(hj) = j , 4 G @ H ( g i 8 hj) = ( i , j ) , and h ( f k ) = k for i E b , j E I N , and k E I M N , then the following diagram is commutative:

’ (1,

(G 0 H,’)

bc@H

( F , -1

CF

$1

’

@ IN,0)

ie

0) so 8 = q5F 0 $ o &iH. Let (it,j,), (i2,j2)E I, 0 I N ; then (IMN,

8((i19jl) 0 ( i 2 , j 2 ) )= h($(4iiH((it~j1) 0(i2~2)))) = 4FF($(4&H(il 0 i2,j1 0j2))) = 4d+(gil ei2@ hjl e j 2 ) )

= 4d$(gilgi26 hjlhjz)) = M$((gi, 8 hjl)(giz0 hj,))) = 4A$(gil Q hjl)$(giz6 hjz)) = 4dfo(i,,jl)f&i,,j,)) = 4dfo(i,,j,)eo(iz,jz)) = 8 ( i l J l )0 W 2 , j 2 ) .

An obvious choice for the isomorphism 8 is lexicographical ordering, 8 ( i , j ) = Ni j , where i E I, and j E I N . Select the elements f k l , fk, E F, where k,, k 2 E I,, are given by k , = N i l j , and k, = Ni, j , for the appropriate i,, i, E I , and j , , j , E IN. Inverting yields i, = k, div N, j, = k, mod N, i, = k2 div N , and j , = k2 mod N. Multiplication of elements fk, and fk,

+

+

in the direct product group F k E I , N , corresponds to

+

G Q H, denoted fk =fklfkz

for

fk

EF

and

k=kl@k2 = (Nil

+ j , ) 0 ( N i , + j,)

= N(il 0 i d

+(j,@j2)

= N [ ( k , div N ) 0 (k, div N)]

+ [ ( k , mod N) 0 (k2 mod N)].

The definition of the direct product of finite groups can be extended easily to handle the direct product of more than two finite groups. Let G,, G,, . . ., Gp be finite groups of order N,, N,, . . ., N p , respectively. Define the direct

16


product of these groups by GI O G2 O * * O G p = { gi, O gi, O . * * O gip:gij E Gj, ij E I N j , j = 1,2, . . .,P } =

{dk:k = 8(i,, is, . . . , i p ) E I N ~ N ~ . . , N ~ }

= D,

where 8: ZNl ZNz O . * . 8 ZNp -,ZN,Nz...Npis an isomorphism. Let i , , fl E ZNl, i,, f,E IN*, . . ., i p , fp E ZNp. The index summation operation for GI 0 * . . Gp is defined by ( i l l i,,

. .., i p ) 0 (i,, f, ..., fp)

= (i,

0 i,, i , 0 f,, ...,i p 0 tp).

In a manner similar to that for the direct product of two groups, the isomorphism 0: IN,@ IN, @ .. . 8 ZNp -P ZNINz...Np can be defined by the lexicographical ordering 8(il, i , , ..., i p ) = i ,

+ (N1)i2+ (N1N2)i3+ + (N,*.-Np-,)ip.

Given an element dk E D, where k E ZN,N2...NP, the indexes i , i, E ZN2,. .., i, E ZNp can be found by the inverse isomorphism,

( :nI: )

ij = kdiv

Nt mod

E

ZNl,

4.

B. Tensor Products of Group Algebras Let G = {gi: i E },Z and H = {hi:j E IN} be two finite multiplicative groups and let G @ H be their direct product. Let IFCG] and F [ H ] be the respective group algebras. Define a = x i E l y a i gEi FCG] and B = x j E I N b i hE iIFCHI. The direct tensor product of elements a and fl is defined by

The direct tensor product operator is distributive over group algebra addition and scalar multiplication: a 0 (B

+ 6 ) = (a 8 P ) + (a 0 61, (a + Y)8 B = (a O B) + (Y 8 P), and (ca) @I /3 = a 6(cB) for c E IF.

Define the direct tensor product of the group algebras IFCG] and l F[aas the set of all formal sums

17


The set IF[C] 0 IF[H] is homomorphic to the group algebra IF[C 0 H ] through an extension of the isomorphism 4: G 0 H + F to a linear homomorphism @: F [ G ] 0 F [ H ] + IF[G 0 H I , where the group F = { f k : k E Z}, is a finite multiplicative group of order M N , and where @ ( x k ak

@pk) =x

k o(ak

6P k )

=x

k @ ( C i e I U C j s I N tlkipkj(gi

=C

k

C j E I Nakipkj#(gi

0

0 hj).

Let a 0 p, y 0 6 E F [ C ] 0 IFCHI, and define their product based on the homomorphism @ by (a 0 P)(r 0

= ( C i e I Mz =xi,kelu

j e I N aibj(gi

x j , f e I Naibjckdf(gi@k

= ( x i , k e I M uickgi@k) = ay

0 hj))(CkeIy

CIS,,

ckdl(gk

0 hf))

6hjf3f)

0 ( x j . I E I N bjdlhj@l)

0 p6.

Let 8: I , 0 IN + I M Nbe the isomorphism on the index sets induced by the # isomorphism and defined by O(i, j ) = k for i E I,, j E I,, k E I,,. Let ak E IF, f k E F , gi E G, and hj E H . An element a E IF[G 0 H I is given by

Similarily, an element b E IF[G 0 H ] is given by

where be(k.1) E IF, g k E G, and hl E H . The product of a and b in lF[G 0 H ] can be computed in IF[C] 0 IF[H] by ab = { C i e I u

[gi

0 ( x j E I N uO(i,j)hj)l}

= C i s I Mx k e I v

k i @ k

6( C j E I N x

[gk

8(

~ I E I bNO ( k , f ) h f ) l }

l e l a~t l ( i , j ) b O ( k , l ) h j @ f ) l

which requires M 2 N Zmultiplications in the field IF. C. Fast Convolution

In the case when H is the cyclic group Z,, then the inner summation from above is a cyclic convolution given by

18


The Cook-Toom algorithm and the Winograd algorithm are two types of fast algorithms for computing cyclic convolutions by reducing the number of multiplications and/or additions (Blahut, 1988). Assume that the best of such fast algorithms requires N,,,, c N 2 multiplications to compute the cyclic convolution. The product ab in I F [ G @ H ] can be computed by M2Nf,,, multiplications in IF[G] Q IF[H]. If the group G is cyclic and the group H is not, reorder the summations to obtain

which requires Mf,,,N2 multiplications. Another approach to reducing the number of multiplications used in computing the product ab in IF[G] 0 IF[H] involves reducing the representations of the elements a and b as elements in lF[G] 0 IFCHI. For example, the representation of a E IFCG 0 H ] as an element in F[G] 0 IF[H] was given by

which has M linear combinations of elements uiE IFCH], i E IM defined by ui = IN j ~ h jThe . group algebra l F [ H ] , where H is a group of order N , is isomorphic to the vector space IFN over the field IF. The group algebra element u i ~ I F I H ]can be considered as the vector v i ~ I F Nwhere vi = (a,(,,,), aO(i,1), . . .,a O ( i , N - I ) ) . Let the span of the set of vectors (vi}, i E ZM, be the set of linearly independent vectors (uo, u l , . . .,uD-l >,where D I M . The vector vi, i E ,Z can be rewritten as vi = x j E I D d i j ufor j dij E IF. The basis vector u j corresponds to the group algebra element uj E IFCHI, given by uj = l k e I N fltjkhk, where $jk E IF and hk E H. The group algebra element ui E IFCHI, i E,,Z is given by

xje


19

The representation in F[G] 6 lF[H] for the element a E F [ G 0 H] consists of D terms of the form mi @ pi, where cli E lF[G], fii E F[H], and i E .,Z A similar representation in F[G] 6lF[H] for the element b E lF[G 0 H] may consist of E terms of the form yj 0 Sj, where yj E IFCG], Sj E FCH], and j E ZE, where E is the rank of the set of vectors vj E IFN, j E I N , given by vj = (b,(j.o,,betj,,), . . . ,be(j,N-l)). The number of multiplications in F[G] 0 FCH] to compute the product ab in F[G 6 H] is D . E . N 2 . If there is no linear dependence between any of the combinations, then D = E = M and the number of multiplications becomes M 2 N 2 , which is the same number obtained by the direct application of the definition for group algebra multiplication.

D. Examples Linear convolution. Let G = { x i : i E Z}and H = { y j :j E Z},both isomorphic to the integers Z. The direct product group algebra R[Z 0 Z]conz x a i j x i y jof which the polynomials in sists of all finite formal sums two variables are a subset. The two-dimensional linear convolution corresponds to products of polynomials and is the multiplication of R[Z 0 Z]. Cyclic convolution. Let G = ( x i : i E Z,} and H = { y j :j E Z,}. The group multiplication for G 0 H is the product of polynomials of two variables, but modulo both x M - 1 and y N - 1. Index summation is ( i l , j l ) 0 (i,, j , ) = ((i, + i,) mod M , ( j , + j , ) mod N ) . The group algebra IRCG 6 H] consists of all polynomials in two variables of degree ( M - 1)(N - l), and two-dimensional cyclic convolution is the product of these polynomials modulo x M - 1 and y” - 1. The template size used in cyclically convolving an image is N x M . Walsh convolution. The direct product group Z; naturally corresponds to Walsh convolution (Crittenden, 1984; Gibbs and Mallard, 1969a, 1969b; Gibbs, 1969c; 1970; Walsh, 1923), despite the fact that the development of Walsh functions was in the context of orthogonal functions (and has relations to wavelets). If the indices i = (i,, . . ., id) and J = ( j , , . ..,j , ) are treated as unsigned binary numbers i , . . .id and j , . . .j d , then the index summationis i 0 j = ( i l . - . i d ) X O R ( j l . . . j.d )

x(i,j)E

V. SEMIDIRECT PRODUCT GROUP ALGEBRAS

The dihedral groups gN(for N > 2), although having two elements z and CJ such that z2 = 1 and oN= 1, clearly cannot be factored into Z,0 Z,, since this direct product group is commutative and the dihedral groups are not

20


commutative. However, the dihedral groups can be factored into what is called a semidirect product group, a more general concept than direct product groups. Such factorization may be useful in developing fast algorithms for convolution in group algebras based on dihedral groups. A . Semidirect Product Groups and Index Notation

Let G and H be two finite multiplicative groups of order M and N , respectively, and let I , and I N be their respective index sets. Denote the group of automorphisms of H by Auto(H). For each g E G we can associate with it an automorphism on H , call it 9. Define the antihomomorphism*: G -+ Auto(H) A by gi,gi,= &jil, for elements g i l ,gi,E G. Note that oo must be the identity automorphism, b0(hj)= hj, for any j E I N , where go is the identity of G. Define the semidirect product of groups G and H by G Q H = {giQhj:giEG,hjEH)

with the following definition of multiplication. For gil,gi,E G and hj,, hj, E H,define (gi,Q hj,)(gi2Q hj2)= (gi,gi,@ Oi2(hjl)hj2).

The set G 6H with the operation defined above is a multiplicative group with identity go 6 h o . The inverse of g i & h j is given by (gi@hj)-' = g;' 6 &l(h,Tl). The multiplication of group elements in G @ H induces an index summation @ on the Cartesian product of index sets I , x I , by ( i l , j l )6( i 2 , j 2 )= ( i l 0 i z , f 2 ( j 1 )~ j , )

for (il,jl), ( i 2 ,j , ) E I , x I,. The Cartesian product I , x IN with index summation @ defined above will be denoted by I , @ I N . The semidirect product group G @ H has order M N and is isomorphic to a group F = {fk: k E I M N } with isomorphism I):G 6 H + F. The isomorphism )I induces the isomorphism 0: ( I , @ I N , 6 )+ ( I M N ,0)by I)(gi 6 hi) If &(gi) = i, M h j ) = j , k &(gi @ hj) = (i, j ) , and Mji)= k, for i E I,, j E I N , and k E I,,, then the following diagram is commutative:

21


B. Semidirect Tensor Products of Group Algebras

Let G = ( g i : i E I,} and H = { h j : jE I,} be two finite multiplicative groups and let G 6H be their semidirect product. Let F [ C ] and IF[E+I be the respective group algebras. Define a = xie,,aigi E F [ C ] and p = Cje,,bihi E IF[H]. The semidirect tensor product of a and p is defined by

a@,P =

1 1 a i b j ( g i 6hj).

i e I Mj e l N

The semidirect tensor product operator has the properties gi 6(h,, + hj2)= 6 hj,) + (gi 6 hjz), ( g i , + gi,) 6 hj = (Si,6 hj) + (Si2 0 hj), and (cg,) 6 hi = gi 6(ch,) for c E IF. Define the semidirect tensor product of the group algebras IF[G] and JF[H] as the set of ail formal sums (gi

IF[C] 6IF[H] =

x

{ i

aj @pi: aj E IF[G], pi E IFCH]

I

.

The set IF[G] 6 IF[H] is homomorphic to the group algebra IF[G 6 H ] through an extension of the isomorphism 4: G 6H + F to a linear homomorphism 0:IF[G] 6IF[H] + H[G 6 H ] , where the group F = { f k : k E I M N } is a finite multiplicative group of order M N . Let (a 6p), ( y 66 ) E IF[G] 6 IFCHI, and define their product based on the homomorphism by

6 b)(y 6 6 ) = ( x i e l , x j e l N aibj(gi 6 h j ) ) ( x k e l Mx l e I N c k d f ( g k 6 = x i , k e I M x j . l e I N aibjckdl(gigk 6 = x i , k a l MC j . l e I N [ ( a i c k g i @ k ) 6( b j d f h k ( j ) h l ) l . In general, (a 6 ~ ) ( 6 y S) z (cry) 6wP). Consider the elements ( g i 6 p), ( g k 66) E F [ G ] 6IFCHI, where gi, g k E G. (a

hl))

dk(hj)hl)

The multiplication of these two elements is given by

6p ) ( g k 6 6, = ( X j e I , bj(gi 6 h j ) ) ( z f e l Nd l ( g k 6 h f ) ) = bjdl(gigk 6 = g i g k 6[ X j , I e I N b j d l d k ( h j ) h l l = gigk 6 [ d k ( x j e I N b j h j ) ( x l e I N d l h l ) l = ( g i s t ) 6( d k ( P ) 8 ) * Alternatively, consider the elements (a 6p), ( c k g k 6 6) E F [ G ] 6 IFCHI, (gi

xj,lslN

where c k

E IF,

dk(hj)hf)

g k E G. The multiplication of these two elements is given by

22


C. Example

The motivating examples for semidirect product groups are the dihedral groups, 9,= { z p d : p E I,, q E I,}, where z2 = 1, 0, = 1, and zcr4 = O " - ~ T . The dihedral group can be written as 9,= Z2 6 Z,. The groups are G = {tp:p E I , } z Z2 and H = ( 0 4 : q E f,} z Z,. The antihomomorA phism : G + Auto(H) is defined by z p ( 0 2 = ) for all q E Z,. The semidirect group product is therefore given by A

(tP1604')(?P2

63

- (z(PI+P2)mod2

fJ42)

6 6 [( - l)pzql + q,] A zP2(a41)042)

= (z(P1+P2)mod2

mod N ) .

The index summation corresponding to this product is ( P i , qi)&z,

42)

=((PI + P J mod 2, Gk1)qz mod N ) =((Pl + P 2 )

mod 2, C(- l)pzql + q 2 1 mod N ) VI.

INVERSES OF

GROUP ALGEBRAELEMENTS

In constructing matrices whose elements are from a group algebra and which have what we call the convolution property, we will need to compute inverses of group algebra elements. This section describes how to do that. Since group algebras generally have nonzero, noninvertible elements, it may be desirable in applications to eliminate these elements. The elimination is possible by constructing quotient rings from the group algebra and ideals containing noninvertible elements. The section includes a discussion of these topics.


23

A . Inverses

Let F [ G ] be a group algebra over the field IF with respect to the multiplicative group G = { gi: i E 1,) whose identity element is go and whose index summation operation is @. As shorthand notation, we will identify field value 1 with group algebra element lg, Ogi. Let a = x i E I N a i gEi IF[G] and p = z j s r N p j g Ej IF[G]. If a p = 1, then fl is said to be a right inverse of a and a is said to be a lefi inverse of p. If a p = 1 = pa, then fi is both a right and a left inverse of a and is simply called the inverse of a, denoted p = a-'. For example, in W [ 9 J

+ IF=;'=;'

but 1 + 0 + a2 has no inverse. To construct a right inverse for a, note that the product of the two elements is

aB =

1(1

kPlN

icBj=k

%pj)gk

1 1

= j e t N i e I N ai@j-JBjgj-

Thus, group convolution can be computed as a matrix product Ab, where A = [ a i a j - , ] is an N x N matrix with row index i and column indexj, and b = [pj] is an N x 1 column vector. For a to have a right inverse, the system Ab = e,, where e, has a 1 in its first component, but 0s in all other components, must have a solution. If A is invertible, then a has a unique right inverse. However, if A is not invertible, it is possible that e, is in the range of A, in which case there are infinitely many right inverses. Similarly, to construct a left inverse for p, note that

The problem reduces to solving aTB = ez, where B = [pi-lej] is an N x N matrix with row index i and column index j , and a = [aj] is an N x 1 column vector. An alternative method for constructing inverses uses the matrix representations discussed in Section 1II.C. Let a E F [ G ] be an invertible element with inverse a-'. Let #(p) be the square matrix which represents p and let I be the identity matrix of the same size. It is true that

I = #(lgo) = #(aa-')

= #(a)#(a-'),

so # ( a p 1 )= #(a)-'. Given a, construct its matrix representation I(a), invert this matrix, and determine p such that #(p) = #(a)-'. This is a system

24


of equations in the unknown coefficients for B. The solution produces fi = a-'. For example, we mentioned earlier that in lR[Q3], (a1 + bz)-' = [a/(a2 - b 2 ) ]1 [ -b/(u2 - b2)32. Using the matrix representations for dihedral groups developed in Section III.C, we have

+

1 $(a1

+ bz) =

["

0

+

U-b

]

0

~

and d(al

+ bz)-'

a+b =

0

-

1

a-b

+

+

as long as a' - b2 # 0. Setting this system equal to c,I + c14(z) cz4(o) c34(za) + c44(a2)+ c54(za2) yields a solution of co = a/(u2 - b'), c1 = -b/(az - b'), and all other ci = 0. For element 1 a , ' a note that its matrix representation is the zero matrix, so it cannot have an inverse within the group algebra. The following function will be useful later. The net of a is defined by

+ +

net(a) =

1 ai ielN

and is a ring homomorphism over group algebra multiplication since

and

If tl is invertible, then net(a) net(a-') = net(aa-') = net(lgo)= 1, which implies that neither net(a) nor net(a-') can be zero. Thus, if a is invertible, net(a) # 0 and net(a-') = (net(a))-'. Note, however, that net(a) # 0 does not necessarily imply that a is invertible. For example, the element (go gl) E lF[Z,], where go, g1 E Z,,satisfies net(g, g l ) = 2, but go g1 is not invertible.

+

+

+

B. Ideals and Quotient Rings

It is possible that nonzero elements of a group algebra are not invertible. To avoid dealing with such elements, it is possible to work with the group algebra modulo the noninvertible elements. The algebraic way to do this is via ideals and quotient rings.


25

Let F[G] be a group algebra over the field IF, where G = {gi:i E I,} is a multiplicative group of order N with identity element go and index summation operation @. Under the operations of addition and multiplication, the group algebra F[G] is a ring. Let I be a nonempty subset of the group algebra ring IF[G] consisting of the elements aB and Pa for every element ~ E I F C G ] and B E I , where the subset I forms a subgroup of F[G] under addition. This set I is called an ideal in the group algebra ring IFTGl. - Let I be an ideal in the group algebra ring IF[G] and let a be any element in IFCG]. The ideal coset for the element a of the ideal I in IFCG], is the same subset of F[G] as the coset a + I when I is considered as the additive subgroup of IFCG]. Since the group algebra ring IFCG] is also a commutative group under addition, the left and right ideal cosets are the same. Let IF[G]/I be the collection of ideal cosets a + I, Vu E IF[G]. This set forms a ring under the operations of addition and multiplication given by (a

+ I ) + ( p + I ) = (a + p) + I , (a

+ I ) ( B + 4 = (UP> + I ,

for the element j?E IF[G]. The ring IF[G]/I is called a quotient ring. Let a be an element from the center of the group algebra ring IFCG]; that is, a E C(IF[G]). Since the element a commutes with every other element in the group algebra ring, the centralizer of the element a, denoted C,(IF[G]), is the entire group algebra ring IF[G]. Furthermore, let a be an element such that its matrix representation, defined in Section III.C, is the zero matrix (it follows that the element a must also be a noninvertible element in IFCG]). Let I be the subset of the group algebra ring F I G ] consisting of all multiples pa of the element a by elements fl E IF[G]. Then it can be shown that this set I is an ideal in the group algebra ring IFCG]. Let ( a ) denote this ideal generated by all multiples of the element a. Example. Let g3= (1, 't, u, zu, u2, d} be the dihedral group of order 6. In the group algebra ring lF[g3], it can be verified that the element 1 + + a2 is not invertible and is in the center of IF[g3]. The element a has matrix representation

The matrix representation of 1

+ u + a2 is

26


$(1

Thus (1

+ 0 + 0 2 ) = $(1) + $(a) + $(u2)

+ 0 + )'a

)r; )r;

cos

(G)

-sin

sin

(f)

cos

is the ideal generated by the element 1 + 0 + 0'.

Example. Let 9 = { 1, i-i, +J, k} be the quaternion group of order 8. In the group algebra ring lF[9],it can be verified that the element 1 (- 1) is not invertible and is in the center of IF[9].The matrix representation of - 1 is defined to be the negative of the matrix representation of the element 1. Then the matrix representation of the element 1 (- 1) is the zero matrix and (1 (- 1)) is the ideal generated by the element 1 (- 1).

+

+

+

+

c2,

Example. Let Z, = {1,[, . . .,lN-'} be the cyclic group of order N. In the group algebra ring lF[Z,], the element CN - 1 can be considered as a polynomial which can be factored into the product of prime polynomials CN - 1 = n k l N Qk([),k = 1, 2, . . ., N, where klN means k divides N with no ([ - d), i = 1,2, .. ., N is a cyclotoremainder and where Qk(c)= mic polynomial over any field IF, gcd(i, k) is the greatest common divisor of i and k, and w, is a kth root of unity in some extension field. It can be verified that the element given by the cyclotomic polynomial Q N ( [ ) is not invertible, is in the center of IF[Z,], and has a zero matrix representation, so < Q , ( [ ) ) is the ideal generated by the element Q&). Now, let F[G] be a group algebra ring over the field IF, where G = { g i : i E I,} is a multiplicative group of order M, and let IF[H] be a group algebra ring over the field IF, where H = { h j : j E I,} is a multiplicative group of order N. Let IF[G] @I IFCH] be the set of all finite sums of the direct tensor products of elements from the group algebras IF[G] and IF[EII. Let I be an ideal in IF[G], J and ideal in IF[HJ, and define the set

n,cd(i,k)=l

ui@Ivi:ui~Iorui~J i=O


27

where n > 0 is a finite positive integer. Let a E F[G] and let /3 E IF[H]. Consider the direct tensor product (a

+ I ) 8 ( P + 4 = (a8 /3) + ( a 8 J) + ( I 8 B ) + ( I 8 4.

If the set K is an ideal in F[G] 6 IFCHI, then this can be reduced to (a

+ I ) 6( P + J ) = (a 8 8) + K .

I:==,

First, let p be an element from the set K and be given by p = ui 6 ui, where n > 0 is a finite positive integer. By its definition, the set K is closed under addition, and the additive inverse of the element p E F[G] 8 F [ H ] is given by - p = ~ ~ (- =ui) 0 o ui, which is also in the set K. These conditions imply that the set K is a subgroup in F[G] 0 IF[H] under addition. Second, let q be an element in F[G] 8 F[H] given by q = (aj 8 /Ij), where aj E IF[G], Pj E IF[H], and rn > 0 is a finite positive integer. Consider the product

cim,o

If ui is in the ideal set I, then ajui is also in I. If ui is not in the ideal set I, then ui is in the ideal set J and Sju, is also in J. Thus p4 is in the set K, which completes the proof that the set K is indeed an ideal in F[C] 8 F [ H ] .

VII. MATRIXTRANSFORMS

Earlier achievements have been made in the construction of discrete matrix transforms based on elements from a group algebra. Loui (1976) developed a generalized Fourier transform based on semisimple algebras, where group algebras are considered as a special case. Based on this general transform, Willsky (1978) defined a class of finite-state Markov processes for nonlinear estimation evolving on finite cyclic groups. The underlying structure of nonlinear filtering was obtained by viewing probability distributions as elements of a group algebra and by taking the transforms of such elements. Matrix transforms with the convolution property were constructed from orthogonality relations, and an efficient method for computing group convolutions was obtained from a generalization of the fast Fourier transform to metacyclic groups. A separate and similar development was also presented by Eberly and Hartung (1987). This development included the use of group algebras as a basis for group convolutions and for the construction of matrices with the convolution property. The discrete Fourier transform, the discrete Walsh

28


transform, and a transform based on the dihedral group were used as examples. Higher-dimensional transforms were constructed where the underlying group is the direct product of other groups. This section begins with a review of this earlier work and a development of the properties of matrices that have the convolution property. The sections that follow present several methods for constructing such matrices with the convolution property: by direct tensor products of matrices, by semidirect tensor products of matrices, and by exhaustive determination of all possible combinations for the matrix entries. The section concludes with a discussion of the application of these matrices with the convolution property to the construction of fast algorithms for computing group convolutions. A. Vector Spaces of Group Algebra Elements

Let F [ G ] be a group algebra over the field IF, where G = { g i : i E Z,} is a multiplicative group of order N with identity element go and index summation operation @. Define F[G]K = IFK[G] = F [ C ] x IF[C] x

*.*

x IF[G]

to be the Cartesian product of K copies of the group algebra IF[G]. Under the usual operations of component-wise addition and multiplication by a scalar, IFKIG] is a K-dimensional vector space over the group algebra lF[G]. Let F R [ G ] and F s [ C ] be R- and S-dimensional vector spaces over the group algebra IF[G], respectively. Define AR~ =S { A = [Uij]:

Uij E

F[G]y i

E ZR,j E Is}

to be the set of all R x S matrices with entries from the group algebra IF[G]. Let x = [ x i ] E lFRIC] and y = [ y j ] E IF'CG], for i E I R , j E 1,. For every A = [ a i j ]E A R xdefine s , the mapping A : IFs[G] -+ IFRIG] by

Let IF"[G] be an N-dimensional vector space and let A N x , be the set of all N x N square matrices with entries from the group algebra IF[G]. Let A = [aij] and B = [bij] be two matrices in A N x , , where aij, bij E IF[G], and i, j E IN. Then the set of matrices A N xisN a ring with addition given by

+ [bij] = [aij + bij]

[ ~ i j ]

and multiplication given by


29

The set of matrices M N x Nis also an associative algebra over the field IF, where scalar multiplication is given by c E IF. c[aij] = [caij], Let x = [ x i ] and y = [ y j ] be two vectors in the IFNIG]vector space, where xi, yj E lF[G], and i, j E I,. The component-wise multiplication of vectors x and y, denoted x o y, is defined by

Cxil 0 C ~ i =l C x i ~ i l . The convolution of vectors x and y, denoted x * y, is defined by

Let C(IF[G])be the center of the group algebra IF[G]. The center of the vector space lFNIG] is the set of vectors that commute under componentwise multiplication, with every vector in IFNIG]and is given by C(IFNIG])= C(IFIG])N. B. Matrices with the Convolution Property

Let IFNIG]be an N-dimemsional vector space over the group algebra IF[G], where G = (gi: i E I,} is a multiplicative group of order N with identity element go and index summation operation 0. Let M N x Nbe the set of all N x N square matrices whose entries are from the group algebra IF[G]. Let u and v be two vectors in C(IFNIG]),and let A be a matrix from the set The matrix A has the conuolution property with respect to the group G if A(u * v) = (Au) 0 (Av).

Theorem 1. For the matrix A = [aij]to haoe the convolution property, it is necessary and sufficient that aimain= a,,,, @,,Vi,m, n E I,.

Proof. Let aimain= a,,,, e n , Vi, m, n E I,; then

30


= A(u*v).

This proves sufficiency. Let the matrix A have the convolution property. Then A(u * V) = (Au) 0 (Av)

C 1 ai.menumvn = m1 1 aimainumvn oINnoIN

m e I Nn e I N

ai,men = airnuin.

This proves necessity.

Theorem 2. Let A = [aij] E A N xbeNa matrix with the convolution property, where i, j E I N . Then: 1. a, = (l)go. 2. (aij)-' = aij-l. 3. [net(aij)lN= 1. 4. For any invertible element y E IFCG], the matrix B convolution property.

= y-'Ay

also has the

Proof: 1. Let aio be the ith entry in the first column of the matrix A with the convolution property. From Theorem 1, aimain= ai,,,@,,

V i , m, n E I N

implies that aim%

= ai,m @O = aim

and aioain = ai.0 en = sin.

Since aimaiO = aimand aiOain= ain,a , must be the group algebra multiplicative identity element (l)go.


31

2. Let j be an element in the index set IN and let j-' also be in the index sel IN such that j ej-' = 0. Let uij be an entry in the matrix A with the convolu tion property. From Theorem 1, a..a..-l = ai,jej-I = 4 0 ij ij and U = Ui,j-Iej = a,. aij-la..

Since a,, is the group algebra multiplicative identity element, this implies that every entry in the matrix A with the convolution property must have a multiplicative inverse in the group algebra, which happens to be given by (aij)-' = aij-1.

3. Let aij be an entry in the matrix A with the convolution property and consider [net(aij)IN.From the multiplicative property of the net function, [net(aij)lN= net(az). From Theorem 1, net@;) = where N j =j 0 j 8 * * * 0j ( N terms). Since the group G is of order N , for any element gkE G, k E I N , the group multiplicative identity element go is obtained from g.: This implies that net(ui,Nj) = and, from Theorem 1, net@,) = net((l)go) = 1. Thus [net(aij)lN= 1. 4. Let y be an invertible element in the group algebra IFCG]. Let the beN a matrix with the convolution property, and let matrix A = [aij] E A N X B = [b,] E .ANxN be matrix given by B = y-'Ay,

so that bij = y-'a,y. Consider the group algebra elements bi, and bin from the same row i of the matrix B. Then bimbin = (~-'airn~)(~-'~in~) = y-'aimyy-'ainy = y-laimainy = 7-lai.m en7

-4.m

en-

32


From Theorem 1, the matrix B has the convolution property. This completes the proof of Theorem 2. C . Examples 1. Matrix Transforms over Dihedral Groups

Let IF6[g3]be a six-dimensional vector space over the group algebra IF[g3],where Q3 = { 1 , 7 , 0 , 7 0 , o’, TO’} is the dihedral group of order 6 with identity element 1 and index summation operation 0 given by i 0 j =( j

+ (-

ly’i) mod 6,

V i ,j

E

16.

Let d t ? 6 x 6 be the set of all 6 x 6 matrices whose entries are from the group Let A = [ a i j ] E - 4 f 6 x 6 be a matrix with the convolution algebra IF[&]. property. From Theorem 1, a.cm a.In = a .~ , m @ n *

vi,m,

z6-

This formula involves only the elements in any given row of the matrix A, so we drop the row index. Consider any row given by [ao a l

u2

~3

a4

~ ~ 1a i,~ W 9 3 1 , i ~ 1 6 .

The formula from Theorem 1 reduces to

aman = urn@,,, Vm,n E 16. It is possible to construct a matrix with the convolution property from elements in the IF[g3]group algebra by determining all possible combinations for the entries in a given row of the matrix. For simplicity, consider the entries in the matrix to be only from the set { +(l)g,: g k E 9 3 , k E 161. From Theorem 1, every entry in the first column of a matrix with the convolution property must be the group algebra multiplicative identity element 1. Thus for every row, uo = 1. Consider all relationships between the row entries u2 and u4. From the formula of Theorem 1,

and U’U’U’

= f%4a4u4= UO.


33

Also, consider the relationships between the row entries u l ,u3,u5. From the formula of Theorem 1,

u l u 2= a1$2 = u3 = ~

=~

4 $ 1

=u2$1

ul@4=u1$4=u5

4 ~ 1 ,

=aZc(l,

U3U2 = U3 $2 = U 5 = U 4 $ 3 = U 4 U 3 ,

U3U4 = U3 $4 ~

5

= U1 = U 2 $ 3 = U 2 U 3 ,

= ~a5Q2 2 = u l = u 4 e 5 = u4u5,

u5u4 = a5Q4= u3 = a2Q 5 = ~

12~5,

and UlUl

= U 1 @ 1 = uo,

‘ 3 u

= u3

$3

= go,

u5u5 = U 5 @ 5 = “0.

The only possible combinations for the row entries u2 and u4 are given by (u2, @4)E

((1, 11, (a,a2),(a2,4)

and are considered separately. Case 1: (u2, u4) = (1, 1). Consider U3 = c ( 1 U 2 = U 4 U l = Ul,

Us = UlU4 = U 2 U 1 = U1.

Thus u1 = a3= u s ,and u 1 E { + l ,+ 7 , +a, fro, +a2, fra2).

This leads to eight possibilities for a given row of a matrix with the convolution property: [l

fl

1

+1

1

fl],

[l

fT

1

+7

1

fr],

[l

fro

1

+To

1

fra],

[I

fra2 1

fro2

1

fza’].

Case 2: ( u z , a4)= (a, 0’). Consider ~ 1= 0 ~

1

= ~~3 = 2 ~(4011 = 02u1,

ulaL= ulu4 = u s = u2al = a a l .

34


Thus a3 = a, a and a5 = a1a2,and since a, a = a2al, a1 E { & T ,

&TO, & T O 2 } .

This leads to six possibilities for a given row of a matrix with the convolution property: [1

&T

0

&TO

CT2

&TO’],

[I

f T 0

0

&TO2

O2

k T

&T

0’

[1

&Ta2 d

1,

&To].

Case 3: (a2,a4) = (a’, a). Consider a1a2= a l a 2 = a j = ~ C X ~= O~

1

4

=~act1, 1

= ~U, = 4 ~2a1 =a2~l.

Thus a3 = a1a2and a5 = a,a, and since a1a2= aal, a1 E { & T ,

&TO,

fTO’}.

This leads to six possibilities for a given row of a matrix with the convolution property: [I

f T

fJ2

&TO2

[1

f T 0

fJ2

&T

[1

&TU’

O2

&TLT

0

fTC],

&Ta’], (r

fT

1.

This approach can easily be extended to construct matrices with the convolution property and with elements from a group algebra based on the dihedral group gM.Let IFN[gM] be an N-dimensional vector space over the group algebra IF[9,,., 3, where

.

= { 1, T, 6,To,

g M

f .

9

OM-’, r a M - l }

is the dihedral group of order N = 2M with identity element 1 and index summation operation given by i@ j =( j

+ (-

ly’i

+ N) mod N,

V i ,j E IN.

Let ANx N be the set of all N x N matrices whose entries are from the group algebra IF [gM]. Any row of a matrix A E A N is given by [ao

0 1 ~

a2

..‘ aNWl], aiE IF[gM], i E I,.

Again from Theorem 1, every entry in the first column of a matrix with the convolution property must be the group algebra multiplicative identity ele-


35

ment. Thus for every row, a0 = 1 . Again for simplicity, the remaining row entries of the matrix are to be selected from the set

{ k ( l ) g k : gk

g M ~

IN}*

Consider all relationships between the even numbered row entries k E I,. It can be verified that (‘2k)M

‘Zk,

=

and c12iu2j= L22k,

Vi, j

E I,.

Then the only possible combinations for the even-numbered row entries, excluding row entry a0, are given by mod M=O, P ~ ~ E I , } . Furthermore, consider the relationships between the odd-numbered row entries a 2 k + 1 , k E I,. Then C12k+la2 t12k+1a4

C12k+l‘2M-2

-

‘2k+3

- a(2k+l)$2 -

- ‘(2k+l)

= ‘(-2)@(2k+l)

= ‘2k+5 = ‘(-4)@(2k+l)

$5

= ‘(Zk+l)$(ZM-2)

= ‘2k-1

= ‘(-(2M-2))

-

- @-2M-2‘2k+l9 = @‘2M-4a2k+19

%3(2k+l)= ‘2’2k+l,

and ‘1%

-

a1 %31

- go,

a3a3

-

a 3 %33

- a07

= ‘(2,-1)%3(2,-1)

= ao.

Q2M-1‘2,-1

It can be verified that for every possible combination of the even-numbered row entries, the a1 row entry can always be selected from the set {+t,

*to, )to2 )...)f z a M - ’ } .

The remaining odd-numbered entries are always determined by the particular selection of the even-numbered row entries and the al row entry. In the special case that all even-numbered row entries are chosen to be 1, then the ctl row entry additionally can assume the values of k 1.

36


2. Matrix Transforms over a Quaternion Group Let IF8[9]be an eight-dimensional vector space over the group algebra lF[9],where 9 = { f 1, fi, kj, fk } is the quaternion group of order 8 with identity element 1 and index summation operation 0 given by the table in Section 1II.A. Let A s xbe 8 the set of all 8 x 8 matrices whose entries are from the group algebra IF[9].Let A = [aij]E Agx8 be a matrix with the convolution property. The elements of any row of the matrix are given by

[ao a,

a2 a3 u4 as a6 a,],

ui~lF[9],i~Z8.

From Theorem 1, aman= a,

@ ,,

Vm,n E I s .

It is possible to construct a matrix with the convolution property from elements in the F [ 9 ] group algebra by determining all possible combinations for the entries in a given row of the matrix. For simplicity, consider the entries in the matrix to be only from the set

{a:gk E 9,k E 1 8 ) . From Theorem 1, every entry in the first column of a matrix with the convolution property must be the group algebra multiplicative identity element 1. Thus for every row, uo = 1. Consider the relationship of row entry a1 with the remaining row entries:

u l a z = u l e Z = u3 = uzel = u z a l , a1a3 = a1 @3

= a2 = u s @ ,= a 3 ~ 1 ,

a 1 ~ 4= a I e 4 = a5 = a 4 @ 1= a4al, a l a s = a1 8 5 = a4 = a s e 1 = a 5 a l , a1u6 = a, @ b = a, = a6@1= a6a1, ~ l a 7= a 1 87 = a6 This implies that a ,

= a 7 @ ,= ~ 7 ~ 1 .

E C(lF[S]).Furthermore,

a l u l = a l e , = uo implies that row entry a ,

=

k 1, where both cases are considered separately.

Case I : a , = 1. Consider

uza2 = a z e 2 = a, = 1, u4u4

= U 4 a 4 = a, = 1,

a6a6

= a6 @ b = a1 = 1,


which imply a2 =

+ 1, a4 = + 1, and t12C14

ct6

=

=

37

& 1. The relationship

@ 4 = a6

and that a2

= a1u2 = u3,

a4 = @,la4 = Us, a6

= alC(6 = a77

imply that selecting row entries a2 and a4 determines the remaining row entries u3, u s , fx6, u7. This leads to four possibilities for a given row of a matrix with the convolution property:

I

1

[ l l

[l

1

[l

1

[l

1 -1

-1

-1

1

1

1

1

-1

-1

-1

-1

1

-1

1

13,

1

13,

-1

-11,

1 -1

-11.

The first two and last two combinations can be rewritten as [1

1 +1

+1

+1

+1

[l

1 +1

+1

T1

f l

1 -1

11, -11.

Case 2: a1 = - 1. Consider a2a2 = a2 $ 2 =

a1 = - 1,

UqU4

= a4@4= a1 = - 1,

agc16

=

@6

= f f l = - 1,

which imply that a2, a4, a6 can be chosen from the set { relationship

+ i, +j, fk}. The

a2a4 = a2@4 = fx6

and that a2 = - a , a 2 = - a 3 , a4 = -u1u4 = - u s , g6 = - @ l a 6 = -a79

imply that selecting row entries a2 and u4 determines the remaining row entries a 3 ,u s , u6, a7. Consider the six possibilities for selecting u2 as three separate cases.

38


Case a: a2 = f i . This implies that row entry a3 = T i and row entry a4 must be selected from

fj, Tj, +k, Tk

+

i or T i, since ~ 2 ~ = 1 4a6 would imply that a6 = k 1). This leads to eight possibilities for a given row of a matrix with the convolution property:

(a4 cannot be

k -k],

[l

-1

+i

Ti

+j

fj

[l

-1

fi

Ti T j

+j

-k

kl,

[l

-1

+i

Ti f k

Tk

-j

j],

[l

-1

+i

Ti f k + k

j

-j].

Case b: a2 = +j. This implies that row entry u3 = T j and row entry a4 must be selected from

+i, Ti, fk, Tk

+

(a4 cannot be +j or T j , since a2a4 = a6 would imply that u6 = 1). This leads to eight possibilities for a given row of a matrix with the convolution property:

Tk

[l

-1

fj

T j +k

[l

-1

-tj

Tj

+k

+k

-i

i],

[l

-1

fj

T j +i

Ti

-k

k],

[l

-1

fj

T j Ti

fi

i

-i],

k -k].

+

Case c: a2 = k. This implies that row entry a3 = T k and row entry a4 must be selected from f i , T i , +j, T j (a4 cannot be fk or T k, since a2a4 = a6 would imply that a6 =

+

1).This leads to eight possibilities for a given row of a matrix with the convolution property:

[l

-1

fk

Tk

fi

Ti

[l

-1

+k

Tk

Ti

+i

[l

-1

+k

Tk

+j

Tj -i

[l

-1

+k

+k

Tj

+j

j

-j],

-j

j], i],

i

-i].


39

D. Direct Tensor Product Matrices Let F'[G] be an M-dimensional vector space over the group algebra IFCG], where G = {gi: i E I,} is a multiplicative group of order M with identity element go. Let IF"[H] be an N-dimensional vector space over the group algebra IFCHI, where H = {hj: j E I,} is a multiplicative group of order N with identity element h,. Let A M xbe M the set of all M x M matrices whose Nthe set of all entries are from the group algebra lF[G], and let A N xbe N x N matrices whose entries are from the group algebra IF[H]. Let IFMNIG0 H] be an MN-dimensional vector space over the group algebra IF[G 0 HI, where G 0 H is a direct product group. From Section IV, G 0 H is isomorphic to the group F = { f k : k E IM,} of order M N and 8: lM 0 1, -+ IMN is an induced isomorphism on the index sets given by k = 8(i,j)for i E I,, j E I N . Let AMN M N be the set of all MN x M N matrices whose entries are from the direct tensor product group algebra lF[G] @I IFCHI. Let A = [ u ~ , ~be~ a] matrix from the set A M xand M ,let B = [bjij,] be a matrix from the set A N xDefine N . C = A 0 B to be a matrix from the set A M N xcomposed M N , from the direct tensor product of matrices A and B, and given by C = [c~(il,j,)~(i2,j2)l = Caiti28 bjlj21. Let matrices A and B both have the convolution property. Let crs and crt be two elements in the same row of the matrix C = A 0 B, and be given by Crs

= %(i,,

jr)e(is,j,),

Crt

= %i,,

jr)8(it,j t ) -

Consider the product CrsCrt

= Ce(i,, j,)e(is,js)%(ir,

jr)e(it,jt).

From the definition for the elements in matrix C = A 0 B, ce(ir,jrp(iS,js)ce(ir,jr)e(it, jr)= (airis8 bjrjs)(airit 0 bjrjt)* From Section IV, this product can be reduced to (airis0 bjrjs)(ai,it0 bj,j,) = (aivisairit 8 bjrjsbjrjt). Since the matrices A and B both have the convolution property, from Theorem 1, (a.W .s a.1.h. @ b.J r l.s bJrlt . ) = (aipiseit @ bjF,jsejr).

Again from the definition for the elements in matrix C = A 0 B, (airsis@it 0 bi,,j, ejJ = c~(i7,jr)~(is @it, j, @it)'

40


Since 8 is an isomorphism over the index summation operation 0,

-

CO(i,,jr)O(i, Qit,j, QjJ- cO(ir,jr)tO(ids) @O(it.jt)l-

By the way the entries c,, and crtwere defined above using the 8 isomorphism, -

%ir.j,)te(is,js) ~ B O ( ~ ~ J~ ) ICp,s @ t *

Thus, it has been shown that C , ~ C , ~= c , , , ~and, ~ from Theorem 1, this implies that the matrix C = A 0 B has the convolution property. E. Semidirect Tensor Product Matrices

Let FM[G] be an M-dimensional vector space over the group algebra F[G], where G = {gi:i E I,} is a multiplicative group of order M with identity element go. Let lFNIH] be an N-dimensional vector space over the group algebra lF[H], where H = {hj: j E I,} is a multiplicative group of order N with identity element h,. Let A M xbeM the set of all M x M matrices whose entries are from the group algebra IFCG], and let A N Xbe N the set of all N x N matrices whose entries are from the group algebra lF[H]. Let IFMNIG6H] be an MN-dimensional vector space over the group algebra lF[G @I HI, where G @I H is a semidirect product group. From Section II.C, G Q H is isomorphic to the group F = {fk: k E I M N } of order M N and 8: I, 6I, + I, is an induced isomorphism on the index sets given by k = O(i,j) for i E I, j E IN. Let A M N xbeM the Nset of all M N x M N matrices whose entries are from the semidirect tensor product group algebra lF[G] 6IF[H]. M ,let B = [ l ~ ~ ,be ~ , a] Let A = [aili,] be a matrix from the set A M x and matrix from the set A N x Define N . C = A @I B to be a matrix from the set composed from the semidirect tensor product of matrices A and B, and given by

c = [~O(i~,j,)O(i~,j~)I = Caili2@I bjlj,I* Let matrices A and B both have the convolution property. Restrict the group algebra element entries in the matrix A to be from the set

{ (1)gi: gi E G, i E I M ) , so that the matrix A is given by A

=

Cai,i,l = Cgs(il,iz)l,

i,, iz E I, gs(il,i2) E G,

where S:I, 6I, -+ IM is a homomorphism. In a similar manner, restrict the group algebra element entries in the matrix B to be from the set {(l)hj:

hj E H,j E IN},


41

so that the matrix B is given by

B = Cbj,j21 5 ChTUl,j2)l,

j l , j 2 E I N , hT(jl,j2)E H,

where T : IN 6I , + I, is a homomorphism. Let cr, and crt be two elements in the same row of the matrix C = A and be given by

From the definition of the elements in the matrix C crs and crfcan be rewritten as

=A

@ B,

Q B, the elements

Then the product of these two elements is given by

which from Theorem 1 would imply that the matrix C = A @I B has the convolution property. Further analysis reveals that the assumption

42


F . Inverses of Matrices with Group Algebra Elements Let IFNIG] be an N-dimensional vector space over the group algebra IF[G], where G = {gi: i E I N }is a multiplicative group of order N with identity N the set of all N x N matrices whose entries are element go. Let A N xbe from the group algebra IFCG]. Let A = [aij] x AN.,be a matrix with the convolution property, where each entry aij is an invertible group algebra element (from Theorem 2). Let A = [a;'] be a matrix in which each entry aijfrom the matrix A is inverted. The transpose of the matrix 2, denoted AT,is defined by AT = [aJ;']. Then the product of the matrices AT and A is given by

Let the matrix B = [b,] E AN be the matrix which results from the matrix product x T A , such that

The entries along the diagonal of the matrix B bii

=

1

Uk,i-lei

=

kcIN

From this, the matrix B

1

UkO

keIN

= J T A can

B

= A T A are given

=

C

by

1 = N.

ksIN

be rewritten as

= ATA = N

9

+ C,

where 9 E AN is the N x N identity matrix and C E AN is an N x N matrix with 0s along the diagonal. If the matrix C is the zero matrix, i.e.,

C

ak,i-I@j=o,

i#j,i,jeZN,

kcIN

then the inverse of the matrix A , denoted A-', can be given by

One direct way to compute the A is to invert the group algebra elements of A , aij, one at a time. Another method is to replace A by a larger matrix with field entries using the matrix representations discussed in Section 1II.C. where z and The idea is illustrated using the dihedral group algebra JF[9,], (T are the generators for the dihedral group and have matrix representations

GROUP ALGEBRAS IN SIGNAL A N D IMAGE PROCESSING

43

The matrix A defined below has the convolution property,

A=

1

1

1

--z

--z

-7

1

a

a2

--z

-70

--a2

1 a2

0

--z

--a2

--zu

1

1

T

-z

T

-z

Ta

-za2

1

1 0

The matrix representation of A is d ( A ) = [d(aij)], the matrix of matrix representations of the entries of A . For our example,

where a = cos(2n/3) and b = sin(2n/3). This matrix can be inverted and used for vector convolution computed in the realm of the matrix representations. G. Fast Computation of the Matrix Transform Let u jE

= [ui]

and v = [uj] be two vectors in C(F”[G]), for ui, uj E IF and i,

IN. From Section III.B, the vector convolution of u and v is defined by u*v

c(c

= kEIN

i@j=k

uiuj)ek

where ek is a vector with a 1 in the kth component and 0s in the remaining components. By this definition, the computation of the vector convolution requires N 2 multiplications in the field IF. However, since the matrix A has the convolution property, by definition

44


A(u *v) = (Au) 0 (Av).

If the inverse of the matrix A can be found, then the vector convolution of the two vectors u and v can be written as

u*v

= AP((Au)

0

(Av)).

The application of the matrix transform A to the vector u yields the vector Au = U = [U,,,], rn E IN, in the vector space IFNIG],which is given by u,,,= C amiui. ielN

Similarily, the vector Av given by

=V =

[V,], n E IN, in the vector space IFNIG] is

V,=

1 anjuj. jEIN

By this definition, the computation of the vectors U = Au and V = Av each requires N 2 multiplications in the field IF. The component-wise multiplication of the vectors U and V requires N multiplications in the field IF, and the application of the matrix transform A-’ to the vector U o V requires an additional N 2 multiplications in the field IF. The computation of the vector convolution of the vectors u and v by the formula

u*v

= A-’((Au)

0

(Av))

requires 3N’ + N or O ( N 2 ) multiplications in the field IF. However, if a “fast” algorithm can be found for computing the matrix transform A and its inverse transform A-’, which requires less than N 2 multiplications in the field IF, then the group convolution of the vectors u and v can be more efficiently computed by this formula on the basis that the matrix A has the convolution property. For example, the cyclic convolution of order N for sequences of complex numbers can be efficiently computed using the formula based on the convolution property of the discrete Fourier transform. The fast Fourier transform (FFT) provides for the computation of the discrete Fourier transform using O(N log N ) multiplications in the field of complex numbers. Thus the computation of the cyclic convolution using the fast Fourier transform, component-wise multiplication, and the inverse fast Fourier transform requires O(N log N ) multiplications in the field of complex numbers. VIII. FEATURE DETECTION USING GROUP ALGEBRAS

We have shown that there is a much larger class of convolutions than cyclic ones from which one can choose when analyzing signals or images. The effectiveness of an algorithm may depend not only on the type of input data, but also on the particular convolution and value function selected.


45

A . Value Functions A value function is any function v: F[G] -+ IF. The group convolution of two elements in F[G] produces another element in the group algebra. For signal and image processing we need to produce an IF-valued response from the convolution. Value functions provide this response. The simplest such function, mentioned in Section I1 where standard convolution by sections was E F[G], discussed, is the coegicient selection function. Given a = zislNctigi the selection function is defined by

select,(a) = cti. For a direct product group algebra IF[G 0 H ] and the selection function is

ct

= x i s , N x j E , M ~ i j (gj), gi,

select,,j(cc) = aij. Other value functions defined on F[G] are net(a) = E i E I N (see ~ Section , VI) and norm(a) = where cl is the conjugate of ct if the field is the set of complex numbers, but just o! otherwise.

Jm,

B. Feature Detection

Many applications exist where one is interested in the acquisition and analysis of signals and images for the detection of certain features. This type of processing can be achieved by segmentation of the data into a description of the various features located in the scene. Surveys of techniques for detecting such features as edges, lines, curves, and spots can be found in Rosenfeld and Kak (1982). We discuss the ideas of feature detection using group algebras. Typically, edge detectors are based on finite-difference schemes. The presence of an edge is determined by comparing the differences of pixel values for neighboring points. To illustrate, consider the edge detector (1,0, - 1) and a two-valued signal ( A , .. ., A , B ,..., B). After convolution, the filtered signal is (0,. . .,0, A - B, 0, . . ., 0). Since A # B, the filter (1,0, - 1) has located the edge between the values A and B. Similarly, consider the edge enhancer (- 1,2, - 1) applied to the same signal. The filtered signal is (0, . . ., 0, A - B, B - A , 0, .. . , 0), so the edge has been detected and enhanced. The characteristic that is common to the standard edge detectors and enhancers is that the sum of the filter weights is 0. For group algebras lF[G], we define the filter f = x i E I N A gto i be an edge detector or edge enhancer if net(f)= = 0. Note that the set I = { f E IF[G]: net(f) = 0} is a twosided ideal of IF[G]. The same type of definition can be constructed for higher-dimensional data sets where the underlying group is a direct product group. For example, an edge detector f = x i E l N x j E l y J j ( ggij ,) E IFCG 0 HI satisfies the condition net(f) = = 0.

xiEIN-f;:

xiEINxjEIMJj

46


C. Application to Signals

Denote the input signal as {si}go and the filter as {fj}j”-J. To compute a group convolution by sections, we need to choose the underlying group for IR[G] and a value function u: IR[G] + IR to apply to the group convolution. Example. Consider IR[Z3], where i @ j = ( i + j ) mod 3 and the value function is select,. The filter is f = 1 - x2 E IR[Z3]. The input is sectioned into blocks of three consecutive values which are treated as coefficients of elements in IR[Z,], say s ( ~=) sk S ~ + + ~ S X ~ + ~ XE’ IR[Z3]. The output sequence {rk}p=ois defined by rk = sk - sL+, = select,(s(k)*f).

+

Example. Let g3be the dihedral group with six elements. Let the filter be f = 1 - t E IR[g3].The sections of input treated as elements of IR[g3]are

dk)= sk + sk+lt + S k + Z a

+

Sk+3to

+ sk+4a2+ sk+5toz,

so that ’(”*.f = (’k - ’ k + l )

+ (sk+4

+ (’k+l

- sk+3)02

- sk)t

+ (sk+5

+ (sk+Z - sk+5)o + (sk+3- s k + 4 ) t o - sk+2)zo2.

If we select the fifth coefficient of each group convolution to generate the output sequence {rk}km,o,then rk = select5(s(k)*f)= s k + 5 - sk+z. In fact, the output sequence can be generated by choosing any function u : IR[g3]+ IR and defining rk = u(sck)*f).

D. Application to Images To illustrate the application to images, we analyze edge detection and enhancement of the standard images of Lena and Mandrill using a 7 x 7 Laplacian of a Gaussian filter which is embedded in an 8 x 8 template by padding the last row and last column with 0s. The images are filtered based on the group convolutions for IR[Z, 0 Z,],IR[9,0 g4],and IR[9 0 91, where Z, is the cyclic group of order 8, g4is the dihedral group of order 8, and 22 is the quaternion group of order 8. Each underlying group is of the form G x G = { ( g i , g j } : i,j E Z,}, so we will reference direct product group elements by ( g i , gj). The digital image is represented as a two-dimensional sequence {si,iz}fl~l~,~~!o of gray levels in the range 0 I silizI 255. Similarly, the filter sequence is { & l j 2 } ~ ; ~ o , j 2 = , , . For all the examples, the assignment of group elements to the 8 x 8 template is


41

The group element (gi, gj) is assigned to the template location in row i and column j . Given tl = crij(gi,gj) E R[G 0 GI, it is represented by an 8 x 8 template whose (i,j) entry is the coefficient uij. The 8 x 8 filter for the Laplacian of a Gaussian, given as group algebra element f = ~ ~ z = o f j l j 2 (gj,) g j l and , represented by a template, is

xi’,=,

To perform a group convolution on a block of input data, we must also interpret the 8 x 8 subimages as group algebra elements. The subimage { s ~ , + ~ , ,i,,~ ,i,+E~Z,> ~ : is interpreted as 7

The group convolution is

7

48


The target point of the convolution of subimage is chosen to be row 4, column 4. The value function u: lR[G x C] -+ IR is chosen to be select 6,6(a) = @6,6. A word of caution is needed. When performing the standard cyclic convolution, one visualizes the output as being obtained by superimposing the filter over the subimage, multiplying corresponding entries, and then adding up the results for a single output. This method works only when the group algebra is IRIZN1x ZN2]and the value function is selectNl-l,N,-l.For other groups or value functions, the same idea works only after a reordering of the filter values; such reordering depends on the underlying group and the selection function used. Consider the general case where the group algebra is G 0 H,G has order Nl and index summation @lG,and H has order N , and The direct product index summation is denoted 0. index summation OH. Selection of the (ml, m,) coefficient of the group convolution is

There are Nl N , index pairs ( i l , i,) and ( j l ,j , ) such that

(4, i 2 ) O ( j 1 , j 2 )

= (il

O d l , i2 ad2) = ( m l , m2).

For each choice of (il, i2),the value of ( j l , j 2 )which satisfies this equation is ( j l , j 2 )= (i?

Ocm,, i;’ ~ ~ m= (i;’, , ) i;’)

o h,m2),

Thus, to pair up subimage template values with filter template values in the “usual” way, the filter weights need to be reordered according to the formula for ( j 1 , j 2 ) . We are using select,,,. The reordering for Z,0 Z,is

In this special case of a filter that is symmetric about its center, the reordering does nothing, so the filter weights are still

T-

0

E-

(96 ' 0 6 )

(96 '16)

('6 ' 0 6 )

('6 '16)

(56 '06)

'16)

( $ 6 '06)

( $ 6'16)

(06 ' 2 6 )

(06 ' 0 6 )

(06 '16)

(16 ' 2 6 )

('6 '06)

(16 '16)

(56

( $ 6' 2 6 ) (56

'26)

('6 ' 2 6 ) (96 ' 2 6 )

9-

('6 '$6)

(06 'E6)

( $ 6' $ 6 ) (56 '$6)

('6 ' $ 6 ) (96 ' $ 6 )

1-

E-

(06 '"6)

(06 '56)

(16 '"6)

( ' 6 '56)

( $ 6' c 6 )

(56 ' $ 6 )

( € 6'"6) (56 'be)

(96 '"6)

(96 ' $ 6 )

('6 '"6)

('6 ' s 6 )

0

('6 " 6 ) (O6 "6)

( $ 6''6) (56''6)

('6 ' ' 6 ) (96

"6)

0

('6 '96) (06 '96)

( $ 6'96) (s6'96)

('6 '96)

(96 '96)

9NISS330Xd 3 9 V M I a N V WNDIS NI SV18391V dnOXD

69

50


The reordering for 22 0 22 is

and the reordered filter weights are 0

0

-3

-1

-3

-6

0

-1

0

0

0

0

0

0

0

0

-3

0

0

- 16

0

40

-3

-I

0

- 16

-8

- 16

- 18

-1

-8

-3

0

0

- 16

0

40

-3

- 16

-6

0

40

- 18

40

128

-6

- 18

0

0

-3

-1

-3

-6

0

-1

-1

0

-8

- 16

- 18

-1

-8

- 16

- 16

Figures 3 through 6 show the Lena image and its corresponding filtered images, and Figures 7 through 10 show the Mandrill image and its corresponding filtered images. An open question is whether or not one can determine, in some quantitative sense, what group convolution and value function should be used to extract certain features in an image. The question is important even when the standard cyclic convolution is chosen: What value functions best detect the attributes of interest in an image? Moreover, given that two groups produce the same qualitative results, can one develop faster convolution algorithms for one group versus the other? We do not address these issues here, but leave them as research problems.


FIGURE3. Original Lena image.

FIGURE

4.

Z,@ z, filtered image.

51

52


FIGURE5. g4@ g4filtered image.

FIGURE 6. 2? @I 9 filtered image.


FIGURE7. Original Mandrill image.

FIGURE8. Z,@ Z, filtered image.

53

54


FIGURE 9. g40 g4filtered image.

FIGURE10. 2 @3 2 filtered image.


55

IX. THEGRASP CLASSLIBRARY

++

AC class library called GRoup Algebras for Signal Processing (GRASP) has been developed as a practical implementation of the arithmetic operations in a group algebra presented in this article. The GRASP class library has object definitions for theoretical concepts of finite groups and the direct products of groups, group algebra rings, and matrices consisting of group algebra elements. This library has replaced the low-level group algebra routines of the complete GRASP system described by Wenzel (1988), which was developed to explore image transformations using convolutions based on a group algebra and to experiment with fast convolutions and matrix transforms based on various group algebras. Now, with the GRASP class library, these concepts can be easily integrated into any application program. The GRASP class library is organized into three main types of object classes: groups, group algebra elements, and matrices of group algebra elements. All group algebra elements are constructed using elements from the field of real numbers and elements from the cyclic group TN, dihedral group gN,the quaternion group 4, or any direct products of these groups. This section briefly describes each of these three types of object classes, including operations which can be performed on object instances. A. Group Classes

The concept of a finite group is implemented through the abstract Group class. The basic operations are outlined as follows: Constructor Group (const enum Type type, const unsigned size)

Constructor for the abstract Group class, which can be called only by a derived class. The enum Type can be one of the following: Group::ProductType for direct product groups, Group::CyclicType for cyclic groups, Gr0up::DihedralType for dihedral groups, and Group::QuaternionType for quaternion groups. The finite group is defined to have the number of elements given by s i z e which must be larger than I. operator( ) virtual const unsigned o p e r a t o r 0 (const unsigned i , const unsigned j ) Returns the result of the index summation i Q j performed in the

group. This is a pure virtual function, which must be supplied by derived classes.


56

operator [I virtual const unsigned operator[] (const unsigned i )

Returns the index of the inverse of specified group element specified such that i 0 i-' = 0. Derived classes may supply more efficient computations of the inverse operator. operator = = const int operator==(const Group & g r o u p ) Returns a Boolean status indicating whether the specified group is of the same enum Type and has the same number of elements. size const unsigned size(void)

Returns the number of elements in the group. The concept of a finite cyclic group is implemented through the CylicGroup class, a finite dihedral group is implemented through the DihedralGroup class, and a quaternion group is implemented through the QuaternionGroup class. Each of these three classes is derived from the abstract Group class. The constructors for these three group classes are outlined as follows: Constructors CyclicGroup (const unsigned N )

Derives a Group class of Group::CyclicGroup type having N elements. DihedralGroup (const unsigned N)

Derives a Group class of Gr0up::DihedralGroup type having 2N elements. QuaternionGroup (void)

Derives a Group class of Gr0up::QuaternionGroup type having 8 elements. Each of the CyclicGroup, DihedralGroup, and QuaternionGroup classes has its own implemention of the purely virtual index summation operator (operator( )) from the derived Group class. The index summation operators for each of these three groups were given in Section 1II.A. The concept of a direct product of groups is implemented through the ProductGroup class, which is also derived from the abstract Group class. The ProductGroup class can consist of any combination of the CyclicGroup, DihedralGroup, and QuaternionGroup classes, including any combination of sizes for the CyclicGroup and DihedralGroup classes. The operations of the ProductGroup class are outlined as follows: Constructor ProductGroup (Group * q r o u p O ,

. . .)


51

Constructs a direct product of groups, where each group is specified in the constructor by a pointer to an instance of the group. This constructor uses variable arguments, with the last group pointer being NULL. The direct product is taken as the order in which the group pointers are specified. At least one group must be specified, and it is possible to construct a direct product group which consists of only a single group and for any of the pointers be to a group which can also be a direct product group. decompose void decompose(unsigned & x , unsigned x i n d e x [ ] ) The direct product group index x is decomposed into an index for

each of the component groups. These component group indexes are stored in the x i n d e x [ 1 array, which must be allocated to store one index for each component group which comprises the direct product group. The formula for direct product group index decomposition is given in Section 1V.A. dim const unsigned dim(void) Returns the number of component groups which comprise the direct product group. operator( ) virtual const unsigned o p e r a t o r 0 (const unsigned i , const unsigned j ) Returns the result of the index summation i@j, where i and j are both first decomposed into indexes of the component groups (using decompose( )), the index summation is performed on each corresponding decomposed index using the index summation operation of the associated group, and then the return index is reconstructed from the component group index summations (using reconstruct( )). operator = = const int operator==(const ProductGroup & p g r o u p ) Returns a Boolean status indicating whether the specified pgroup is comprised of the same ordering of the same component groups. reconstruct const unsigned reconstruct(unsigned x i n d e x [ l ) A direct product group index is returned based on the reconstruction from the indexes in the x i n d e x [ 1 array for each of the component groups. The formula for direct product group index reconstruction is given in Section 1V.A.

58


B. Group Algebra Class

The concept of a group algebra is implemented through the GrpAlg class. Unlike the Group class, in which an instance was the entire group, an instance in the case of the GrpAlg class is simply a group algebra element. A group algebra element is comprised of an array of coefficients from the field of real numbers, where each coefficient is associated with an element from the underlying group. The basic operations in the GrpAlg class are outlined as follows: Constructors GrpAlg (ProductGroup *pgroup)

Construct a group algebra element over the specified direct product group pointed to by pgroup. All the coefficients for the element are initialized to zero. GrpAlg ( c o n s t GrpAlg & x ) Constructor by making a copy of the group algebra element x. The constructed group algebra element is over the same direct product group and also has the same coefficients as the element x. inv c o n s t i n t i n v (GrpAlg * x ) Attempts to compute the right inverse of the group algebra element. If the group algebra element is right-invertible, then the inverse element value is stored in x and the function returns a nonzero (true) value. Otherwise, if the group algebra element is not right-invertible, the contents of x are unchanged and the function returns a zero (false) value. The group algebra element and x must pass the GrpAlg::sametype( ) test. The method implemented for computing a right-inverse of a group algebra element is given in Section V1.A. net const f l o a t netfvoid)

Returns the net value of the group algebra element using the definition given in Section V1.A. norm c o n s t f l o a t norm(void)

Returns the length value which results from viewing the coemcients of the group algebra element as a vector. operator( ) f l o a t & o p e r a t o r 0 ( c o n s t u n s i g n e d igroup0, . . . I Accesses the coefficient of the particular group algebra element as referenced by the indexes for the component groups in the direct product. This routine uses variable argument and expects the total


59

number of arguments to equal the number of groups comprising the direct product group of the element. operator[] float & operator [ I (const unsigned i ) Access the coefficient of the particular group algebra element as referenced by the index into the direct product group. operator = const GrpAlg operator=(const GrpAlg & x ) Assignment of the coefficients from x to the group algebra element. The group algebra element and x must pass the GrpAlg::sametype( ) test. operator= = const int operator==(const GrpAlg & x ) Returns a Boolean status indicating whether the coefficients of the group algebra element and those of x match (within some small tolerance). The group algebra element and x must pass the GrpAlg::sametype( ) test. operator + const GrpAlg operator+(const GrpAlg & x ) Returns a group algebra element which is the sum of the group algebra element and x using the definition given in Section 1II.B. The group algebra element and x must pass the GrpAlg:sametype( ) test. operatorconst GrpAlg operator-(void)

Returns the additive inverse of the group algebra element. operator* const GrpAlg operator*(const GrpAlg & x ) Returns a group algebra element which is the product (or group convolution) of the group algebra element and x. This computation is based on the definition given in Section III.B, where the a corresponds to the group algebra element and corresponds to x, where the ordering matters. The group algebra element and x must pass the GrpAlg::sametype( ) test. const GrpAlg operator*(const float s )

Returns a group algebra element which is the result of multiplying the group algebra element by the scalar s. operator/ const GrpAlg operator/(const float s )

Returns a group algebra element which is the result of dividing the group algebra element by the scalar s.

60


operator+ = & x) Returns the group algebra element to which has been added the group algebra element x. The group algebra element and x must pass the GrpAlg::sametype( ) test. sametype const int sametype(const GrpAlg & x ) Returns a Boolean status indicating whether the group algebra element and x are constructed over the same direct product group. size

const GrpAlg operator+=(const GrpAlg

const unsigned size(void)

Returns the number of elements in the direct product group over which the group algebra element is defined. There are two simple classes which are derived from the GrpAlg class: the GrpAlgO class and the GrpAlgl class. Both of these derived classes have a single constructor which has the same format as the constructor for the GrpAlg class, i.e., a single pointer to a direct product group, and neither class adds any new operations. The GrpAlgO class constructor simply initializes a specific group algebra element which is the additive identity, and the GrpAlgl class constructor simply initializes a specific group algebra element which is the multiplicative inverse. C. Group Algebra Matrices Class

The concept of a matrix comprised of group algebra elements is implemented through the GrpAlgMat class. Only a limited set of matrix operations have been implemented, and these are outlined as follows: Constructor GrpAlgMat (unsigned n r o w s , unsigned n c o l s , Pr o duct Gr o up * p q r oup

Construct a matrix of dimensions nrows by n c o l s consisting of elements from the group algebra over the specified direct product group pointed to by pqroup. Each group algebra element in the matrix was constructed using the constructor function for the GrpAlg class. convprop const int convprop(void)

Returns a Boolean status indicating whether the matrix of group algebra elements has the convolution property as defined by Theo-


61

rem 1 in Section VI1.B. The matrix must have the same number of rows and columns as the number of elements in the direct product group for the group algebra elements. operator( ) GrpAlgMat & o p e r a t o r 0 (unsigned r , unsigned c ) Accesses the group algebra element in the matrix at row r and column c. operator = const GrpAlgMat operator=(const GrpAlgMat & x ) Assigns the group algebra elements from the matrix x to the matrix of group algebra elements. If the matrix of group algebra elements and matrix x fail both the GrpAlgMat::sametype( ) and GrpAlgMat::sameshape( ) tests, then the matrix of group algebra elements is converted into one having the same type and shape as matrix x. operator + const GrpAlgMat operator+(const GrpAlgMat & x ) Returns a matrix of group algebra elements which is the sum of the matrix of group algebra elements and the matrix x. The matrix of group algebra elements and the matrix x must pass both the GrpAlgMat::sametype( ) and GrpAlgMat::sameshape( ) tests. operatorconst GrpAlg operator-(void)

Returns the additive inverse matrix of group algebra elements. operator* const GrpAlgMat operator*(const GrpAlgMat & x ) Returns a matrix of group algebra elements which is the result of multiplying the matrix of group algebra elements by the matrix x. The number of columns in the matrix of group algebra elements must equal the number of rows in the matrix x and the matrix of group algebra elements, and the matrix x must pass both the GrpAlgMat::sametype( ) and GrpAlgMat::sameshape( ) tests. const GrpAlgMat operator*(const GrpAlg & x ) Returns a matrix of group algebra elements which is the result of multiplying each group algebra element in the matrix by the group algebra element x. The group algebra elements in the matrix and x must pass the GrpAlg::sametype( ) test. const GrpAlgMat operator*(const float s )

Returns a matrix of group algebra elements which is the result of multiplying each group algebra element in the matrix by the scalar s.

62


sameshape const int sameshape(const GrpAlgMat & x ) Returns a Boolean status indicating whether the matrix of group algebra elements and the matrix x have the same number of rows and columns. sametype const int sametype (const GrpAlgMat & x ) Returns a Boolean status indicating whether the matrix of group algebra elements and the matrix x are constructed of group algebra elements over the same direct product group.

D. Example Usage The following example program demonstrates the usage of the object classes from the GRASP class library including some of their operations. An explanation of the code is included in the embedded comments. #include #include #include #include #include #include

C s t d i o . h> < s t d l i b .h> t s t d a r g . h> "group. h" " g r p a l g . h" " gmat r i x h "

.

main ( i n t a r g c , c h a r * * a r g v ) (

// C o n s t r u c t a d i r e c t p r o d u c t of groups 24 x 25. ProductGroup Z4x25(&CyclicGroup(4),&CyclicGroup(5),

NULL);

// C r e a t e two group a l g e b r a e l e m e n t s over t h e group 2 4 x 24 // which h a s 2 0 e l e m e n t s . Each element i s i n i t i a l i z e d t o // t h e a d d i t i v e i d e n t i t y element. GrpAlg a ( & 2 4 x Z 5 ) ; GrpAlg b(&Z4xZ5);

// S e t i n d i v i d u a l c o e f f i c i e n t s w i t h i n e a c h group a l g e b r a e l e m e n t . a(2.3) = 2; b(3,ZI = 3 ;

// A complex group a l g e b r a e x p r e s s i o n . a = a*-(b*4);

// Attempt t o compute i n v e r s e of group a l g e b r a element a and // s t o r e t h e r e s u l t i n group a l g e b r a element b. i f (a.inv(&b)) (


63

// C o n s t r u c t a group a l g e b r a element t o s t o r e p r o d u c t // of g r o u p a l g e b r a e l e m e n t s a and b . GrpAlg c ( & 2 4 x 2 5 ) ; c = a * b ;

// Compare t h e p r o d u c t r e s u l t t o t h e m u l t i p l i c a t i v e // i d e n t i t y e l e m e n t f o r t h e same g r o u p a l g e b r a . P r i n t // a '1' i f t h e y a r e e q u a l ; a ' 0 ' o t h e r w i s e . printf

l " % d \ n " , ( c == G r p A l g l ( & Z 4 x Z S ) ) ;)

1 // C o n s t r u c t a s q u a r e m a t r i x h a v i n g number of rows and columns / / e q u a l t o t h e number of e l e m e n t s i n t h e g r o u p 24 x 25. GrpAlgMat m ~ 2 4 x 2 5 . s i z e 0 , 2 4 x 2 5 . s i z e 0 , & 2 4 x 2 5 ) ;

// I n i t i a l i z e t h e m a t r i x t o c o n s i s t of t h e m u l t i p l i c a t i v e // i d e n t i t y g r o u p a l g e b r a e l e m e n t a l o n g t h e d i a g o n a l and // t h e a d d i t i v e i d e n t i t y g r o u p a l g e b r a e l e m e n t e l s e w h e r e . f o r ( u n s i g n e d r=O; r < 2 4 x 2 5 . s i z e O ; + + I ) ( f o r ( u n s i g n e d c=O; c < 2 4 x 2 5 . s i z e O ; + + c ) { i f ( I == c ) m ( r , c ) = G r p A l g l ( & Z 4 X 2 5 ) ; e l s e m ( r , c ) = GrpAlgO(&Z4xZ5);

1 1

// I f t h e m a t r i x h a s t h e c o n v o l u t i o n p r o p e r t y , p r i n t a '1'; // o t h e r w i s e , p r i n t a ' 0 ' . printf

("%d\n", m . c o n v p r o p 0 ) ;

X. GRASP CLASSLIBRARYSOURCECODE A . gr0up.h # i f n d e f -GROUP-H# d e f i n e -GROUP-Hc l a s s Group (

// a b s t r a c t c l a s s

public: -Group ( v o i d ) ( )

// d e s t r u c t o r

// R e t u r n number of e l e m e n t s i n t h e g r o u p c o n s t u n s i g n e d s i z e ( v o i d ) ( r e t u r n N;)

// I n d e x summation o p e r a t o r - p u r e v i r t u a l f u n c t i o n v i r t u a l c o n s t u n s i g n e d o p e r a t o r O l c o n s t u n s i g n e d i , c o n s t u n s i g n e d j)=O;

64


// Return the index of the inverse group element virtual const unsigned operator[l(const unsigned i); / / Are the groups identical? virtual const int operator==(const Group &group) {return (type==group.type) &6 (N==group.N);)

protected: // Distinquish between group types enum Type {ProductType, CyclicType, DihedralType, QuaternionType} type; unsigned

// number of group elements

N;

// Constructor Group (const enum Type settype, const unsigned size);

1; class CyclicGroup : public Group ( public: // Constructor and destructor CyclicGroup (const unsigned N ) -CyclicGroup (void) { )

:

Group(Group::CyclicType,N)

()

// Index summation operator const unsigned operator0 (const unsigned i, const unsigned j); // Return the index of the inverse group element const unsigned operator[](const unsigned i) [return (N-i)%N;)

1; class DihedralGroup : public Group { public: // Constructor and destructor DihedralGroup (const unsigned N) "DihedralGroup (void) { }

:

Group(Group::DihedralType,Z*N)

// Index summation operator const unsigned operatorO(const unsigned i, const unsigned j); 1; class QuaternionGroup : public Group { public: // Constructor and destructor QuaternionGroup (void) : Group(Group::QuaternionType,8) -QuaternionGroup (void) { }

{)

// Index summation operator const unsigned operator0 (const unsigned i, const unsigned j); }:

class ProductGroup : public Group ( public: // Constructor and destructor ProductGroup (Group * g o , ); "ProductGroup (void);

...

// last must be 0

()

65

GROUP ALGEBRAS IN SIGNAL A N D IMAGE PROCESSING

// Index summation operator const unsigned operator0 (const unsigned i, const unsigned j);

// Are the groups identical? const int operator==(const ProductGroup &group); // Return number of groups in the direct product const unsigned dimcvoid) {return ngroups;) // Decompose index of product group into index of component groups void decompose(unsigned & x, unsigned xindex[l); // Reconstruct product group index from that of component group indexes const unsigned reconstruct(unsigned xindex[l); private : unsigned Group

ngroups;

// number of groups

**group;

// array of ptrs to groups in product

};

#endif

B. groupcpp #include #include #include

<stdlib.h> "group.h" " 1ib h "

.

// Constructor for a group Group::Group (const enum Type settype, const unsigned size) { // Make sure the size is larger than 1 if (size < 2 ) ( error ("Group constructor: size must be larger than 1"); 1 type=settype; N = size;

// fill in these elements

1 // Index of the inverse of the group element const unsigned Group::operator[l(const unsigned i) ( for (unsigned j=O; j C N ; ++j) ( if ( 0 == (*this)(i,j)) return j ; 1

1 // Index summation operator for Cyclic group const unsigned CyclicGroup::operatorO(const unsigned i, const unsigned j) return (i+j)%N;

1

(

66


// Index summation operator f o r Dihedral group const unsigned Dihedra1Group::operatorO (const unsigned i, const unsigned j ) return ( N + j + ((j61) 3 -i : i)) % N; 1 static unsigned QuaternionIndexSum~81IB1 = // ( 0 , 1, 2 , 3 . 4 , 5. 6, 7 ) , // (1, 0 , 3 , 2 , 5, 4 , 7 , 6 1 , // { 2 , 3 , 1, O r 6, 7 , 5, 4 ) , // ( 3 , 2 , 0 , 1, 7 , 6, 4 , 5 1 , // { 4 , 5 , 7 , 6, 1, 0 , 2 , 3 1 , ( 5 , 4 , 6, 7 , 0 , 1, 3 , 2 ) . (6, 7 , 4 , 5 . 3 , 2 , 1, O ) , ( 7 , 6, 5, 4 , 2 , 3, 0 , 11,

(

{

go=+1

gl=-1 gZ=+i g3=-i 94=+1

// +=-I // 96=+k / / g7=-k

1; // Index summation operator for Quaternion group const unsigned QuaternionGroup::operatorO(const unsigned i, const unsigned j ) return QuaternionIndexSum[il[jl; I

static unsigned GroupListSize (Group **GroupList) {

// Initialize f o r computing the total number of elements in the // direct product of the groups. unsigned N = 1; // Multiply the size of each group in the list f o r (unsigned i=O; NULL ! = GroupList[il; ++i) ( N *= GroupList[il->size(); 1

return N;

1

// Constructor f o r the direct product of groups ProductGroup::ProductGroup (Group * g o , )

...

:

Group(Group::ProductType,GroupListSize(&gO)) (

Group **GroupList

= &go;

// Count the number of groups in the list for (ngroups=O; NULL !=GroupList[ngroupsl; ++ngrOUps); group = new Group*[ngroupsl ; f o r (unsigned i=O; i < ngroups; ++i) group[il

1 // Destructor f o r the direct product of groups ProductGroup:: -ProductGroup (void) ( delete group;

>

=

GroupList[il;

(


// Are the direct product groups identical? const int ProductGroup::operator==(const ProductGroup &pgroup)

67

(

// First of all, are the number of groups the same? if (ngroups ! = pgroup.ngroups) return 0; // Verify that each of the component groups in the direct product // match - in order for (unsigned i=O; i < ngroups; ++i) ( if ( ! (group[il == pgroup.groupli1)) return 0; 1

return 1;

// direct product of groups is identical

// Index summation operator for direct product of groups const unsigned ProductGroup::operator()(const unsigned i, const unsigned j ) (

// Allocate buffers for the decomposition of the product group // indexes into indexes of component groups unsigned *xindex = new unsigned[ngroupsl; unsigned 'yindex = new unsignedlngroupsl;

// Decompose each product group index into indexes of component groups decompose (i, xindex); decompose ( j , yindex); // Compute the index summation in the component groups for (unsigned k=O; k < ngroups; ++k) { xindexIk1 = ~*qroupIkl)(xindexIkl,yindexlkl);

1 // Construct product group index from a list of component group indexes unsigned I = reconstruct (xindex); delete xindex; delete yindex; return

// done with temporary arrays

I;

1 // Decompose each product group index into indexes of component groups void ProductGroup::decompose(unsigned L x, unsigned *xindex) ( for (unsigned i=ngroups-1; i > 0; --i ) { xindexlil = x % grouplil->size(); x /= groupIi1->size(); 1 xindexlil = x;

1 // Construct product group index from a list of component group indexes const unsigned ProductGroup::reconstruct(unsigned *xindex) (


68

unsigned x = xindex[Ol; for (unsigned i=l; i C ngroups; ++i) ( x = x * group[il->size0 + xindex[il;

1 return x:

C. grpa1g.h #ifndef --GROUP-ALGEBRA-H-#define --GROUP-ALGEBRA-H-include

"gr0up.h"

class GrpAlg

(

public:

// constructors and destructors GrpAlg (ProductGroup 'pgroup); GrpAlg (const GrpAlg 6 x); -GrpAlg (void); // Access routines GrpAlg & operator=(const GrpAlg & float & operator[l(const unsigned float & operatorO(const unsigned const unsigned size(void) (return const int sametype(const GrpAlg & const int operator==(const GrpAlg // Addition, const GrpAlg const GrpAlg const GrpAlg const GrpAlg const GrpAlg const GrpAlg

// main constructor // COPY // destructor

X); i) (return c[il;) ig0, );

....

N;}

x); &

XI;

convolution, scalar multiply operations on elements operator+(const GrpAlg & x); // add // convolve operator*(const GrpAlg & x); operator*(const float s ) ; // scalar mult operator/(const float 5 ) ; // scalar divide // unary negate operator-(void); operator+=(const GrpAlg & XI; // accumulate

// Other operations on group algebra elements const float net(void); const float norm(void); const int inv(GrpA1g x); protected: unsigned float ProductGroup 1;

// sum coeffs // vector length // mult inverse?

N; // number of elements in group algebra *c; // array of coefficients *pgroup;// ptr to the direct product of groups

class GrpAlgO : public GrpAlg { // additive inverse element public: GrpAlgO (ProductGroup *pgroup) : GrpAlg(pgroup) ( ) -GrpAlgO (void) ( ) j;

// // // // // //

assign index DG index groups size same GrpAlg? equal element


class GrpAlgl : public GrpAlg { // multiplicative inverse element public : GrpAlgl (ProductGroup *pgroup) : GrpAlg(pgroup) (c[OI=l;) -GrpAlgl (void) ( )

1; #endif

D . grpalg.cpp #include #include #include #include #include #include

<stdlib.h> <stdarg.h> <math.h> ”grpa1g.h” ” 1ib h ”

.

// Constructor f o r a group algebra over a direct product of groups GrpA1g::GrpAlg (ProductGroup *set-pgroup) ( // Make sure the product group is valid if (NULL == set-pgroup) ( error (“GrpAlg constructor: invalid product group“); 1 pgroup = set-pgroup; N = pgroup->size(); c = new float [NI :

// initialize to the zero element f o r (unsigned i=O; i < N ; +ti) c[il

=

0.0;

1 / / Copy constructor GrpA1g::GrpAlg (const GrpAlg

&

x) (

pgroup = x.pgroup: N = x.N; c = new float IN1 ; for (unsigned j = O ; j < N ; + + j ) C[ll = X.C[ll;

1 // Destructor GrpA1g::-GrpAlg (void) ( if (c) delete c ; 1

// array of coefficients

// Given the index values of the groups in the direct product, compute // the index value in the isomorphic direct product group float & GrpAlg::operator()(const unsigned ig0, ...) { va-list gindexes; // variable group index arguments va-start (gindexes,igO);// initialize variable argument processing

69

70


// allocate buffer for indexes in component groups unsigned *index = new unsigned[pgroup->dimOl; index[Ol

=

// start off with the first index

ig0;

// leading index-major ordering f o r (int i-1; i < pgroup->dim(); ++i) ( index[il = va-arg(gindexes,unsigned);

i va-end(gindexes1;

// done with variable arguments

// convert indexes in component g r o u p s to index in product group unsigned dindex = pgroup->reconstruct(index); delete index; return ~[dindexl ;

// done with temporary buffer // return element from direct product group

}

// Are the two elements from the same group algebra? const int GrpAlg::sametype(const GrpAlg & x) { return (*pgroup == *(x.pgroup));

1 // Assignment operator GrpAlg & GrpAlg::operator=(const GrpAlg & x ) ( If not the same group type, destroy the previous object if (!sametype(x)) ( error ("GrpAlg::operator=: types do not match");

1 // Then just copy over the new coefficients for (unsigned j=O; j < N; ++j) c[jl = x.c[jl;

return *this;

> // Do the two elements have the same values from the same group algebra? const int GrpAlg::operator==(const GrpAlg & x) { // Error if the types do not match if (!sametype(x)) { error ("GrpAlg::operator==: types do not match"); 1

// Stop when a value is different for (unsigned i=O; i < N; ++i) ( if (fabs(c[il-x.c[i]) > FLT-EPSILON) return 0; }

return 1;

// element values are the same

1 // Group algebra addition of elements const GrpALg GrpAlg::operator+(const GrpAlg

&

x) (


// E r r o r if the types do not match if (!sametype(x)) ( error ("GrpAlg::operator+: types do not match"); )

GrpAlg r(x);

// make a copy of x

for (unsigned i=O; i < N; ++i) r.c[il += c[il; return

I;

1

// Group algebra additive accumulation Const GrpAlg GrpAlg::operator+=(const GrpAlg

x) (

&

// E r r o r if the types do not match if (!sametype(x)) ( error ("GrpAlg::operator+=: types do not match"); 1 for (unsigned i=O; i < N; ++i) c[il += x.c[il; return *this;

1 // Group algebra multiplication of an element by a scalar const GrpAlg GrpAlg::operator*(const float s ) { GrpAlg r(*this);

// make a copy of this

for (unsigned i=O; i < N; ++i) r.c[il

*=

s;

return I; )

// Group algebra division of an element by a scalar const GrpAlg GrpAlg::operator/(const float s ) ( GrpAlg r(*this);


for (unsigned i-0; i < N; ++i) r.c[il /= s ; return r ;

1

// Additive inverse of a group algebra element const GrpAlg GrpA1g::operator-(void) ( GrpAlg r(*this); f o r (unsigned i=O; i

return

1

I;


< N; ++i) r.c[il

=

-r.c[i];

72


// Group algebra multiplication (convolution) of elements const GrpAlg GrpAlg::operator*(conSt GrpAlg & X) ( // E r r o r if the types do not match if (!sametype(x)) { error ("GrpAlg::operator*: types do not match");

I GrpAlg r(pgroup);

// zero element for computing convolution

for (unsigned i=O; i < N; ++i) { for (unsigned j = O ; j < N ; ++j) { unsigned k = (*pgroup)(i,j); r [ k l += c[il * x.c[jl; 1 1 return r ; }

// Compute the multiplicative inverse o f a group algebra element const int GrpAlg::inv(GrpAlg * x) { // Error if the types do not match if (!sametype(*x)) { error ("GrpAlg: :inv: types do not match"); 1 unsigned i, j; // loop indexes

// Get the group algebra multiplicative identity element GrpAlgl one(pgroup); // Establish a two dimension array for Gauss Jordan Elimination float **a = new float*"]; for (i=O; i < N; ++i) a[il = new float[N+lI; // Fill in the left NxN matrix with the group algebra element // represented a s a linear transform. for (i=O; i < N; ++i) { for ( j = O ; j < N; + + j ) { / / This works out to be i + j-(-l) where // '+' is the index sum operation a[il [ j] = c[ (*pgroup)(i,('pgroup) j l ) 1 ;

I }

// Fill in the last column vector with the elements from / / the group algebra multiplicative identity element for (i=O; i < N; ++i) a[il[NI = one.c[il; // Perform the Gauss Jordan elimination which returns the rank / / of the matrix which is less than N if the left NxN matrix // is linearly dependent. int is-inverse = (N == GaussJordan(a,N,N+l));


73

// I f t h e e l e m e n t i s i n v e r t i b l e , t h e n copy t h e c o e f f i c i e n t s // o f t h e i n v e r s e e l e m e n t . f o r (i=O; i

c[il = a [ i l [ N l ;

N;

// R e l e a s e d y n a m i c a l l y a l l o c a t e d b u f f e r memory f o r ( i = O ; i < N ; + + id)e l e t e a [ i l ; d e l e t e a;

// r e t u r n b o o l e a n s t a t u s

return is-inverse;

1 // Compute t h e n e t of a g r o u p a l g e b r a e l e m e n t const f l o a t GrpAlg::net(void) f l o a t sum=0.0;

{

// f o r c o m p u t i n g sum o f c o e f f i c i e n t s

f o r (unsigned i=O;

i

+ f 1

(1

"]ro

after replacing rI and tan a, according to Eqs. (3), (4), and (1). Considering two different electron trajectories marked by the additional index numbers I and 11, respectively, magnification M caused by the EMS is defined by the quotient

M

= rS,ll 'r,a

-

'S.1

(7)

- rr.1

which, using Eqs. ( 2 ) and (6), finally results in the expression

The result is the same if the electron trajectories do not have the same meridian plane. Thus,

r, = Mr,

(9) express the relation between image plane and object plane in the shadow imaging mode if one takes into account that a point P(x,y) in the surface plane of the specimen correlates with that electron trajectory which is characterized by a point of reversal having the same coordinates xr and y,. Taking into account the ratio of the radial electron velocity component v, and the initial electron velocity vo given by or

Vr -

xs = Mx,

= tan a1 =

VO

y, = Myr

rr

f + 2zd ~

coordinate z, of the point of reversal, which coincides with its distance from the corresponding object point at the specimen surface, is obtained from energy considerations and results in z, =

r,"

(f+ 22,)'

zd

An additional specimen bias V,, usually negative, changes Eq. ( 1 1) to

where V,, as mentioned above, is the positive value of the high voltage applied to the electron gun.

102

REINHOLD GODEHARDT

Now a slight radial deflection of the electron will be included in the consideration. Following Wiskott (1956) in assuming that the electron interaction with the disturbing field can be described by a small additional velocity Avr transferred to the electron at the point of reversal, the electron trajectory remains unchanged until the electron has passed this point. The trajectory branch of the reflected electron, however, is changed, resulting also in changed quantities a,, F,, a",, and 7,. The change of a, to a, is described by tan

=

vr ~

+ Avr = tan a , + Av,

-

YO

VO

and with Eq. (13) the changed coordinate F, is determined by

The equation tan ii,

= tan a2

+ (1 +

7)::

- -

is obtained by replacing a , , r , , and m,, by a,, F,, and Eq. (4). Applying the same to Eq. (5) results in

a,, respectively, in

Omitting the last item in the parentheses, as it can be neglected, and using M = 2zt/f, the displacement Ars = F, - r, in the image plane is finally expressed by 1 Ars = - M ( f 2

+ 2 ~ Avr ~ ) VO

Before expressions for a contrast estimation are derived further based on this equation, information concerning the EMS will be given. In the publication of Lukjanov et al. (1968), and also in most of the other publications dealing with the subject, the focal length f used in the previous equations is assumed to be that of a thin electron diverging lens given by

f = 42,

(18)

However, this value is valid only for rd 1 and let there exist i E { 1,2, . .., n} such that EXi # fa. Then there exists j E { 1,2, . . . ,n} such that P X j # U . The following result has also been shown by Grzymala-Busse (1988):

Theorem 2. Let n > 1 and let E X i # P X j # U for each j E { 1,2, . . ., n } .

0 for

each i E { l , 2, ... , n}. Then

We define a partition x = {XI,X , , . . . , X,,} on U as roughly definable, strong if and only if EXi # 0 for each i E { 1,2, . . . , n } . Due to Theorem 2, for each block X i of roughly definable, strong partition x on U we have that P X i # u. Table IV illustrates this case. A partition x = { { c l ,c2, c3}, {c4, c5}, {c6, c7, c8}} is roughly definable, strong for P = A . The partition x = {XI,X , , ...,X,,} on U will be called internally undefinable, weak if and only if E X , = /a for each i E { 1,2, ... ,n } and there exists j E { 1,2, ...,n } such that P X j # U . For Table VI, partition x = { { c l } ,{c3, c9}, { c l l } ) ,and P = A , E((C1))

=

0,

P ( { c l } )= { c l , c3}

P((C3, .9}) = 0,

P((c3,c9}) =

u

_ p ( { c l l ) )= fa, F ( { c l l } )= { C S , C l l } ,

so x is internally undefinable, weak. A partition x = {XI,X,, . . ., X . } on U will be called internally undefinable, strong if and only if E X , # 0 and P X i # U for each i E { 1,2, . . .,n } . Table VIII illustrates this case for partition x = ( { c l ,c12}, (c3, c5}, (c7, c8}}

170

JERZY W. GRZYMALA-BUSSE TABLE VIII DECISION TABLE ___

~

Attributes Capacity Vibration 5 5 2

cl c3 c5 cl c8 c12

medium medium low low low low

4 2 4

Decision Quality low medium medium high high low

TABLE IX DECISION TABLE

Decision

low low

c13

and P

= A,

medium low

since

P((CLC12)) =

!a,

P({cl, ~ 1 2 )= ) {cl, c3, c7, c12}, P((C7,c8j) =

P ( { c 3 , c 5 } )=

0,

P((c3, c5))

{ ~ l~, 3~, 5~,8 } ,

=

0,

P((c7, ~ 8 ) =) (c5, c7, c8, c12}, so x is internally undefinable, strong. Theorem 1 may be given in another form (Grzymala-Busse, 1988): Theorem 1’. Let n > 1 and let P X i = U for each i E { 1,2, . .., n } . Then EXj = 0for each J E {1,2, ...,n}.

Due to Theorem l’, condition that p X i = U for each i E { 1,2, . ..,n } is so strong that analogs of an externally undefinable sets do not exist for partitions on U. Moreover, there exists only one kind of partition x = {XI, X , , . . . , X , } on U for which P X i = U for each i E { 1,2, . . ., n } , given by the following definition. The partition x = { X , , X,, . .. , X , } on U , n > 1, is called totally undefinable i f and only if F X i = U for each i E (1,2, . ..,n } . In Table IX, a partition

171

ROUGH SETS

x = ( { c l ,c13), {c3, c9>>is totally undefinable for P = A, because P ( { C l , c31) =

0,

F((c1, ~ 1 3 )=) U ,

P ( ( c 3 , c9)) =

0,

P((c3, ~ 9 >=) U .

11. ROUGH SETTHEORY AND EVIDENCE THEORY

Evidence theory, also called Dempster-Shafer theory, was introduced by A. P. Dempster (1967, 1968) and developed further by G. Shafer (1976). It is a generalization of probability theory. A. Basic Properties

In probability theory, probabilities should be assigned to elements of sample space. Probability for any event, a subset of the sample space, may be computed by Kolmogorov's addition axiom. In evidence theory this is not so. Let 0 be a frame of discernment (the notion that corresponds to the sample space of probability theory). The basic idea of evidence theory is a basic probability assignment, i.e., a function m: 2' -+ [0, 13 such that 1. m ( @ ) = 0,

The number m ( X ) is called a basic probability number of X . Thus, basic probability numbers are assigned to subsets of the frame of discernment, as opposed to probability theory, where probabilities are assigned to singleton sets. In evidence theory it may happen that m ( { x } )# 0 and m ( X ) = 0, where x E X . In probability theory that is impossible; if probability p(x) # 0, then P ( X ) z 0. Two additional functions, a belief function and a plausibility function, are commonly used in evidence theory. Both functions are defined from the basic probability assignment. A function Bel: 2' -+ [O, 11 is called a belief function over 0 if and only if Bel(A)=

1 m(B). BEA

A function P1: 2@+ [0, 11 is called a plausibility function over 0 if and only if Pl(A) = 1 - Bel(1). A subset X of the frame of discernment is called a focal element if and only if m(X)> 0.

172

JERZY W. GRZYMALA-BUSSE

In evidence theory, two independent pieces of evidence may be combined using Dempster’s rule of combination by so-called orthogonal sum. Say that the two pieces of evidence are described by their basic probability assignments m and n. Focal elements of m are A , , A , , . . .,A,, and focal elements of n are B,, B,, . . ., Bl. Then, for a focal element X = Ai n Bj, where i E (1,2, ..., k } and j E (1,2, ..., 1 } ,

m 0 n(X)=

1

m(Ai) * n(Bj)

1

m(Ai)*n(Bj) -1-

ij AinBj=X

C

ij AinBj=X

ij AinB,#0

m(Ai) * n(Bj)

m(Ai).n(Bj)’ ij A i n B j = pI

where m 0 n denotes the orthogonal sum of m and n. The main difference between evidence theory and rough set theory is philosophical-rough set theory presents an objective approach to the real world, while evidence theory uses a subjective approach. Also, in rough set theory the basic tool are sets (lower and upper approximations of a concept, computed directly from input data), while evidence theory relies on functions, e.g., basic probability assignment, given by experts. However, there exists a simple correspondence between rough set theory and evidence theory (Skowron and Grzymala-Busse, 1994). Say that a real-world problem, categorizing motorcycles ml, m2, . ..,m12, is presented in Table X. The frame of discernment 0 is the set of all values of the decision d = Quality. For the rest of this section we will assume that we are interested only in the entire set A of attributes and not in any subset P of the set A. Let [XI denote a block of the partition U/IND(A), where x E U.In the example from Table X,such blocks are (ml, m8, m10), (m2, m6, m l l ) , (m3, m12}, (m4,m7), TABLE X DECISION TABLE

ml m2 m3 m4 m5 m6 m7 m8

m9 m10 mll m12

-

medium low medium low low low low medium medium medium low medium

high medium medium low high medium low high low high medium medium

low low low low low low low medium medium high high high

ROUGH SETS

173

{m5}, {m9>.We may define two functions, f : U -+ 2' and g: U + 2', defined as follows: f(x) =

M Y 9

4 I Y E CxI>,

g(x) = { Y E Ulf(x) = f(Y)L

In our example,

f(m1) = {low,medium, high} = f(m8) = f(mlO), f(m2) = {low,high}

= f(m3) = f ( m 6 ) = f(ml1) = f(m12),

f(m4) = {low} = f(m5) = f(m7), f(m9)

=

{medium},

and g(m1) = {ml, m8, m10} = g(m8) = g(mlO), g(m2) = {m2, m3, m6, m l l , m12) = g(m3) = g(m6) = g(ml1) = g(m12), g(m4) = {m4, m5, m7} = g(m5) = g(m7), g(m9) = {mg}.

Any decision table determines a basic probability assignment on 0 with {f(x)lx E V} as focal elements. In the example from Table X,focal elements are {low},{medium}, {low,high}, and {low,medium, high}. We may define a function m: 2e --+ [O, 11, associated with the given decision table, as follows. First, for any 0 E 2e, we test whether 0 is equal to f(x) for some x E U. If so,

If not, m(0) = 0. It is not difficult to recognize that the above function m is a basic probability assignment. In our example,

174


Thus we have established a relation between rough set theory and evidence theory: For every decision table we may create a basic probability assignment. There exists a converse relation, i.e., we may create a decision table for a basic probability assignment with relational numbers as values. We will show how such a decision table is created in two parts. In the first part of the construction, let us create a decision table for a given set of focal elements. Say that our frame of discernment is the set {low, medium, high} and that the focal elements are {low), {medium, high], and {low, medium, high}. We will create a decision table such that the set ( f ( x ) l x E V } will be identical with the set {{low},{medium, high}, and {low, medium, high}}. We will use a generic attribute, denoted a, with positive integers as generic values. The first step of construction of the corresponding decision table is presented in Table XI. All values of all focal elements are listed in the column denoted by decision d. Also, examples are numbered 1,2, . . . , 6. The second step of construction of the decision table is presented in Table XII, where values of the attribute a are inserted. The values are consecutive positive integers, constant for each focal element. The obtained decision table satisfies the requirements, as we may show by the following computations. The blocks of the partition U/IND({a)) are {l}, {2,3), and (4, 5 , 6 } . Function f is given by f(1) = {low}, f(2) = f(3) = {medium, high), f(4) = f(5) = f(6) = {low, medium, high}. Thus, focal elements are {low}, {medium, high}, and {low, medium, high}. TABLE XI DECISION TABLE-FIRSTSTEP OF THE CONSTRUCTION Attribute

Decision

175

ROUGH SETS TABLE XI1 DECISION TABLE-SECOND STEPOF THE CONSTRUCTION

medium high 5

3

medium high

Function g is as follows: g(1) = {1),

g(2) = g(3) = { 2 , 3 ) , g(4) = g(5) = d 6 ) = ( 4 , 5 6 ) .

Moreover, the basic probability assignment m is defined m ( ( l o w } )= i , m( (medium, high)) = f, m( {low, medium, high}) =

4.

Now, in the second part of the construction, let us assume that we want to create a decision table with the same focal elements but with another basic probability assignment. This may be accomplished by adding duplicates to the decision table constructed in the first part. In this particular example, let us assume that we want to change the current basic probability assignment into a new one, denoted n, such that n( ( l o w ) ) = 0.2, n( {medium, high)) = 0.4, and n({low,medium, high)) = 0.4. The ith row of Table XI1 should be duplicated xi times, where i = 1,2, . . .,6. It is not difficult to recognize that n ( { l o w } )=

XI

x1

+ x2 + + x6' "'

176


Attribute a 1

2

3 4 5

6 I 8 9 10

11 12 13 14 15

Decision d

1 1 1

low low low

2 2 2 2 2 2

medium medium medium high high high

3 3 3 3 3 3

low low medium medium high high

Hence we should solve the following system of equations:

+ + + + = 0.4 + + + x4 + + = 0.4 ( X I + + + x6). XI

x2

x3

XS

x6

= 0.2

(XI

x2

*"

x6)9

(XI

x2

' ''

x6)7

x2

' * *

The simplest solution of the above system of equations is x1 = x2 = x3 = 3 and x4 = x5 = x6 = 2. The decision table obtained as a result of this construction is presented in Table XIII.

B. Dempster Rule of Combination We may combine two decision tables into a single decision table the same way as two basic probability assignments are combined into a single one using the Dempster rule of combination. Moreover, if the two basic probability assignments computed from the two tables are m and n, respectively, then the basic probability assignment for their combination is m 0 n. The construction is straightforward and will be explained by analysis of the following example. Say that two decision tables are given, Tables XIV and XV. First we need to compute the values of function f for both tables. Then we construct new tables with decision values replaced by values of the function f. The corresponding tables are presented as Tables XVI and XVII. The

177

ROUGH SETS

Attributes Weight Power

Decision Quality

I

I

medium low medium medium low

1

2 8 10 11

low low medium high high

medium medium high high medium

TABLE XV DECISION TABLE Size

medium red

Attributes Weight Power

medium medium high high

f (4 {low, medium, high}

{low, medium, high} {low, medium, high}

Attributes 13 14 15

16

Size

Color

big big medium big

red blue red red

f (4 {medium, high} {medium} {high} {medium, high}

178

JERZY W. GRZYMALA-BUSSE TABLE XVIII COMBINATION OF TABLESXVI AND XVII Weight medium medium medium medium low low low medium medium medium medium medium medium medium medium low low low

Attributes Power Size high high high high medium medium medium high high high high high high high high medium medium medium

big big medium big big medium big big big medium big big big medium big big medium big

Color red blue red red red red red red blue red red red blue red red red red red

{medium, high} {medium} {high} {medium, high} {high} {high} {high} {medium, high} {medium} {high} {medium, high} {medium, high} {medium} {high} {medium, high} {high} {high) {high}

combination of Tables XVI and XVII is presented as Table XVIII. Every row of Table XVIII is computed partly from Table XVI and partly from Table XVII. For every row x of Table XVI and for every row y of Table XVII such that the intersection off(x) and f(y) is a nonempty set, we create a new row of the Table XVIII; such row will be denoted by (x, y). Attributes of Table XVIII are all attributes of the first table followed by all attributes of the second table. Attribute values for row (x, y) are copied from row x of the first table and y of the second table. Table XVIII is not quite a decision table-there is no column designated as a decision. Instead of a decision there is a column denoted as f(x) n f ( y ) . However, we may transform this table into a decision table by replacing the column name ‘‘f(x) nf(y)” by a name of the new decision and values of f(x) nf(y) by single decision values, evenly split between rows. The resulting decision table that corresponds to the Table XVIII is presented as Table XIX. Note that the basic probability assignment rn for Table XVI may be computed as follows:

f(1) = f(8) = f(l0) = {low, medium, high}, f(2) = f(1 1 ) = {low, high},

179

ROUGH SETS

Attributes

1

14 16 17

I

Weight

Power

Size

medium medium medium medium low low low medium medium medium medium medium medium medium medium low low low

high high high high medium medium medium high high high high high high high high medium medium medium

big big medium big big medium big big big medium big big big medium big big medium big

Color red blue red red red red red red blue red red red blue red red red red red

Quality

I

medium medium high medium high high high medium medium high high high medium high high high high high

g(1) = d 8 ) = g(10) = { 4 8 , lo}, 9(2) = 9(11) = (2, 1I},

The focal elements are {low, high} and {low, medium, high}, and

m({low,high}) = 3, m( { low, medium, high}) = 3. On the other hand, the basic probability assignment n for Table XVII is

f(13) = f(16) = {medium, high}, f(14)

=

{medium},

fU5) = {high}, g(13) = g(16) = (13, 16},

d14) = { 14}, g(15) = {15},

the focal elements are {medium}, {high},and {medium, high}, and

180


n( {medium}) = $, n( {high}) = $, n( {medium, high}) =

4.

Focal elements o the orthogonal sum m 0 n are {mediurr ,, {high}, and {medium, high}. The orthogonal sum m 0 n is described as follows: m 8 n({high})

m((low, high}).n((high})i m((low, high}).n({medium,high)) + m((low, medium, h$h)).n((high) 1 - m({/ow,high}).n({medium}) - 0.4.0.25 + 0.4.0.5 + 0.6.0.25 - 0.45 - 0.5, 1 - 0.4.0.25 0.9 m Q n({medium})

0.6.0.25 0.15 - m({low,medium, high}).n({medium})= _ _=-= 1 - m({low, high)).n((medium))

0.9

0.167,

0.9

m @ n({medium,high})

0.6.0.5 - m({low, medium, high}),n({medium,high}) 1 - m({/ow,high}).n({medium})

0.9

- -0.3 _ - 0.333. 0.9

On the other hand, we may compute the same probability assignment for Table XIX: f(1) = f(4) = f(8)

= f(l1) = f(12) = f(l5) =

(medium, high},

f(2) = f ( 9 ) = f(13) = {medium}, f(3) = f(5) = f(6) = f(7)

= f(l0) = f(14) = f(16) = f(17) = f(l8) =

{high},

g(1) = g(4) = g(8) = g(l1) = g(12) = ~ ( 1 5= ) {1,4, 8, 11, 12, IS}, g(2) = g(9) = g(13) = {A 9, 13}, g(3) = g(5) = g(6) = g(7) = g(10) = g(14) = g(16) = g(17) = g(18) = (3, 5, 6, 7, 10, 14, 16, 17, l8},

focal elements are {medium}, {high}, and {medium, high}, and the basic probability assignment is m({medium}) = m({high}) =

= 0.167, = 0.5,

m( {medium, high}) = & = 0.333.

Thus, the basic probability assignment computed from Table XIX is identical with the basic probability assignment computed as the orthogonal sum of m and n.

ROUGH SETS

181

C . Numerical Measures of Rough Sets

A few numbers characterizing a degree of uncertainty may be associated with a rough set. Though they are sometimes useful for evaluation, such numbers have limited value, because the most important tool of rough set theory is sets-lower and upper approximations of a given concept X , a subset of the universe U . There are two universal measures of uncertainty for rough sets, introduced in Grzymala-Busse (1987); see also Grzymala-Busse (1991). We will assume that P is a subset of the set A of all attributes. The first measure is called a quality of lower approximation of X by P , denoted y ( X ) ,and equal to

IEXl

~

IUI * In Pawlak (1 984) y ( X ) was called simply quality of approximation of X by P. The quality of lower approximation of X by P is the ratio of the number of all certainly classified elements of U by attributes from P as being in X to the number of elements of U . It is a kind of relative frequency. There exists a nice interpretation of y ( X ) in terms of evidence theory. To introduce this interpretation, we need another definition of the frame of discernment 0 than in Section II.A, namely, we will consider 0 as the set of all concepts, i.e.,

0 = U/IND({d}), where d is a decision. Thus focal elements are blocks of U/IND( {d}), and the basic probability assignment m is defined by

1x1

m(X)= -

IUI’

where X is a concept, i.e., a block of U/IND({d}). With this interpretation, y-( X ) becomes a belief function, as was first observed by Grzymala-Busse (1987). The second measure of uncertainty for rough sets, a quality of upper approximation of X by P , denoted y(X), is equal to

The quality of upper approximation of X by P is the ratio of the number of all possibly classified elements of U by attribute from P as being in X to the number of elements of U . Like the quality of lower approximation, the quality of upper approximation is also a kind of relative frequency. It is a plausibility function of evidence theory if we accept the above interpretation of the basic probability assignment (Grzymala-Busse, 1987).

182


Sometimes yet another measure of uncertainty of rough sets is used, called an accuracy of approximation of X by P, and defined as IfX I IFXI' The above measure has secondary value because it may be easily computed from y ( X ) and Y(X). Yet anot6er measure of uncertainty for rough sets, called directly rough measure, is explained in Section III.C.1.

111. SOME APPLICATIONS OF ROUGH SETS

Rough set theory has been broadly applied in many real-world domains. The detailed record of applications can be found in many reports published in journals and conference proceedings, especially in Slowinski (1992), Ziarko (1994), and Lin (1995). Roughly speaking, the most useful areas of applications are data analysis (or attribute analysis, e.g., seeking for the most essential attributes, elimination of redundant or weak attributes, finding attribute dependencies, estimation of attribute significance) and induction of rules (regularities) from raw data (given as decision tables). Most of methods of attribute analysis using rough set theory are quite straightforward, with the exception of estimating attribute significance. This topic is explained below. Rule induction is covered in the following section. A. Attribute Significance

Rough set theory may be used for analysis of significance of attributes. In this approach, an extension of the notion of the quality of lower approximation of set X , associated with a subset P of the set A of all attributes, is used. The set X , usually a concept, is replaced by the set of all concepts U/IND({d}), where d is a decision. A quality of lower approximation of U/IND({ d } ) ,associated with P, is defined as follows:

where U/IND({d}) = {XI,X , , . . ., Xn}. A significance of the attribute a is computed as the difference between the quality of lower approximation of U/IND( { d } ) associated with the set A of

183

ROUGH SETS

all attributes and the quality of lower approximation of U/IND( { d } ) associated with the set A - { a } . The more important attribute a is the more quality of U/IND({d)) is reduced by deleting a and thus the difference is bigger. Let us illustrate the notion of significance of attributes by the following analysis of Table IV. In Table IV, A = {Capacity, Vibration), d = Quality, and U/IND({d)) = { (cl, c2, c3), (c4, c5), {c6,c7, cS}). The quality of U/IND({d}) associated with A is IA(lcl,c2, c3))l

+ IA((c4, c5))I + IA((c6, c7, c8})l I UI

- I{cl,c3}1

+ I(C4)l + l{c7}1 -- 2 + 1 + 1 = 0.5. I Ul

8

The quality of U/IND({d}) associated with A

-

{ a } , where a = Capacity,

is IA - (a)({cl, ~ 2 ~, 3 } ) 1+ IA - {a)({c4, c5})l

+ IA - {a)({c6, c7, d})l

I UI

Thus the significance of attribute Capacity is 0.5 - 0 = 0.5. Gunn and Grzymala-Busse reported (1994) results of analysis of the attributes solar energy, transmission, El Niiio, CO, trend, and CO, residual, influencing the earth’s global temperature. For data covering years from 1958 to 1990, the ranking of the attributes was computed using statistical methods (ANOVA) and then using rough set theory. Results, cited in Table XX, show surprising similarity in final ranking of the attributes. Similar analysis of attribute significance was done by Pawlak et al., (1986) and by Slowinski and Slowinski (1990). TABLE XX COMPARISON OF ATTRIBUTE RANKING The most important attribute Statistics (ANOVA)

Rough set theory

CO, residual

I

The least important attribute

(0.001)

El Niiio (0.002)

Solar energy (0.119)

Transmission (0.119)

CO, trend (0.943)

CO, residual (0.545)

Solar Energy (0.424)

El Niiio (0.364)

CO, Trend (0.212)

Transmission (0.152)

184


B. LERS: A Rule Induction System Rough set theory is used in a few systems of knowledge acquisition and machine learning from examples. The main purpose of all of these systems is rule induction, i.e., searching for regularities in the form of rules in the original raw data. Induced rules represent knowledge-information on a higher level than that from original raw data. Rules are more general, since more new, unseen cases (examples) can be classified by rules. Also, rules may be inserted into a knowledge base of the expert system. In this section we will study one of the rule induction systems, called LERS (Learning from Examples based on Rough Sets); see Grzymala-Busse (1992). A prototype of LERS was first implemented at the University of Kansas in 1988. The prototype was coded in Franz Lisp under UNIX. The current version of LERS is coded in C and may run under UNIX or VMS. Other systems of rule induction using rough set theory include RoughDAS and Roughclass (Slowinski, 1992; Slowinski and Stefanowski, 1992), created at the Poznan University of Technology in Poland, and Datalogic (Szladow and Ziarko, 1993; Ziarko, 1992), developed at the University of Regina in Canada. Some applications of these systems are listed by Pawlak et al. (1995). The user of system LERS may use four main methods of rule induction: two methods of machine learning from examples (called LEMl and LEM2, where LEM stands for Learning from Examples Module), and two methods of knowledge acquisition (called All Global Coverings and All Rules). Two of these methods are global (LEM1 and All Global Coverings); the remaining two are local (LEM2 and All Rules). In general, the input data are preprocessed before rule induction. For example, numerical values are replaced by intervals, in the process called discretization. The most important reason for discretization is that rules induced directly from numerical data are too detailed. New cases cannot be classified using rules with such detailed conditions. Also, during preprocessing, one of the methods of dealing with missing attribute values should be used when necessary. Input data file is presented to LERS in the form illustrated in Table XXI. In this table, the first line (in the sequel, any line does not need to be one physical line-it may contain a few physical lines) contains declarations of variables: a stands for an attribute, d for decision, x means “ignore this variable.” The list of these symbols starts with “(” and ends with “)”. The second line contains declarations of names of all variables, attributes, and decisions, surrounded by brackets. The following lines contain values of all variables.

185

ROUGH SETS

TABLE XXI LERS INPUT DATAFILE

< [

a Capacity

5 4 5 5 2 4 4

2

X

X

a

Interior fair fair good fair fair excellent good good

Noise medium medium medium low medium low low low

Vibration medium medium medium low low medium low low

d Quality low low low medium medium high high high

) ]

TABLE XXII DECISION TABLE

c5

c6 c8

medium medium medium low low medium low low

L

Before running any of the four methods of rule induction, LERS checks input data for consistency. If the input data are inconsistent, lower and upper approximations for each concept are computed. In local methods, rules are induced directly from lower and upper approximations of the concept. In global methods, the original decision table is substituted by 2k new decision tables, where k is the number of concepts (i.e., the number of different decision values). In each new decision table, the original decision is replaced by a new variable, called either lower substitutional decision or upper substitutional decision, and with values equal to the lower or upper approximation of a concept, respectively. Rules are induced only for the lower or upper approximations of the concept; all remaining examples in a table are considered to be negative and-as such-may be all marked by a special symbol, say "*." The three tables with lower substitutional decisions are presented as Tables XXII-XXIV, and the three tables with upper substitu-

186

JERZY W. GRZYMALA-BUSSE TABLE XXIII DECISION TABLE Decision Quality

Attributes Capacity Vibration

5 4 5 5 2 4

cl c2 c3 c4 c5

c6 cl c8

4

-

2

*


*

* medium

* * * *

TABLE XXIV DECISION TABLE I

I

I


c6 C7 c8

I

Decision Quality

medium medium medium

* * *

low medium low low

* *

*

high

*

TABLE XXV DECISION TABLE

I ~


Cl c2 c3 c4 c5 c6 c7 c8

5 4 5 5 2 4 4 2


~~

Decision Quality

low low low

* *

low

*

*

187

ROUGH SETS


cl c2 c3 c4 c5 c6 Cl

c8

5 4 5 5 2 4 4 2

cl c2 c3 c4 c5 c6 cl c8

-

* *

* medium medium

*

* medium

TABLE XXVII DECISION TABLE

-


Decision Quality

Attributes Capacity Vibration 5

4 5 5 2 4 4 2


Decision Quality

* high

* *

high high high high

tional decisions are presented as Tables XXV-XXVII; all of these tables are computed from Table IV. Note that all six tables (Tables XXII-XXVII) are consistent. In general, rules induced from lower approximation of the concept, for reasons explained in Section I.E, are called certain. Rules induced from the upper approximations of the concept are called possible. Rules induced from training data may be used for classification of new, unseen cases. Usually certain and possible rules are used independently, together with additional information such as the number of training cases that a rule describes, the number of training cases for which the rule was successful (i.e., the rule correctly classified these cases), etc. (Grzymala-Busse and Woolery, 1994).

188

1. L E M l -Single


Global Covering

In the LEMl option LERS may or may not take into account priorities, associated with attributes and provided by the user. For example, in the medical domain, some tests (attributes) may be dangerous for a patient. Also, some tests may be very expensive, while-at the same time-the same decision may be made using less expensive tests. LERS computes a single global covering using the following algorithm: Algorithm SINGLE GLOBAL COVERING input: A decision table with set A of all attributes and decision d; Output: a single global covering R; begin compute partition U/IND(A); P := A; R := 0; If U/IND(A) 5 U/IND( {d}) then begin for each attribute a in A do begin Q := P - {a}; compute partition U/IND(Q); If U/IND(Q) I U/IND({d}) then P := 0 end {for} R := P end {then} end {algorithm}.

The above algorithm is heuristic, and there is no guarantee that the computed global covering is the simplest. However, the problem of looking for all global coverings in order to select the best one is of exponential complexity in the worst case. LERS computes all global coverings in another option, described below. The option of computing of all global coverings is feasible for small input data files and when it is required to induce as much rules as possible. After computing a single global covering, LEMl computes rules directly from the covering using a technique called dropping conditions (i.e., simplifying rules). In the case of Table 111 (this table is consistent), the following rules are induced by LEM 1:

189

ROUGH SETS

(Noise, medium) & (Vibration, medium) (Quality, (Interior, fair) & (Vibration, low) + (Quality, (Interior, excellent) + (Quality, (Interior, good) & (Vibration, low) + (Quality, -+

low), medium), high), high).

On the other hand, with the following attribute priorities, Capacity: 0, Interior: 1, Noise: 0, Vibration: 1 (the larger number the higher priority), the following rules are induced by LEMl from the same Table 111: (Interior, fair) 8 (Vibration, medium) + (Quality, low), (Interior, good) & (Vibration, medium) + (Quality, low), (Interior, fair) & (Vibration, low) + (Quality, medium), (Interior, excel lent) + (Qua1ity , high) , (Interior, good) & (Vibration, low) + (Quality, high).

It is obvious that these rules were induced from the global covering {Interior, Vibration}. The LEMl applied to data from Table IV (this table is inconsistent) will induce the following rules (all priority numbers are set to zero). Certain rules: (Capacity, 5) 8 (Vibration, medium) + (Quality, low), (Capacity, 5 ) & (Vibration, low) + (Quality, medium), (Capacity, 4) & (Vibration, low) -+ (Quality, high).

Possible rules: (Vibration, medium) -+ (Quality, low) with rough measure 0.75, (Capacity, 5 ) 8 (Vibration, low) + (Quality, medium) with rough measure 1, (Capacity, 2) -+ (Quality, medium) with rough measure 0.5, (Capacity, 4) + (Quality, high) with rough measure 0.667, (Capacity, 2) -+ (Quality, high) with rough measure 0.5.

Note that possible rules, induced by LERS, are additionally equipped with numbers, called rough measures. In general, say that X denotes the original concept, described by a rule, and that the same rule actually describes examples from a set Y, a subset of the universe U . If the rule is not certain, set Y does not need to be a subset of X . A rough measure, associated with the rule, is defined as follows: I X n YI

190


The interpretation of the rough measure is that it is a kind of relative frequency, which may be interpreted as a conditional probability P ( X I Y ) . For example, for the possible rule (Vibration, medium) + (Quality, low)

induced by LEMl algorithm from Table IV, the concept X is {cl, c2, c3}, the set of all examples, described by the rule, is {cl, c2, c3, c6}, so the rough measure is I(c1, c2, c3} A (el, c2, c3, c6}( 3 - ((cl, c2, c3}( = = 0.75. I{cL c2, c3, c6}1 ({cl,c2, c3, c6}l 4

Obviously, all rough measures, associate with certain rules, are equal to 1. The rule’s rough measure is helpful in classification of new cases, because it contains information about the quality of the rule. The larger the rough measure, the better the rule is. 2. LEMZ-Single

Local Covering

As in LEMl (single global covering option of LERS), the option LEM2 (single local covering) may take into account attribute priorities assigned by the user. In the LEM2 option, LERS uses the following algorithm for computing a single local covering: Algorithm SINGLE LOCAL COVERING Input: A decision table with a set A of all attributes and concept 6; Output: A single local covering T of concept B; begin G := B;

lr

:=a;

while G # 0 begin T := 0; T(G) := {(a, v)la E A, v is a value of a, [(a, v)] n G # whlle T = $3or [TI LZ B begin select a pair (a, v) E T(G) with the highest attribute priority, if a tie occurs, select a pair (a, v) E T(G) such that [[(a, v)] n GI is maximum; if another tie occurs, select a pair (a, v) E T(G) with the smallest cardinality of [(a, v)]; if a further tie occurs, select first pair; T := T u {(a, v)}; G := [(a, v)] n G;

a};

ROUGH SETS

191

T(G) := {(a,v)l[(a, v)l n G # 0}; T(G) := T(G) - T; end (while} for each (a,v) in T do if [T - {(a,v)}] s 6 then T := T - {(a,v)}; T := T u {T}; G := B [TI;

u

TET

end {while}; for each T in T do [S] = 6 then T := ?r - {T}; if

u

SET- { T]

end {algorithm}.

Like the LEMl option, the above algorithm is heuristic. The choice of criteria used was found as a result of many experiments (Grzymala-Busse and Werbrouc, 1991). Rules are eventually induced directly from the local covering; no further simplification (as in LEM1) is required. For Table 111, LEM2 induces the following rules: (Noise, medium) & (Vibration, medium) + (Quality, low), (Interior, fair) 8. (Vibration, low) + (Quality, medium), (Noise, low) & (Capacity, 4) --t (Quality, high), (Capacity, 2) & (Interior, good) + (Quality, high). 3. AZl Global Coverings

The All Global Coverings option of LERS represents a knowledge acquisition approach to rule induction. LERS attempts to compute all global coverings and then all minimal rules are induced by dropping conditions. The following algorithm is used: Algorithm ALL GLOBAL COVERINGS input: A decision table with s e t A of all attributes and decision d; Output: the set IR of all global coverings of {d}; begin IR := 0; for each attribute a in A do

compute partition U/IND( {a}); k : = 1; while k IIAl do begin

for each subset P of the set A with (PI = k do

192

JERZY W.GRZYMALA-BUSSE

if (P is not a superset of any member of lR) and ( WIND( (a}) I U/IND( (d))

n

aE P

then add P to IR, k:= k + 1 end (while) end {algorithm}.

As we observed before, the time complexity for the problem of determining all global coverings (in the worst case) is exponential. However, there are cases when it is necessary to induce rules from all global coverings (Grzymala-Busseand Grzymala-Busse, 1994). Hence the user has an option to fix the value of a LERS parameter, reducing in this fashion the size of global coverings (due to this restriction, only global coverings of size not exceeding this number are computed). For example, from Table IV, the following rules are induced by the All Global Coverings option of LERS: (Capacity, 5) & (Noise, medium) + (Quality, low), (Capacity, 4) & (Noise, medium) -+ (Quality, low), (Interior, fair) & (Vibration, medium) + (Quality, low), (Interior, good) & (Vibration, medium) + (Quality, low), (Noise, medium) & (Vibration, medium) + (Quality, low), (Capacity, 5) & (Noise, low) + (Quality, medium), (Capacity, 2) & (Noise, medium) + (Quality, medium), (Interior, fair) & (Vibration, low) + (Quality, medium), (Interior, excellent) + (Quality, high), (Capacity, 4) & (Interior, good) + (Quality, high), (Capacity, 2) & (Interior, good) + (Quality, high), (Capacity, 4) & (Noise, low) + (Quality, high), (Capacity, 2) & (Noise, low) + (Quality, high), (Interior, good) & (Noise, low) -+ (Quality, high), (Interior, good) & (Vibration, low) + (Quality, high).

4. All Rules

Yet another knowledge acquisition option of LERS is called All Rules. The name is justified by the way LERS works when this option is invoked: All rules that can be induced from the decision table are induced, all in their simplest form. The algorithm takes each example from the decision table and induces all potential rules (in their simplest form) that will cover the example and not cover any example from other concepts. Duplicate rules are deleted. For Table IV, this option will induce the following rules:

ROUGH SETS

193

(Capacity, 4) & (Interior, fair) + (Quality, low), (Capacity, 5) & (Interior, good) + (Quality, low), (Capacity, 5) & (Noise, medium) + (Quality, low), (Capacity, 4) & (Noise, medium) + (Quality, low), (Capacity, 5) & (Vibration, medium) --+ (Quality, low), (Interior, good) & (Noise, medium) + (Quality, low), (Interior, fair) & (Vibration, medium) + (Quality, low), (Interior, good) & (Vibration, medium) + (Quality, low), (Noise, medium) & (Vibration, medium) -+ (Quality, low), (Capacity, 2) & (Interior, fair) + (Quality, medium), (Capacity, 5) & (Noise, low) + (Quality, medium), (Capacity, 2) & (Noise, medium) + (Quality, medium), (Capacity, 5) & (Vibration, low) + (Quality, medium), (Interior, fair) & (Noise, low) + (Quality, medium), (Interior, fair) & (Vibration, low) + (Quality, medium), (Noise, medium) & (Vibration, low) + (Quality, medium), (Interior, excellent) + (Quality, high), (Capacity, 4) & (Interior, good) + (Quality, high), (Capacity, 2) & (Interior, good) + (Quality, high), (Capacity, 4) & (Noise, low) + (Quality, high), (Capacity, 2) & (Noise, low) + (Quality, high), (Capacity, 4) & (Vibration, low) --+ (Quality, high), (Interior, good) & (Noise, low) + (Quality, high), (Interior, good) & (Vibration, low) + (Quality, high), (Noise, low) & (Vibration, medium) -+ (Quality, high).

C. Real- World Applications of LERS The LERS approach to rule induction was used at NASA Johnson Space Center for developing an expert system that may be used in the medical area on board the space station Freedom. Also, LERS was used for rule induction useful for enforcing Sections 311, 312, and 313 of Title 111, the Emergency Planning and Community Right to Know, as a part of the project funded by the U.S. Environmental Protection Agency. Another application of LERS was medicine, the traditional area of application for machine learning. First, the system was used for the choice of warming devices for patients recovering after surgeries (Budihardjo et al., 1991). Second, the LERS system induced rules to assess preterm labor risk for pregnant women (Grzymala-Busse and Woolery, 1994). The error rate of existing diagnostic methods is high (62-83%), since the risk is very difficult

194


to diagnose. The system LERS induced rules from training data (half of a file) and then classified unseen cases (the other half of a file) using these rules for three files. The resulting classification error for unseen cases was much smaller (10-32%). The LERS system was also successful in rule induction in such diverse areas as linguistics (Grzymala-Busse and Than, 1993) and global warming (Gunn and Grzymala-Busse, 1994).

IV. CONCLUSIONS Rough set theory offers an alternative approach to handling imperfect data. The closest other known approach is evidence theory. The main difference between rough set theory and evidence theory is that the former is objective (lower and upper approximations of the set are computed directly from input data), while the latter is subjective (basic probability assignment is given by an expert). Though the number of papers or reports about rough sets is close to a thousand, and the scope of rough set theory is dynamically expanding, there is room for much more research, as it is clear from activities in workshops devoted to rough sets (Ziarko, 1994; Lin, 1995). Logical fundamentals of rough set theory (close to modal logics), further developments of rough set theory itself, and real-world applications, to name a few, are being explored extensively.

REFERENCES Budihardjo, A., Grzymala-Busse, J. W., and Woolery, L. (1991). In “Proc., IV Int. Conf. on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems” (Koloa, Kauai, Hawaii), p. 735. Chan, C.-C., and Grzymala-Busse, J. W. (1989). In “Proc. of the ISMIS-89, IV International Symposium on Methodologies for Intelligent Systems” (Charlotte, NC), p. 281, North Holland, Amsterdam, New York, Oxford, Tokyo. Dempster, A. P. (1967). Ann. Math. Stat. 38, 325. Dempster, A. P. (1968). J . Roy. Stat. SOC.,Ser. B 30,205. Dubois, D., and Prade, H. (1992). In “Intelligent Decision Support. Handbook of Applications and Advances of the Rough Set Theory” (R.Slowinski, ed.), p. 203, Kluwer, Dordrecht, Boston, London. Grzymala-Busse, D. M., and Grzymala-Busse, J. W. (1994). In “Proc. of the XIV Int. Avignon Conf.” (Paris), p. 183. Grzymala-Busse, J. W. (1987). “Rough-Set and Dempster-Shah Approaches to Knowledge Acquisition Under Uncertainty-A Comparison.” Manuscript.

ROUGH SETS

195

Grzymala-Busse, J. W. (1988). J. Intelligent & Robotic Systems 1, 3. Grzymala-Busse, J. W. (1991). Managing Uncertainty in Expert Systems.” Kluwer, Dordrecht, Boston, London. Grzymala-Busse, J. W. (1992). In “Intelligent Decision Support. Handbook of Applications and Advances of the Rough Set Theory” (R. Slowinski, ed.), p. 3, Kluwer, Dordrecht, Boston, London. Grzymala-Busse, J. W., and Than, S. (1993). Behav. Res. Meth., Instr., & Comput. 25, 318. Grzymala-Busse, J. W., and Werbrouc, P. (1991). “On the Best Search Method in the LEM and LEM2 Algorithms.” Department of Computer Science, University of Kansas, TR-91-16. Grzymala-Busse, J. W., and Woolery, L. (1994). J. Am. Med. Inform. Asso. 1,439 Gunn, J. D., and Grzymala-Busse, J. W. (1994). Human Ecol. 22, 59. Lin, T. Y.,ed. (1995). “Proc. of RSSC‘95 Workshop” (San Jose, CA), Springer-Verlag, Berlin, New York. Pawlak, Z. (1982). Znt. J . Comput. Inform. Sci. 11, 341. Pawlak, Z. (1984). Int. 3. Man-Machine Stud. 20,469. Pawlak, Z. (1991). “Rough Sets. Theoretical Aspects of Reasoning about Data.” Kluwer, Dordrecht, Boston, London. Pawlak, Z., Grzymala-Busse, J. W., Slowinski, R., Ziarko, W. (1995). “Rough Sets.” Accepted for publication in Commun. A C M . Pawlak, Z., Slowinski, K., and Slowinski, R. (1986). Int. J. Man-Machine Stud. 24,413. Shafer, G . (1976). “A Mathematical Theory of Evidence.” Princeton University Press, Princeton, NJ. Skowron, A., and Grzymala-Busse, J. W. (1994). In “Advances in the Dempster Shafer Theory of Evidence” (R. R. Yaeger, M. Fedrizzi, and J. Kacprzyk, eds.), p. 193, John Wiley, New York. Slowinski, K., and Slowinski, R. (1990). Int. J. Man-Machine Stud. 32,693. Slowinski, R. (ed.), (1992). “Intelligent Decision Support. Handbook of Applications and Advances of the Rough Set Theory.” Kluwer, Dordrecht, Boston, London. Slowinski, R., and Stefanowski, J. (1992). In “Intelligent Decision Support. Handbook of Applications and Advances of the Rough Set Theory” (R. Slowinski, ed.), p. 445, Kluwer, Dordrecht, Boston, London. Szladow, A., and Ziarko, W. (1993).A1 Expert 7,36. Ziarko, W. (1992). In “Intelligent Decision Support. Handbook of Applications and Advances of the Rough Set Theory” (R. Slowinski, ed.), p. 61, Kluwer, Dordrecht, Boston, London. Ziarko, W., ed. (1994). “Rough Sets, Fuzzy Sets and Knowledge Discovery. Proc. of RSKD’94 Workshop” (Banff, Canada), Springer-Verlag, Berlin, New York.



Theoretical Concepts of Electron Holography FRANK KAHL AND HARALD ROSE Institute of Applied Physics, Technical University of Darmstadt. Darmstadt. Germany

. . . . . . . . . . . . A. Image Intensity and Contrast Transfer . . . . . . . . . B. Low-Dose Imaging . . . . . . . . . . . . . . .

I. Introduction

.

. .

. .

. . . . .

.

.

.

11. The Scattering Amplitude . . . . . . . . 111. Image FormationinTransmission Electron Microscopy

. .

. .

.

. . . . . . . . . . . . . . . . . . . . .

197 200 203 203 207 213 213

.

218 221 221 228 232 232 240 244 246 252

IV. Off-Axis Holography in the Transmission Electron Microscope . . . A. Formation of the Off-Axis Hologram . . . . . . . . . . B. Spectral Signal-to-Noise Ratio in Off-Axis Transmission Electron Microscope Holography . . . . . . . . . . . . . . . . . . . . V. Image Formation in Scanning Transmission Electron Microscopy . . . . A. Bright-Field Imaging . . . . . . . . . . . . . . . . . B. Spectral Signal-to-Noise Ratio . . . . . . . . . . . . . . VI. Off-Axis Scanning Transmission Electron Microscope Holography . . . A. Basic Principles . . . . . . . . . . . . . . . . . . B. Formation of the Scanning Transmission Electron Microscopy Hologram C. Fourier Transform of the Hologram . . . . . . . . . . . . D. Spectral Signal-to-Noise Ratio . . . . . . . . . . . . . . VII. Conclusion . . . . . . . . . . . . . . . . . . . . . Appendix: Fresnel Diffraction in Off-Axis Transmission Electron Microscope Holography . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

.

. . .

. .

.

. .

. .

253 256

I. INTRODUCTION

Electron microscopy is well suited for obtaining structural information on an atomic scale. The maximum information that can be obtained by electron microscopy is the complete knowledge of the elastic and all inelastic complex scattering amplitudes of the specimen. The complete reconstruction of the amplitude and the phase of each scattering amplitude is possible only if the information about these quantities is contained in the recorded image. Since only the absolute value of the electron wave function can be measured, the phase information of a single wave or a package of incoherent waves is irretrievably lost in the image plane of a conventional transmission electron microscope (TEM) or the detector plane of a scanning transmission electron microscope (STEM). Holography preserves the phase by superposing a coherent reference wave onto the elastically scattered wave (Gabor, 1949, 1951; Hawkes and Kasper, 197

Copyright 0 1995 by Academic Press, Inc.

All rights ofreproduction in any form reserved.

198

F. KAHL AND H.ROSE

1994). As a result, information about both the amplitude and the phase of the scattered wave is contained in the interference pattern. Under certain conditions the phase and the amplitude of the scattered wave can be reconstructed from such interference patterns. Since electrons are fermions, two electrons cannot occupy the same quantum state (Sakurai, 1985). Therefore, two different electrons cannot interfere with each other. Owing to this behavior, the reference wave must be a coherent partial wave of the wave package which describes a particular electron. All holographic techniques are restricted to the analysis of elastic scattering processes because the reference wave can only interfere with the elastically scattered part of the wave. Hence we disregard in the following considerations all inelastic scattering processes. In-line holography is realized in conventional bright-field imaging in the TEM and the STEM (Hanssen, 1971,1982; Hawkes and Kasper, 1994) in the case of coherent illumination. The coherent part of the wave at the exit plane, located directly beneath the specimen, can be written as a superposition of the undisturbed plane wave t,bo and the elastically scattered wave rc/,. Both of them pass through the imaging system of the TEM and interfere with each other at the image plane, where they form a Fraunhofer hologram. The current density jiat the image plane has the form

ji

I $ I=~ l$o12 + 2 R ~ { $ ~ Z+Il$s12,

(1)

q0

where the bar indicates the conjugate complex value. Here and $s describe the unscattered part and the scattered part of the wave, respectively, at the recording plane; k = 27t/l, l, and m, are the wave number, the wavelength, and the mass of the electron, respectively. In Fourier space we obtain

+ G$J + F T I $ ~ I ~ .

F T ( ~ , ) FTI$,I~+ F T { & , ~

(2)

The first term on the right-hand side of this equation represents the zero peak, the middle term represents the twin images and the third term is the quadratic or “spurious” term. The two twin images contain the elastic scattering amplitude linearly, while the spurious term comprises the quadratic terms of the elastic scattering amplitude and of the inelastic scattering amplitudes. The zero peak is produced by the unscattered part of the wave. The linear twin image can be used to reconstruct the elastic scattering amplitude. Unfortunately, this is possible only if the twin images can be separated in Fourier space from the zero peak and from the quadratic terms, which are called spurious terms in accordance with Gabor (1949). The main disadvantage of all in-line holographic procedures is the overlap of the twin images and the quadratic term (Fig. 1). As a consequence, these terms cannot be separated from each other. However, for a weak phase object, the quadratic term becomes negligibly small owing to the generalized optic theorem

199

THEORETICAL CONCEPTS OF ELECTRON HOLOGRAPHY

peak

‘quadratic

terms

FIGURE1. Locations of the central peak, the twin images, and the quadratic term in the Fourier image of an in-line hologram.

Tt;I:o;;

k

first twin image

quadratic term

FIGURE 2. Locations of the central peak, the twin images, and the quadratic term in the Fourier image of an off-axis hologram.

(Glauber and Schomaker, 1953; Glauber, 1958). In this case the elastic scattering amplitude can be reconstructed, as demonstrated by Gabor (1949). To overcome the disadvantages of in-line holography, Leith and Upatnieks (1962) proposed off-axis holography. Subsequently this method was successfully employed in electron holography by H. Lichte (1991a) and A. Tonomura (1986, 1993). Off-axis holography uses a tilted reference wave which separates the twin images from each other and from the quadratic term, as shown in Fig. 2. Owing to the successful application of off-axis holography in the TEM, T. Leuthner and H. Lichte (1989) suggested that this technique could also be

200

F. KAHL AND H.ROSE

employed in the STEM. Unfortunately, they assumed an unrealistic point probe, thus neglecting the aberrations and the finite size of the aperture. However, at atomic resolution these effects must be taken into account. In the following sections we first outline the theory of image formation in TEM and of TEM holography, as realized by Lichte and Tonomura. In the second step we briefly restate the theory of bright-field imaging in STEM and subsequently develop the theory of off-axis holography in the STEM. Finally, we compare the performance of TEM off-axis holography with that of the STEM and with that of STEM in-line holography. AMPLITUDE 11. THESCATTERING For determining the intensity distribution of a hologram, it is sufficient to know the electron wave far from the object. Neglecting inelastic scattering processes, the wave function adopts the asymptotic form (Sakurai, 1985) eikr

v%r) x

$as

+ f(ki, k,)T

= eikir

(3)

at a distance r which is large compared with the diameter Do of the object. The complex elastic scattering amplitude f(ki, k,) depends on the wave vectors ki and k, of the incident and the scattered electron, respectively (see Fig. 3). Accordingly, the amplitude of the scattered spherical wave is a function of both the direction of incidence and the direction of scattering. In the incident plane wave

FIGURE3. Illustration of the asymptotic behavior of the elastically scattered electron wave (3).


optic ax is

I I

3

20 1

,, X

/

/object

plane

t FIGURE4. Definition of the angles determining the propagation of the electron in front of and behind the specimen.

case of elastic scattering the absolute value of the wave vector is preserved: We assume that the specimen is located in a plane perpendicular to the optic axis, as shown in Fig. 4.The optic axis is chosen as the z coordinate of the x, y, z coordinate system. For electron energies larger than about 100 keV, most electrons are deflected by small angles so that backscattering can be neglected. In this case the small-angle approximation

+ kO, x ke, + ke,

ki % ke,

k,

can be applied. The angles 0,8,

+ sin 4iey), 8 = B(cos 4 f e , + sin 4,eJ

0 = O(cos diex

4i,and 4,

(5)

(6)

are defined in Fig. 4. For the

202

F. KAHL A N D H. ROSE

following considerations we assume a tilted incident plane wave of the form ,,$(r) = e i k , i = e i k z e i k p e . (7) Behind the object plane z = zo, the incident plane wave is modulated due to the interaction with the specimen. For a thin object this modulation can be described by a generalized complex transmission function T(@,P O ) ? (8) which generally depends on both the angle of incidence 0 and the position vector po = x,e, + y,e, at the object plane. For the wave function just behind the object plane we obtain $(po, 0 )= T(@,p o ) e i k p e e e i k z o .

(9) It is advantageous to separate this wave function into an undisturbed plane wave and a wave resulting from the interaction of the incident wave with the specimen: @) = e i k p o e e i k z o (T(@, p,) - l } e i k p ~ e e i k z ~ . (10)

+

The term T - 1 is zero outside the area of the specimen with the diameter Do. Hence at a distance d = z - z, which fulfills the condition

the Fraunhofer approximation (Hawkes and Kasper, 1994)

can be applied for the partial wave containing T - 1 . Here p denotes the position in the plane z. Since the unscattered wave extends over the entire object plane, the Fraunhofer condition (11) is not satisfied. Therefore, the Fresnel approximation eik(p- p J 2 / 2 d

$(P? z ) = eikdI $ ( P o 9 z o )

iAd

d2Po

(13)

must be used to propagate the undisturbed plane wave to the plane z. As a result we obtain eikz,eikd(l

$(,,, z ) = e i k r , e i k d ( l - 8 2 / 2 ) e i k e p

+

d

+e2/2)

F(@,

where

{ T(@,Po) - l}e-ik(e-e’Pod 2P o

(14)


203

represents the scattering amplitude and e Z Pj

the angular position at the observation plane z. The first term on the righthand side of (14) is the small-angle approximation of the undisturbed wave exp(ikir) at this plane. Since the factor (17)

d is the small-angle approximation of the spherical wave eikr

-

eik

Jdz+pt

T - J d W ’ it follows from (3) that the function F ( 0 , 0) represents the small-angle approximation of the elastic scattering amplitude f (ki, kf). The transmission function of a very thin phase object does not depend on the angle of incidence 0. In this case the scattering amplitude ~ ( 0 e) ,= F ( e - 0 )

(19)

is a function of the scattering angle 8 - 0. It should be noted that for T(O, p,) = T(p,), the approximation (15) coincides with the Glauber highenergy approximation (Moliere, 1947; Glauber, 1958).

111. IMAGE FORMATION IN TRANSMISSION ELECTRON MICROSCOPY

A . Image Intensity and Contrast Transfer

Within the frame of validity of Gaussian dioptrics, the propagation of an electron wave from the plane z, to the plane z is given by the Fresnel transformation (13), provided that the space d = z - z, between the two planes is free of electromagnetic fields. In this paraxial approximation the action of a thin lens located at the plane zl is described by the lens transmission function (Goodman, 1968)

T ( ~ ,=) e-ik(pf/2f),

(20)

where f denotes the focal length of the lens. To simplify the following considerations we assume a pointlike field emission gun. This ideal point source furnishes spatially coherent illumination. Moreover, we consider only paral-

204

F. KAHL AND H.ROSE

lel axial illumination: *.

=

@.

(21)

In this case the wave function +e at the exit plane directly behind the specimen has the form

The specimen plane is located at a distance g = zl - z, in front of the lens plane zl. The objective aperture is placed in the diffraction plane z, which coincides with the back focal plane of the objective lens, as shown in Fig. 5. The distance between this plane and the image plane is denoted by 1. The action of the aperture can be described by an aperture function A = A(p,). point source -+- - - - - - - - - - - - condenser

specimen

- - - - - _ - - _ - _source plane

-B

incident plane wave

-+ I---A f

gauss I an object p l a ne

T

9

objective lens-

1

lens plane

----

I

aperture -+ ---

diffraction plane -back focal plane

l

tz optic a x i s FIGURE 5. Image formation in the TEM.


205

Considering the effect of this aperture and that of the lens (20) the wave function at the image plane zi can be written as a sequence of three Fresnel propagations between the planes z, + zl -,z, --f z i : +i(pi, z i )

= eik(zi-zo)

JJ/ T(o,Po)

eWpt- pd2/28

eik(~.-~t)2/2/ e-ik(P:/2.f)

ilzg

ilf

A(PJ

The integration over pI can be performed analytically. The magnification of the image is defined as

where n denotes the number of intermediate images located between the object plane z, and the final image plane zi. For mathematical simplicity it is advantageous to replace the position vector pa by the aperture angle

The second relation holds true in the case (MI >> 1. For the arrangement shown in Fig. 5 we have n = 0. Taking further into account the thin-lens equation 1

1

1

we obtain

This result shows that the three Fresnel transforms of the expression (23) are equivalent to a sequence of two Fourier transforms. In the following we assume a circular aperture with an aperture function

A(e) = A ( O )=

1:

0:

for 0 Ie,, else,

206


where 0, denotes the limiting aperture angle. By inserting the relation (10) into the integrand of (27) and considering the expression (15), the wave function at the image plane (27) takes the well-known form (Rose, 1977)

Unfortunately, the assumption that the source is monochromatic and the lens system ideal does not yield a realistic description of the image intensity. To be more accurate, we must consider the energy spread of the source, the defocus Af of the specimen, and the aberrations of the objective lens. Assuming isoplanatic conditions, we can neglect all off-axial aberrations. The defocus Af

= Z,

- Z,

(31)

of the objective lens is positive if the specimen plane z , is located between the lens and the Gaussian object plane 2,. Owing to the energy spread of the source, the actual energy E of an electron differs by the amount AE=E-Eo

(32)

from the mean energy E , of the beam. The influence of the defocus, of the energy deviation, and of the spherical aberration can be taken into account by replacing the aperture function A ( 8 ) by the generalized complex aperture function

X(e, AE) = A(e)e-iy(e,AE),

(33)

where (34) is the phase shift produced by these perturbations (Schemer, 1949; Goodman, 1968; Hawkes and Kasper, 1994); C, and C, denote the constants of the spherical and the axial chromatic aberration, respectively. Substituting (33) for A(6) in relation (30), the wave function at the image plane zi is found as

The corresponding current density is given by ji(&,AE) = jol+i(g,AE)l 2

THEORETiCAL CONCEPTS OF ELECTRON HOLOGRAPHY

207

Here j , denotes the constant current density at the object plane. The actual current density is the average over all possible energy deviations BE of the electrons:

(k)

ji

-

=

1 :(

)

j i -, AE p(AE)dAE,

(37)

where p(AE) is the normalized energy distribution of the source. B. Low-Dose Imaging

In the case of low-dose imaging, the noise resulting from the CCD camera or the photographic plate is negligibly small compared with the quantum noise. To account for the effect of quantum noise, we subdivide the image into N x N square pixels of equal area (Ma,,)’ = (Ma)’, where a,, = a is the lateral length of the corresponding object elements. The signal N,, recorded by the pixel (I, m), I, m = 1, 2, ..., N , is assumed to be proportional to the number of electrons that have hit the pixel area during the illumination time t. 1 . Average Image Intensity

The mean value (N,,) of the signal N,, is defined as e

e

B(p,, - p ) j i ( p ) d’p.

(38)

The proportionality factor 4 represents the quantum efticiency. The vector Mp,, points to the center of the pixel (I, m) at the image plane. The box function B(p) is defined as

(0:

else.

For reasons of simplicity, we refer the image coordinates back to the object plane by means of the relation p = - Pi

M’

Employing relation (38), the mean value of the discrete Fourier transform

208

F. KAHL AND H. ROSE

of the digitized image intensity is obtained as

With the aid of the convolution theorem we derive the expression

provided that the number of pixels is sufficiently large. The functions N and J are given by N(n) = 4

sin(kal2,/2) sin(kaRJ2) kaR, kaR, '

's

(44) (45)

J(Q) = - j,(p)e-ik"P d2p,

A2

respectively. The range 0 IIQl/A I Q,,,/L of the transferred spatial frequencies is limited by the aperture at the diffraction plane. If the size a of the pixels satisfies the sampling condition

we obtain (r"((n)) = M 2

TJ(n)N(Q) e

- M Z"N(C2) s s j i ( p , AE)p(AE)ePik"Pd2p dAE

a2 e

(47)

for the significant region 0 I R I R,,,. This result is in agreement with the sampling theorem proposed by Whittaker (1915) and Shannon (1949). Considering the expression (36) for j i ( p , A), we can perform the integration over p analytically:

i

i (I(Q))= FjoN(S2) d(Q) + lF(O, n)A(R)

e

1-

- - F(0,

a

-Cl)A(R)

s

s

e-iYcn*AE'p(AE) dAE

eiY(R,m)p(AE) dAE


209

Here d(Q) is the two-dimensional delta function which accounts for the central peak, the next two terms represent the twin images, while the last integral forms the quadratic term. Although the elastic scattering amplitude F is separated from the transfer characteristic of the microscope in the expressions for the two twin images, a reconstruction of the scattering amplitude is not possible for an arbitrary specimen because the twin images overlap with the quadratic term and with each other. In the special case of a weak phase object, the quadratic term is negligibly small. The scattering amplitude F of a weak phase object (19) depends only on the scattering angle 8 - 0 and satisfies the Friedel law:

F(e - 0 )= F(O - e).

(49)

In this case the second and the third terms in expression (48) can be combined into a single term. As a result we obtain the weak phase-object approximation

where

9(n)= 9 ( R ) = - A @ )

s

sin y(R, AE)p(AE)dAE

(51)

is the so-called phase-contrast transfer function (PCTF). To incorporate the effect of the energy spread, a Gaussian energy distribution is assumed

p(AE) =

~

e-( 1 / 2 ) ( A E b d 2 .

(52)

For this distribution the integral in expression (51) can be evaluated analytically, yielding

z(Q) = - A ( Q ) ~ - ( ~ / ~ ) [ ( ~ U E / ~ E Osin ) C yo(Q), ,~~R~

(53)

yo(Q) = y(Q, AE = 0).

(54)

with

If we employ the normalized units listed in Table I, the results are valid for

210

F. KAHL AND H. R O S E

TABLE I NOTATION OF VARIABLES AND IMAGING PARAMETERS IN CONVENTIONAL AND I N NORMALIZED FORM, RESPECTIVELY

EM Specific

Normalized

Af 6, @d

A := Af/AN 9, := e,pN 9, := 0 , / 6 ,

a

B := a/dN

Biprism angle

6

8 := spN

Detector coordinate Fourier coordinate Spatial frequency

0

9 := Q/BN o := Q/6, o := ( Q / l ) d N

Parameter/Function Defocus Aperture angle Detector angle Pixel size

n

Normalization Factor AN=&

6,

=s *

dN =

a

AN

Source parameters

dN

any microscope. In particular, the PCTF takes the well-known form

P(w) =

-A(w)e-(1/2)&2m4

sin Y O ( O ) ,

(55)

where It

y0(w) = -(w" - 2Aw2).

2

For a weak phase object and parallel illumination, the maximum transferred spatial frequency (Born and Wolf, 1975) is Qmax

A

-

$0

(57)

A.

The function sin yo oscillates rapidly beyond the first zero w1 = ,/% of its argument (56). Since a reconstruction of the scattering amplitude becomes very difficult for spatial Frequencies higher than wl, we take w1 as the maximum normalized aperture angle so.If the microscope operates at the Scherzer focus A = 1.21 (Scherzer, 1949), the first zero of yo occurs at w1 = 1.56. The maximum normalized pixel size CZ is therefore given by d = 1/(2w1)= 0.32. For a specific microscope with the imaging parameters C, = 1.4 mm,

A

C, = 1.2 mm, =

1.21,

E , = 100 keV,

so = w1 = 1.56,

a, = 0.5 eV,

a = 0.32,

(58)


1.00

21 1

----_

0.75

0.50 0.25

0 1

I 1 IPV'

VV

-0.25

FIGURE6. Effective PCTF of a TEM with imaging parameters listed in (58).

is plotted in Fig. 6 for the section coy = 0. The scattering amplitude of weak phase objects can be reconstructed from a single micrograph up to the information limit, apart from the transfer gaps centered at the zeros of the PCTF. The information limit is defined as the maximum spatial frequency coI that can be distinguished from the noise. This limit results from the incoherent aberrations which determine the envelope of the PCTF. Using an amorphous carbon foil, the PCTF can be derived from the diffractogram of the micrograph of an amorphous carbon foil (Thon, 1971). Unfortunately, the rapidly decreasing distance between the zeros of the PCTF causes an increasing density of the transfer gaps. However, by using defocus series, reconstruction of the scattering amplitude over the whole range up to the information limit is possible (Schiske, 1968; Coene et al., 1992; Saxton, 1994). It should be noted that the restriction to weak phase objects is necessary only to suppress in the diffractogram the overlapping quadratic term. Offaxis holography separates the twin images from each other and from the quadratic term. In this case the restriction to weak phase objects is no longer mandatory.

2. Spectral Signal-to-Noise Ratio In order to obtain a statistically significant signal, a certain number of scattered electrons per object element must be collected. The signal-to-noise ratio (SNR) of the image depends on the dose

212

F. KAHL AND H. ROSE

which represents the number of electrons incident on the specimen area during the illumination time t. The spectral SNR (Berger, 1993; Berger and Kohl, 1993) is defined as

If K quantities Ni E lR are statistically independent, the variance of any linear combination

is given by K

var{Nge,}=

(ail’var{Ni}. i=l

Since the signals of the individual pixels are statistically independent, we obtain for the variance of the Fourier-transformed image (41) the expression 1

var{r((n)} = -

A4

N-1

C

var{Nl,}.

t,nr=o

(64)

For moderate doses the number Nl, of electrons detected by the pixel (I, m)is approximately given by a Poisson distribution with var{Nlm)

=

0 therefore remains unscattered. An electrostatic biprism (Mollenstedt and Duker, 1956; Mollenstedt, 1992) is inserted at a distance lb = vl in front of the intermediate image plane. In the case v 0. Accordingly, the wave function for xb 2 0 contains the whole information about the object. The biprism tilts each of the two partial waves through an angle 6 toward the optic axis. Assuming a wire with diameter db for the biprism, the tilted wave fronts overlap at the intermediate image plane within a stripe of width 261, - db. This overlap region contains the off-axis hologram. To record a hologram of a specimen area L x L, the angle 6 and the position parameter v of the biprism must satisfy the condition

26Vl - db = 261, - db.

IMilL

Here Mi is the magnification at the intermediate image plane. If we replace the image distance 1 by means of the equation M.=

'

1

--

f'

condition (68)can be written as L I26fV - -.db lMil

(68)

214


point source + _ _ _ _ _ _ _ _ _ _ _ -

-- - - - - - - - - _ - source plane

condenser+ incident plane wave gaussian Object plane

lens plane -----

diffraction plane -back focal plane

_ _ _ _ _ _ _ - _ _ _ biprism plane

_ - _ - - - - - intermediate

image plane

FIGURE7. The formation of an off-axis hologram in the TEM.

To determine the condition for the spatial separation of the twin images and the quadratic term, we first calculate the current density in the hologram in the absence of any specimen. The angular tilt vector

6 = 6ex

(71)

of the biprism points in the x direction if we assume that the wire extends along the y axis. In this case the reference wave and the plane object wave form a simple hologram consisting of interference fringes in the y direction. In the limiting case v -+ 0, we find j(o)(pi) - jo leikdxi

Mi'

+ e-ikdxi

I2 = -4j0cos2(6kxi). Mi'


215

interference

detector plane

cur rent den sit y

'!j

Fourier t r a n s f o r m

FIGURE8. Principle of off-axis holography.

The Fourier transform of this expression, I

l(Q)=

's

A

j(0)(pi)e-i"npl d2pj

demonstrates that two sideband peaks are located symmetrically about the central peak (R,= 0, R, = 0) at the positions Ry= 0, R, = 26, and R, = -26, respectively, as shown in Fig. 8. When the specimen is present,

216

F. KAHL AND H. ROSE

the corresponding twin images (sidebands) are centered about the peaks of these sidebands. The sidebands do not overlap with the central term if the maximum angular radii R,,,,, and R,,,, of the central image and the sidebands, respectively, satisfy the condition

26 2 Rc,max + Rs,max-

(74)

We restrict our considerations to Fraunhofer holograms, which are formed in the vicinity of the image plane conjugate to the object. The effect of the biprism on the wave function can be taken into account by the transmission function

acting at the plane z b . Considering parallel coherent illumination (21), the wave function at the image plane zi is found as

X

X

Since the transmission function (75) is discontinuous at x b = - d b / 2 and xb = db/2, we must separate the integration over the biprism plane in two integrations, one over the half-plane x b > db/2,the other over the half-plane x b < -db/2. The wave at the half-plane x b < -db/2 is determined by the undisturbed plane reference wave. The transmission function T can therefore be replaced by unity for the integration over this half-plane. Moreover, we assume that the biprism is located at a short distance above the image plane zi.In this case we have v 38,

f

+ 0,

(153)

is fulfilled. For reliable reconstruction of the complex scattering amplitude, the contrast of the stripes in the image must be as high as possible. The magnitude of the contrast depends on the configuration of the detector. For example, a circular detector whose diameter is large compared with the periodicity of the current density at the detector plane largely cancels out the oscillations of the image intensity. To maximize the contrast of the stripes in the image, the detector must be subdivided into stripes which run parallel to the stripes of the current density. The width of each stripe must be one-half of the periodicity interval,

of the current density (150) at the detector plane. The sign of the stripe detector function,

must alternate with the number angular width of each stripe,

II

of the stripes, as shown in Fig. 16. The

represents half the diffraction angle of a line grid whose periodicity coincides with the distance d, = 2fd

(157)

238

F. KAHL AND H.ROSE s t r i D e detector

t s t r i o e detector

I

I I

I

I

I I

I

I

FIGURE16. Principle of the stripe detector. The periodicity of the stripes matches the periodicity of the current density. The sign of the detector function alternates with the number of stripes. Each stripe covers either a region of constructive interference or a region of destructive interference.


239

between the two probes at the object plane. For probe positions R , = nAR, the maxima of the current density at the detector plane match the stripes of the detector, which are weighted by 1. In this case a strong image signal of positive sign is obtained, as shown in Fig. 16a. Probe positions with R, = (n 1/2)AR yield a strongly negative image signal (Fig. 16b). In order to discuss the actual arrangement of the stripe detector, we assume a STEM with a voltage of100 kV, a limiting aperture angle 8, x 13 mrad, and a distance d, = 100 A between the probes. For these values we obtain from (156) an angle AOS x 1.85 x rad. Moreover, the brightfield area must be subdivided into

+

+

stripes. Hence if we use a CCD array with 512' pixels, each stripe of the detector should extend over three pixel rows. To satisfy this condition for a pixel size of 20 pm and a focal length f = 2 mm, we need a magnification

lMil=

3 x 140 x 20pm z 162 200s

(159)

Any enlargement of the probe distance d, results in smaller stripes, making the construction of the detector even more difficult. The maximum number of stripes is limited by the pixel number N , of the CCD array. Hence we obtain, for the maximally allowed probe distance,

For probe distances larger than a few hundred angstroms, the effect of the off-axial coma must be considered. The coma prevents the unambiguous reconstruction of the scattering amplitude. Therefore, the maximum distance d, between the two probes is limited by both the minimum size of the detector pixel and the coma. The probe distance must be chosen in such a way that (a) each interference stripe at the detector plane remains broader than the size of the pixel and (b) the coma does not exceed a distinct limit. To establish the appropriate location of the pivot point, we assume a microscope with f = 2 mm, 8, = 0,= 13 mrad, and a probe distance d6 = 100 A. Considering further the relation S = d6/(2f), we find from (153) that the pivot point must be located at a distance ( > 20.8 m beneath the objective aperture plane. It should be mentioned that in this case the pivot point is located outside the microscope.

240


B. Formation of the Scanning Transmission Electron Microscope Hologram

So far we have assumed an ideal point source yielding an illumination wave which propagates in the direction of the optic axis. However, any real source has a finite size. Electrons that are emitted from a nonaxial point of the source enter the biprism plane with an additional tilt angle S. Accordingly, the “iso-phase axis” of the two plane waves behind the biprism is tilted by this angle. The total tilt angle of the object wave is -6+u+B,

(161)

while the corresponding angle for the reference wave is given by (162)

+6+u+S.

These plane waves are transformed into two convergent waves by the objective lens. The center of the resulting probes at the object plane are displaced by the distance

=fS

(163)

from the centers of the spots formed by the axial plane wave. Simultaneously, the intersection point of the “iso-phase axis” with the aperture plane is shifted by ss from the nominal position -let, as shown in Fig. 17. Here s is the distance between the biprism plane and the objective aperture plane. Considering these displacements, we obtain for the wave function at a plane just behind the aperture the expression

+ eik(+6+a+ p)paeik(+8+a+

P)(ia-sp)

1.

( 164)

To incorporate the phase shift caused by the aberrations of the objective lens we must multiply the expression (164) by the phase term

XP, P,,

AE) = .(e, PA - r(e, AE)

( 166)

introduced by the objective lens consists of the familiar term y given in (34) and the term (Rose, 1989) K(e, p,) = ~ , 0 3 p , cos(a,

+ a,

- a,)

( 167)

which accounts for the effect of the third-order coma. The coma of magnetic


tilted -1 "iso-phase axis"

---

@

24 1

-

"iso-phase axis"

I I

._----

- - - - - -- -- - - - - - - -

- - - -- - - - - - - - - - - - I

I I I I

.. .. .. ...

.. ..

2nd scanning coil

'1

-c

-------__-

aperture plane

I?'SP

l a j

I;

FIGURE 17. Course of the iso-phase axis for producing an off-axis STEM hologram in the cases of an axial (-) and a tilted (-----) incident plane wave. The iso-phase axis coincides with the (virtual)ray which intersects the center of the biprism.

242

F. KAHL AND H. ROSE

lenses contains a radial coefficient and an azimuthal coefficient which accounts for the effect of the Larmor rotation. The angles ag and a, are the polar angles of the vectors 8 and pol respectively. In the case aK = 0, the axis of the coma figure points in the radial direction. In order to obtain the wave function at the detector plane, we must propagate each of the two partial waves (164) from the objective aperture plane to the detector plane by employing the procedure outlined in Section V.A. Since the diameter of the probes is in the order of a few angstroms, we can replace the vector p, in the phase term (167) of the coma by the positions pp = R - f S and pr = R f 6 of the centers of the specimen probe and the reference probe, respectively. Employing this approximation, we derive for the wave function at the detector plane the expression

+

where

for the object wave. Assuming a single point source and using relations (170), (171),(172), and (148), we obtain for the angular current density at the detector plane the relation


ei[~(O, -fa+ R+u)-K(B', - f 6 + R + a ) ]

e -ik(-fS+R+e)(O-e')

243

d28 d (173)

where I , is the current of the specimen probe at the object plane. In the absence of a specimen ( F = 0), the angular current density,

forms stripes which are distorted by the coma in the outer region of the detector. This distortion complicates significantly the reconstruction of the scattering amplitude from the hologram. It may therefore be necessary to limit the size of the specimen area to such an extent that the coma can be neglected. The maximum phase shift K,,, resulting from the coma is found from (167) to be K,,,

%

kcK@d,.

(175)

In order to estimate the influence of the coma we assume an objective aperture angle 6, = 13 mrad, a typical value (176)

IcKl

for the coma coefficient, a probe distance d , = 20 nm, and an acceleration voltage of 100 kV.The resulting maximum phase shift K,,,

%

0.08

(177)

is negligibly small. Since this phase shift increases only linearly with the probe distance d,, the influence of the coma can still be tolerated if the probe distance is increased to about 30 nm. In the following we consider only object fields whose diameters are smaller than 30 nm. In this case we can neglect the phase shift K in expression (173) and obtain

244

F. KAHL AND H. ROSE

For an extended polychromatic source, the current density per solid angle at the detector plane is given by jd(@,

R) =

ss

jd(@, R, a, E)p(AE)q(a)d 2 0 dAE.

(179)

Here q ( a ) and p ( E ) determine the spatial and the energy distribution of the effective source formed by the statistically independent constituent point sources. C . Fourier Transform of the Hologram

The considerations concerning the Fourier transform of the off-axis hologram are outlined in Section V.A. We again assume that the number of pixels N is sufficiently large and that condition (46) of the sampling theorem is satisfied. In this case the mean value of the Fourier transform of the image intensity,

s

I(R) = 2 jd(O,R)D(O) d 2 0 , qze recorded during the illumination time z = N 2 z , is found to be

The first term,

Z(n)= Q(Q)e-ikfan

(180)


245

denotes the amplitude of the central intensity, while

D(@)A(IO - fiJ)A(@)F(@ -

1

a, 0)P(0,fi)eikZS6edZO

(183) with li! = 25/f6 A f2 describes the amplitude of each of the two sidebands. The angular radius 8, of the central term and the radius 19,of each sideband term are given by

respectively. Hence if the imaging parameters satisfy the separation condition (153),

the sidebands are spatially separated and do not overlap with the central term. To reconstruct the scattering amplitude, we need to consider only the amplitude

+ 2fs)

=

?$&(a

+ 256)

of the first sideband, which is centered about the position

5 f

iz,= - 2 - 6

in the diffractogram. Expression (183) shows that the scattering amplitude can only be separated from the properties of the microscope for thin phase objects. Since in this case the scattering amplitude F(O - f2, 0 )= F(SZ) does not depend on the detector angle 0, the sideband term can be written as

--

Z,(Q)

=qzp

e

~

j

" e-ikf*fi{ Q (2 f 6 ) 6(a)

718,"

-2F(kli!)g(fi)).

2

A(@)D(@)eZikfge d2@

(188)

Here A ( ! @ - fiI)A(@)D(O)P(O,li!)ezikfgedZO (189)

246


is the sideband transfer function (SBTF). The phase factor exp( - ikf 6fi) need not to be corrected, for reasons outlined in Section 1V.A. It should be noted that the envelope function Q(f2 + 2s/f 6) is rotationally symmetric with respect to fi,, = -2s/f 6. To estimate the effect of this shift, we assume a 100-kV STEM with d, = 20 nm,

f = 2 mm,

s = 100 mm,

C, = 1.2 mm.

(190)

For these values the center of the envelope function is displaced by the angle S

S

GI, = 2-6 = - d

f

x 0.5 mrad

(191)

f 2 ,

from the origin fi = 0 of the diffractogram. Assuming an optimized threering detector (142) with a cutoff angle

Od + 0, = 20, x 26 mrad,

(192)

we find that the relative shift of the center of the damping envelope with respect to the radius of the sideband amounts to only 1.9%.Since this shift is very small, the effect of the envelope function is quite similar to that in conventional STEM. The factor P(0,fi)is identical with the term (122) for conventional STEM imaging. Therefore, the energy spread affects offaxis STEM hologram in the same way as it does for the phase-constrast image. D. Spectral Signal-to-Noise Ratio

The noise reduces the information which can be extracted from the STEM hologram. For reasons of simplicity we restrict our considerations to weak scatterers. In this case the number of electrons scattered out of the illumination cone is negligibly small. Hence the current through the bright-field area of the detector is approximately equal to the current 1, of the specimen probe. Without an object, the angular current density in the detector plane is given by ("'

=

2,4(@)&{ 747 1

+ Re[Q(2+.>]

- Im[Qk+o)]

sin[Zks(/e

-

f

""'1)

p(E)q(a)d E d 2 0

~ 0 ~ [ 2 k 6 ( f @-

?)I}.

y)] (193)


247

For a bright-field detector with D'(0)

=

1,

(194)

the contributions of the cosine and the sine term to the variance (133) cancel out as well as the terms that are linear in the scattering amplitude owing to the validity of the generalized optical theorem (Glauber and Schomaker, 1953; Rose, 1977). Neglecting the quadratic terms which account for the electrons scattered out of the bright-field region, we obtain for the variance of the sideband intensity of the diffractogram the expression

Using this expression, the spectral SNR of the sideband intensity for R # 0 is found to be

Comparison of this relation with the corresponding expressions (67), (93), and (138) suggests that the effective SBTF should be defined as follows:

This function can be used to compare the spectral SNR of off-axis holography in STEM with that in the TEM and with those of TEM and STEM bright-field imaging. Assuming a Gaussian function for both the energy distribution (52) and the radiation characteristic of the effective source (140), we obtain for the effective SBTF in normalized units (Table I) the expression

1 e-~eZ(10-m12-92)2ein(9x/A9,)

d29

(198)

Here Ass denotes the normalized stripe width (156) of a stripe detector for maximum stripe contrast. Apart from the factor exp (in

&)

248


stripe detector

annular detector

hologram detect o r -1

FIGURE18. Combination of a stripe detector and an annular detector forming an optimum detector for off-axis STEM holography.

the real part of the integrand in (198) coincides with the integrand of the effective PCTF (141) for standard STEM bright-field imaging. The strong oscillations caused by the phase term (199) cancel out the whole integral if an annular phase-contrast detector with only a few rings is used. To avoid this disastrous behavior we choose a detector configuration with a detector function D ( 8 ) = Ds($x)Da($)* (200) Such a detector consists of rings which are subdivided into stripes, as shown in Fig. 18. The factor D,($,),which matches the interference pattern of the intensity at the object plane in the absence of an object, is obtained from (155) by replacing 0,by $,ON. The factor Da(9) accounts for the annular subdivision of the detector. To elucidate the effect of the factor D,($,)on the effective SBTF, we have plotted in Fig. 19 the real part and the imaginary part of the product D,(9,)exp(in$,/A$,). The curves demonstrate that the real part is always positive, while the imaginary part has the form of a sawtooth function. Since all other factors of the integrand vary slowly over a region of many oscillations, the imaginary part of the integral in (198) is approximately zero. The mean value 2/n w 0.63 of the real part demonstrates that the stripe detector (155) counterbalances to a large extent the damping effect of the oscillating phase factor (199). Considering the effect of the stripe detector, we can substitute the factor 2/n for the product D,(9,)exp(in9,/A9,) in the integrand of the expression (198). The resulting approximation, e-zd:(m+2s// 1

%eff(9)

=

8)2

iJzn2$, 9,

jI

A( 3 - 0 I )A($)Da($) 1

e - i [ ~ ~ ( l m - 8 1 ) - ~ ~ ( 9 ) l e - ~ ~ z (mi218-

W 2d 28

(201)


FIGURE19. Real part (-)

and imaginary part (-----)

249

of the product Ds(9,)exp(in$,/A&).

for the effective SBTF is rotationally symmetric if we disregard the displacement -2&s/f of the envelope function Q. In this case the real part of 9s,eff coincides with the effective PCTF of the standard bright-field STEM imaging (139), apart from the factor 1/(& % 0.23. The optimum transfer function (201) will be obtained when the factor Do($)of the detector function (200) coincides with the detector function (142) for optimum phase contrast. The optimum hologram STEM detector therefore combines the features of both the stripe and the ring detector, as shown in Fig. 18. In this case the real part of the SBTF has the same shape as the PCTF (139),apart from the constant factor 0.23. To illustrate this behavior, we have plotted in Fig. 20 the real part of the SBTF and the PCTF for a 100-kV STEM with C, = 1.2 mm, C, = 1.4 mm, f = 2 mm, and a monoenergetic point source. For comparison, the corresponding curves for a realistic source with source parameters a, = 0.5 eV,

a, = 43 A,

1 IM,I = 80

(202)

are depicted in Fig. 21. To completely survey of the complex SBTF for a monoenergetic source, we have plotted in Fig. 22 the absolute value, the real part, and the imaginary part of the SBTF, and in Fig. 23 the corresponding phase. The SBTF for off-axis STEM holography and the PCTF of standard STEM bright-field imaging are less sensitive with respect to the energy spread of the source than the SBTF or the PCTF of the TEM. On the other hand,

250

F. KAHL AND H. ROSE

0

1

2

3

W

FIGURE 20. PCTF of the optimized three-ring detector (-) and the real part of the SBTF of the three-ring stripe detector (-----) as functions of the normalized spatial frequency w. In both cases a monoenergetic point source has been assumed.

0.301

0

1

2

3

w

FIGURE 21. PCTF of the optimized three-ring detector (-) and of the real part of the SBTF for the three-ring stripe detector (-----) for a gaussian source with an energy spread (variance) a, = 0.5 eV and a source radius (variance) us = 43 A.

the performance of the STEM is strongly affected by the finite extent of the effective source. Accordingly, coherent bright-field imaging in STEM can only be achieved with a field-emission gun. Another critical point in off-axis STEM holography concerns the extreme accuracy required for the stripes of


0.08

251

1

0.06

0.04

0.02

0 FIGURE 22. Absolute value (-), real part (---), and imaginary part (----) the SBTF in the case of a three-ring stripe detector and a monoenergetic point source.

of

1.6 1

FIGURE 23. Phase of the SBTF for the three-ring stripe detector.

the detector. To demonstrate this behavior, we have plotted in Fig. 24 the absolute value of the SBTF for a relative deviation of the stripe width of OX, O.l%, and 1%. The curves reveal that a relative accuracy of at least 0.1% is necessary in order to prevent an appreciable reduction of the SBTF. This accuracy should be attainable in practice without difficulties. Finally,

252

F. KAHL AND H. ROSE

0.08

1

0.06

0.01

0.02

0 0

1

2

3

w

FIGURE 24. Effect of an inaccurate stripe width on the SBTF in the case of a combined three-ring stripe detector and a monochromatic point source for a relative deviation of 0% (-), 0.1% (---), and 1% (----) of the stripes.

it should be noted that a shift of the interference pattern by the angle AOJ2 maximizes the imaginary part of the SBTF, while it suppresses the real part.

VII. CONCLUSION In the case of conventional bright-field imaging in TEM and STEM (in-line holography), the reconstruction of the scattering amplitude is restricted to weak phase objects. When no focus series is available, the reconstruction of the scattering amplitude up to the information limit is hampered in the TEM by the gaps of the PCTF. An important advantage of the STEM equipped with a configured brightfield detector is the possibility of extracting specific information about the object from a single exposure by subsequent combination of the signals from the detector elements in various ways. The PCTF obtained by an optimum three-ring detector has a triangular shape without any transfer gaps up to the maximum transferred spatial frequency. Due to this triangular shape, the speckles are largely suppressed in the bright-field image of the STEM. The PCTF of the STEM is less sensitive with respect to deviations of C, and Af than the PCTF of the TEM for parallel illumination. On the other hand, the


253

spatial coherence of the source must be higher for the STEM than for the TEM. Accordingly, a field-emission gun must be used in the STEM. Since only half of the elastically scattered electrons contribute to the modulation of the bright-field intensity, the spectral SNR is about three times lower than that in TEM employing parallel illumination. Off-axis holography in the TEM is not restricted to weak phase objects. However, an accurate knowledge of both the defocus Af and the coefficient C, is mandatory for reconstructing the complex scattering amplitude of arbitrary thin objects up to the information limit. Moreover, off-axis holography has an about three times lower spectral SNR than conventional TEM brightfield imaging. The optimum detector configuration for off-axis STEM holography is obtained by subdividing the three-ring detector for conventional phasecontrast imaging into stripes. The absolute value of the resulting SBTF has the same shape as the PCTF in the conventional STEM. Hence the resolution will be the same in both cases. Unfortunately, for a given dose the achievable maximum spectral SNR is about 10 times lower than that of the conventional TEM. Therefore, this technique seems to be applicable only for radiation-resistant objects. STEM holography is restricted to thin phase objects. Since this fundamental disadvantage cannot be overcome, the field of application of STEM holography is strongly limited. APPENDIX: FRESNEL DIFFRACTION IN OFF-AXIS TRANSMISSION ELECTRON MICROSCOPE HOLOGRAPHY Within the frame of validity of the Fresnel approximation, we obtain for the wave function (76) at the intermediate image plane

where the functions

[/-

$(+)(pi) = 9

IMilfAv

(6vl -

2(1 - v)

(204)

- xi)]

and

I Mi If J v

2(1

x)]

- v) + xi + 1 - v

d28

(205)

254

F. KAHL AND H.ROSE

with S ( X )

1 2

= -{1

+ (1 - i) [%(x) + i Y ( x ) ] }

are expressed in terms of the Fresnel integrals (Abramowitz, 1970),

Considering the asymptotic behavior

2

1 1 Y ( x ) x - - - cos 2 nx

,

nx

(3

xz)

(208)

of these functions for large arguments x >> 1, we derive the relations ei(n/4)

@(x) x 1- 7 ei(n/2)x2 J 2nx

for x >> 1 and lim 9 ( x ) = 0.

(209)

x + -m

To simplify the considerations, we analyze the behavior of @+) and $(-Iin the absence of a specimen (F = 0). The argument of .9in (204)is positive for db xi < 6vl - ___ 2(1 - v)’

and that of 9in (205)is positive for Xi>

-6vl+-

db

2(1 - v)’

Since the extension of the Fresnel fringes surrounding the edges xi = 6vl - [db/2(1 - v ) ] and xi = -6vl + [db/2(1- v ) ] shrinks with decreasing v, we restrict our considerations to v > 1, we derive from (218) and (209) the condition lMil v < (27rp(*)L&,,,)2-.

f2

In the following we require that 90% of the area of the hologram should not be affected by the edge fringes. This inner zone corresponds to the region defined by 0.95 2 p(*) 2 0.05. Assuming L = 200 A,

E,,,

f = 2 mm, we obtain

= 0.03,

IMil = 50,

E , = 100 keV,

256

F. KAHL AND H. ROSE

Equivalent considerations show that in this case $(+)(p(+))and $(-)(p(-)) are negligibly small in the region p(*) < - 0.05 outside the hologram. For this example, the wave function in the intermediate image plane can be approximated by (77), apart from the stripes -0.55L/IMil < x i < -0.45L/IMil and 0.45L/IMiI < x i < 0.55L/IMiI centered about the edges of the hologram. Assuming a typical diameter d, = 1 gm for the wire of the biprism, it follows from condition (70) that the biprism angle must be chosen as

REFERENCES Abramowitz, M., and Stegun, I. A. (1970). “Handbook of Mathematical Functions,” 9th ed., p. 295, Dover, New York. Batson, P. E. (1988). In “Treatise on Materials Science and Technology, Vol. 27,” p. 337. Academic Press, New York. Berger, A. (1993). Ph.D. thesis, TH Darmstadt, Germany. Berger, A., and Kohl, H. (1993). Optik 92 (4), 175. Born, M., and Wolf, E. (1975). “Principles of Optics,” 5th ed., p. 483. Pergamon Press, New York. Coene, W., Janssen, A,, de Beeck, O., and Van Dyck, D. (1992). Phys. Rev. Lett. %, 3743. Dinges, C., Kohl, H., and Rose, H. (1994). Ultramicroscopy 55,91. Eusemann, R., and Rose, H. (1982). Ultramicroscopy 9, 85. Gabor D. (1949). Proc. Roy. Soc. A 197,454. Gabor D. (1951). Proc. Phys. Soc. B64,449. Glauber, R. J. (1958). In “Lectures in Theoretical Physics, Boulder, Vol. 1,” p. 315. Interscience, New York. Glauber, R. J., and Schomaker, V. (1953). Phys. Rev. 89,667. Goodman, J. W. (1968). “Introduction to Fourier Optics,” McGraw-Hill, San Francisco. Hammel, M. (1992). Ph.D. thesis, TH Darmstadt, Germany. Hammel, M., and Rose, H. (1995). Submitted for publication. Hanssen, K.-J. (1971). In “Advances in Optics and Electron Microscopy, Vol. 4” (V. E. Cosslett, ed.), p. 1. Academic Press, New York. Hanssen, K.-J. (1982). In “Advances in Electronic and Electron Physics, Vol. 59,” p. 1. Academic Press, New York. Hanssen, K.-J., and Trepte, L. (1971). Optik 33, 166. Hawkes, P. W., and Kasper E. (1994). “Principles of Electron Optics, Vol. 3, Wave Optics.” Academic Press, New York. Helmholtz, H. V. (1886). Crelles J. 100,213. Kunath, W., Zemlin, F., and Weiss, K. (1985). Ultramicroscopy 16, 123. Leith, E. N., and Upatnieks, J. (1962). J . Opt. SOC.Am. 52, 1123. Leuthner, Th., and Lichte, H. (1989). Phys. Stat. Sol. (a) 116, 113. Lichte, H. (1991a). In “Advances in Optics and Electron Microscopy, Vol. 12” (T.Mulvey, ed.), p. 25. Academic Press, New York.


257

Lichte, H. (1991b). UItramicroscopy 38, 13. MoliBre, G. (1947). 2. Naturforschung. 2a, 133. Mollenstedt, G. (1992). In “Advances in Optics and Electron Microscopy, Vol. 12” (T. Mulvey, ed.), p. 1. Academic Press, New York. Mollenstedt, G., and Diiker, H. (1956). Z. Physik 145, 377. Reimer, L. (1984). “Transmission Electron Microscopy,” Springer Series in Optical Sciences. Springer-Verlag, Berlin. Rose, H. (1974). Optik 39,416. Rose, H. (1975). Optik, 42,217. Rose H. (1977). Ultramicroscopy 2,251. Rose, H. (1989). “Image Formation in Electron Microscopy,” Lecture notes, TH Darmstadt, Germany. Sakurai, J. J. (1985). “Modern Quantum Mechanics,” Addison-Wesley, New York. Saxton, W. 0.(1994). Ukramicroscopy 55, 171. Scheinfein, M., and Isaacson, M. (1986). J. Vac. Sci. Techno!.B4,326. Scherzer, 0.(1949). J. Appl. Phys. 20,20. Schiske, P., “Proc. 4th Eur. Conf. on Electron Microscopy” (Rome), Vol. 1 , p. 145. Shannon, C. E. (1949). Proc. IRE 37,lO. Thon, F. (1971). In “Electron Microscopy in Material Science,” p. 570. Academic Press, New York. Tonomura, A. (1986). In “Progress in Optics, Vol. XXIII” (E. Wolf, ed.), p. 183. Elsevier, New York. Tonomura A. (1993). “Electron Holography,” Springer Series in Optical Sciences, SpringerVerlag, Berlin. Whittaker, E. T. (1915). Proc. Roy. SOC. Edinburgh, Sect. A 35, 181. Zeitler, E., and Thomson, M. G. R. (1970). Optik 31,258.


.

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOL. 94

Quantum Neural Computing SUBHASH C. KAK Department of Electrical and Computer Engineering. Louisiana State University. Baton Rouge. Louisiana

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Mind-Body Problem B. Emergent Behavior . . . . . . . . . . . . . . . . . . . . . C. Indivisibility and Quantum Models . . . . . . . . . . . . . D . A Historical Note . . . . . . . . . . . E. Overview of the Chapter I1. Turing Test Revisited . . . . . . . . . . . . . A . TheTest . . . . . . . . . . . . . . . . . B. On Animal Intelligence . . . . . . . . . . . . . . . . . . . . . . . C. Gradations of Intelligence D. Animal and Machine Behavior . . . . . . . . . 111. Neural Network Models . . . . . . . . . . . . . A . TheBrain . . . . . . . . . . . . . . . . . B. Neurons and Synapses . . . . . . . . . . . . C . Feedback and Feedforward Networks . . . . . . . D . Iterative Transformations . . . . . . . . . . . IV. The Binding Problem and Learning . . . . . . . . . A . The Binding Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Learning . . . . . . . . . . . . . C. Chaotic Dynamics D . Attention, Awareness, Consciousness . . . . . . . . . . . . . . . . . . . E. Cytoskeletal Networks . . . . . . . . . . . F. Invariants and Wholeness V . On Indivisible Phenomena . . . . . . . . . . . . A . Uncertainty . . . . . . . . . . . . . . . . B. Complementarity . . . . . . . . . . . . . . C. What Is a Quantum Model? . . . . . . . . . . VI . On Quantum and Neural Computation . . . . . . . . A . Parallels with a Neural System . . . . . . . . . B. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . C. More on Information D . A Generalized Correspondence Principle . . . . . . VII . Structure and Information . . . . . . . . . . . . A . A Subneuron Field . . . . . . . . . . . . . B. Uncertainty Relations . . . . . . . . . . . . VIII . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 259

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

260 261 262 262 264 265 266 266 267 269 270 272 272 273 276 278 284 284 285 288 289 291 292 293 294 295 296 298 298 301 302 304 305 305 307 309 310

Copyright 8 1995 by Academic Press. Inc. All rights of reproduction in any form reserved.

260

SUBHASH C. KAK

I. INTRODUCTION

This chapter reviews the limitations of the standard computing paradigm and sketches the concept of quantum neural computing. Implications of this idea for the understanding of biological information processing and design of new kinds of computing machines are described. Arguments are presented in support of the thesis that brains are to be viewed as quantum systems, with their neural structures representing the classical measurement hardware. From a performance point of view, a quantum neural computer may be viewed as a collection of many conventional computers that are designed to solve different problems. A quantum neural computer is a single machine that reorganizes itself, in response to a stimulus, to perform a useful computation. Selectivity offered by such a reorganization appears to be at the basis of the gestalt style of biological information processing. Clearly, a quantum neural computer is more versatile than the conventional computing machine. Paradigms of science and technology draw on each other. Thus Newton’s conception of the universe was based on the clockworks of the day; thermodynamics followed the heat engines of the nineteenth century; and computers followed the development of the telegraph and telephone. From another point of view, modern computers are based on classical physics. Since classical physics has been superseded by quantum mechanics in the microworld and animal behavior is being seen in terms of information processing by neural networks, one might ask if a new paradigm of computing based on quantum mechanics and neural networks can be constructed. In recent years proposals have been made by Beniof (1982),Deutsch (1985, 1989), Feynman (1986), and others for the development of computers based on quantum mechanics. In these schemes the quantum mechanical basis states are the logic states of the computer and the computation is a unitary mapping of these states into each other. Hermitian Hamiltonians are specified that define the interactions. From another perspective, these computers let the computation of several problems proceed simultaneously, as in the evolution of a superimposition of states. By itself that is no better than several computers running in parallel. However, if one were to imagine a problem where the partially evolved computations of some of these superimposed states are used to find the solution, then it might be that such a machine offers improved speed in the sense of complexity theory. The idea of a quantum computer has not yet been shown to be practical (Landauer, 1991; Gramss, 1994). Furthermore, these proposals deal only with the question of the physics underlying basic computation; they do not consider the question of how a computation process leads to intelligence.

QUANTUM NEURAL COMPUTING

26 1

The proposal for a quantum neural computer is based on an explicit representation of the unity of the computation process; it defines computation as a behavior in relation to other systems. In other words, we introduce computation in relative terms, a perspective that appears to be appropriate for computation related to artificial intelligence (AI) problems. This shift in perspective is like the shift urged by Mach in his criticism of the notion of absolute space that had been assumed by Newton. According to Mach, space should be eliminated as an active cause in a system of mechanics; a particle moves not relative to space but rather to the center of all the other masses in the universe. Likewise, one says that intelligent computation can be defined only in terms of the computational behavior related to other such systems. A . The Mind-Body Problem

In order to be able to design machines that are equal to the capabilities of brains, it is essential to understand the nature of brain behavior. Our first hurdle is the mind-brain problem. The brain is defined by its physical and chemical properties, and it is assumed to function in ways that can be predicted by physics and chemistry. In contrast, the mind is an abstract entity that consists of sensations, beliefs, desires, and intentions. How is the physiochemical body related to a nonphysical mind? In identity theories, the mind is a part of the physical properties of the brain, as is true of the electrical recordings from the brain. Mental states then should be seen as brain states, and statements about cognition can be reduced to statements about behavior or a disposition to behave. However, identity theory does not explain how the activity in the brain assumes a unity which constitutes an awareness of the self. If stable brain states are identified as memories, as is done in many current models, can the problem of self be reduced to the problem of a bundling of these memories? Experiments with split-brain patients have shown that memory does not require consciousness. The converse, that consciousness does not require memory, has been known for a longer time. The split-brain subject appears to experience consciousness via the dominant cerebral hemisphere. By contrast, the minor cerebral hemisphere does not enable the subject to have conscious experience, although it subserves memory, nonverbal intelligence, and concepts of spatial relations. We know ourselves as unique indivisible minds, and not as a unique brain. In dualist theory, the mind has a reality independent of the body. Dualists cite introspection by which knowledge of things such as pleasure, pain, hap-

262

SUBHASH C. KAK

piness, emotions, and so on, is gained as proof of the existence of mind. The argument against dualism is that the recognition of the color green can be reduced to photons of a certain frequency striking the retina. Another objection is that a nonphysical entity cannot influence a material object without violating conservation laws. B. Emergent Behavior

Mind, with its concomitant intelligence, is often taken to be an emergent property of the complexity and the organization of the interconnections between the neurons. The concept of emergent property comes from chemistry, where the properties of water are taken to be “emerging” from the properties of hydrogen and oxygen. The properties of water could not originally be predicted from the properties of hydrogen and oxygen; it is assumed that in future, given the properties of atoms and the rules of their combinations, the properties of water would be completely explainable from the properties of its constituents. The notion of emergent behavior is a consequence of a reductionist approach to phenomena. Such an approach is a reasonable foundation for scientific theories. However, the workings of the mind appear to be so much different from the substratum of theories about the brain that doubts have been expressed about it being an emergent phenomenon. Neuroscience is also based on a reductionist agenda. It is believed that once the physiochemical basis of the activity and the interactions of the nerve cells are understood, mental events will be explained in terms of physiochemical events taking place in the nerve cells. But how awareness can emerge from principles of physics or chemistry is not obvious at this point. Neither does current theory explain how awareness could arise from complexity alone. The idea of emergent behavior does not rule out the need to add new laws at the higher level. Does the mind have unique properties that are not reducible to physical laws? In going to higher levels related to brain behavior, we confront the tyranny of complexity: The explosion of combinatorial possibilities may make it impossible to answer whether the higher-level behavior is reducible to that of lower levels. C. Indivisibility and Quantum Models

Reductionist approaches to brain function do not capture the richness of biological information processing. According to the complementarity interpretation of quantum theory, any indivisible phenomenon must be described


263

by a wavefunction. This is why “elementary” particles, which in turn have “subparticles”as constituents, are described by wavefunctions. One can even speak of a wavefunction for a macro object and indeed for the entire universe. If consciousness is taken to be indivisible, one has no choice but to model it in a quantum mechanical fashion. Evidence for the unity of self-awareness or consciousness is provided by many neuropsychological experiments. These include split-brain research as well experiments on dissociation of behavior from its awareness, as in prosopagnosia, amnesia, and blindsight (Weiskrantz, 1986). Schrodinger (1965), Penrose (1989), and other scientists have argued that, as a unity, consciousness should be described by a quantum mechanical type of wavefunction. No representation in terms of networking of classical objects, such as threshold neurons, can model a wavefunction. Therefore, current computing machines, which are based on mechanistic laws of information processing, are unlikely to lead to machines that would match human intelligence. It is sometimes argued that since quantum mechanics is not needed to describe neural processes, therefore it should not enter into any higher-level descriptions of cognitive processes or of consciousness. The noisy environment characterizing nerve impulse flow should drown out any quantum mechanical behavior. On the other hand, it has been experimentally determined (see Baylor et al., 1979, Schnapf and Baylor, 1987) that the retina does respond to single photons although the process that leads to-such response is not understood. This was determined by directing dim flashes of light into one eye of a subject sitting in total darkness. The subject perceived a flash when only seven photons were absorbed. Since a population of about 500 rods in the eye absorbed the photons in a random spatial pattern, there was no likelihood that any single rod had absorbed more than one photon. The response itself is a macroscopic nerve signal. Now consider social computing (Kak, 1988), that is, processing that takes place in a society of individuals without a conscious notion of a collective self. Current machines are not capable of matching such performance. Social computing is clearly less powerful than computing that is accompanied by awareness, so current computing models fall considerably short of the capabilities of biological systems. There also exist speculations regarding consciousnessin quantum physics. It has been argued that the collapse of the wavefunction occurs owing to its interaction with consciousness. Physicists such as Wigner have argued that science, as it stands now, is incomplete since it does not include consciousness. In a variant of the Schrodinger cat experiment, Wigner (1967) visualized two conscious agents, one inside the box and another outside. If the inside agent makes an observation that leads to the collapse of the wave-

264

SUBHASH C. KAK

function, then how is the linear superposition of the states for the outside observer to be viewed? Wigner argued that in such a situation, with a conscious observer as a part of the system, linear superposition must not apply. Clearly, these questions are a restatement of the ancient debate regarding the notion of free will in a deterministic or causal universe. Regardless of its origin, consciousness appears to bring entirely new, and paradoxical, aspects into the nature of physical reality. Thus, in Section V.C.l we describe a scenario dealing with quantum mechanics, due to John Archibald Wheeler, that admits the possibility of current actions influencing the past if commonsensical, mechanistic interpretations are sought. Wheeler has likened the physical universe to a gigantic information system whose unfolding depends on how we question it.

D. A Historical Note The ancient Vedic science of cognition, based primarily on consciousness examining itself, which has been traced in India to at least 2000 B.C. (Kak, 1994c), led to a rich conceptual structure for the mind. The individual was seen as a system, with the body, the energy field, mind, intellect, and emotion as different hierarchical levels. The mind itself was seen as a system consisting of the sense organs, an emergent characteristic of T-ness, associative memory, logic, and an underlying universal principle of consciousness. The whole conception was sometimes viewed as a dualistic system of matter and consciousness, but more often as a unitary (monistic) system where a universal consciousness is the groundstuff of reality, with individual awareness as an emergent characteristic related to brain organization and behavioral associations. Other, more advanced structural models spoke of consciousness centers and agents. Animal and human minds were taken to be different in the sense that humans possess a language that is richer than that of animals; animals were also taken to be sentient, conscious beings. Details of this fascinating tradition, which is not widely known in the West, may be found in Sinha (1958), Aurobindo (1970), Dyczkowski (1987), and Kak (1993c, 1994b). We d o not know whether the categories of the Vedic cognitive science can be related to current neurophysiological discoveries. It is significant that the tradition claims to define a science of consciousness, and the description is in terms of a holistic framework that is reminiscent of that of quantum mechanics. It is therefore not surprising that, at least for Schrodinger, one of the creators of quantum mechanics, Vedic ideas were a direct source of inspiration (Schrodinger, 1961; Moore, 1989, pp. 170-173). Schrodinger (1965) also believed that the Vedic conception provided the resolution to the paradox of consciousness. Walter Moore, Schrodinger’s biographer, summarizes:


265

The unity and continuity of Vedanta are reflected in the unity and continuity of wave mechanics. In 1925, the world view of physics was a model of a great machine composed of separable interacting material particles. During the next few years, Schrodinger and Heisenberg and their followers created a universe based on superimposed inseparable waves of probability amplitudes. This new view would be entirely consistent with the Vedantic concept of All in One (Moore, 1989, p. 173).

E . Overview of the Chapter

This chapter presents several perspectives on the problem of machine and animal intelligence so as to set the framework for introducing the notion of a quantum neural computer. Our emphasis is on concepts and not on practical considerations. We do not speak about a design for a quantum neural computer, but the examination of this concept is very useful in understanding the limitations of current techniques of machine intelligence. We begin by revisiting the Turing test, and suggest that a new test that measures gradations of intelligence should be devised. Recent research on the cognitive capabilities of some animals, which validates the notion of levels of intelligence, is reviewed. This research establishes that certain cognitive abilities of animals, such as abstract generalization, are beyond the capabilities of conventional computers. This is followed by a survey of the state of current neural network research. We consider new neuron and network models including one where, in analogy with quantum mechanics, the neuron outputs are taken to be complex. We also review rapid training of feedforward networks using prescriptive learning, chaotic dynamics in neuron assemblies, models of attention and awareness, and cytoskeletal microtubule information processing. Recent discoveries in neuroscience that cannot be placed in the reductionist models of biological information processing are examined. We learn from these studies that all biological systems cannot be viewed as connectionist circuits of components or systems; there exist dynamic structures whose definition is, in part, related to the environment and interaction with other, similar systems. Although there are biological structures that are well modeled by artificial neural networks, in general biological systems define a concept of interdependence that is much stronger than the notion of connectionism that has been used in artificial neural networks. This is based on a recursive relationship between the organism and the environment. We call such interdependence connectionism in its strong sense. One may postulate systems that are equivalent in their connectionist complexity to biological systems.

266

SUBHASH C. KAK

We define a quantum neural computer as a strongly connectionist system that is nevertheless characterized by a wavefunction. In contrast to a quantum computer, which consists of quantum gates as components, a quantum neural computer consists of a neural network where quantum processes are supported. The neural network is a self-organizing type that becomes a different measuring system based on associations triggered by an external or an internally generated stimulus. We consider some characteristics of a quantum neural computer and show that information is not a locally additive variable in such a computer. Models for a subneuron field that could explain the unity of awareness are reviewed. Uncertainty relations connecting the programs for structure and environment are described. Issues related to the design of intelligent systems, different in their programmatic structure from those found in nature, are discussed. The chapter argues that quantum neural computing has parallels with brain behavior. Owing to this perspective, a considerable part of this chapter is devoted to pointing out the limitations of the conventional computing paradigm and the characteristics of quantum systems. We adopt the philosophical point of view that quantum reality reveals itself through questions, as embodied by different experimental arrangements. It is natural, then, first to explore the information theoretic issues related to the concept of a quantum neural computer. 11. TURINGTESTREVISITED If current computers are so fundamentally deficient in comparison with biological organisms at certain kinds of tasks, one might ask why the discipline of computer science has not generally recognized it. It appears that a preoccupation with comparisons with human thought distracted researchers from considering a rich set of possibilities regarding gradations of intelligence. The prestige of the famed test for machine intelligence proposed by Alan Turing (1950), which now bears his name, was partly responsible for such a focus. A . The Test

The Turing test proposes the following protocol to check if a computer can think: (1) The computer together with a human subject are to communicate, in an impersonal fashion, from a remote location with an interrogator; (2) The human subject answers truthfully, while the computer is allowed to lie to try to convince the interrogator that it is the human subject. If in the


267

course of a series of such tests the interrogator is unable to identify the real human subject in any consistent fashion, then the computer is deemed to have passed the test of being able to think. It is assumed that the computer is so programmed that it is mimicking the abilities of humans. In other words, it is responding in a manner that does not give away the computer’s superior performance at repetitive tasks and numerical calculations. The trouble with the popular interpretation of the Turing test is that it focused attention exclusively on the cognitive abilities of humans. So researchers could always claim to be making progress with respect to the ultimate goal of the program, but there was no means to check if the research was on the right track. In other words, the absence of intermediate signposts made it impossible to determine whether the techniques and philosophy used would eventually allow the Turing test to be passed. In 1950, when Turing’s essay appeared in print, his test embodied the goal for machine intelligence. But a statement of a goal, without a definition of the path that might be taken to reach it, detracts from its usefulness. Had specific tasks, which would have constituted levels of intelligence or thinking below that of a human, been defined, then one would have had a more realistic approach to assessing the progress of AI. Perhaps this happened because the dominant scientific paradigm in 1950, following old Cartesian ideas, took only humans to be capable of thought. The rich tradition of cognitive philosophy that emerged in India about 4000 years ago (Kak, 1993b) where finer distinctions in the capacity to think were argued, was at that time generally unknown to A1 researchers. The dominant intellectual ideas ran counter to the folk wisdom of all traditions regarding animal intelligence. That Cartesian ideas on thinking and intelligence were wrong has been amply established by the research on subhuman intelligence of the past few decades. Another reason why Turing’s test found a resonance in the intellectual climate of his times was the then prestige of operationalism as a scientific philosophy. According to the operationalist view, a machine is to be seen as having thoughts if its behavior is indistinguishable from that of a human. The ascendancy of operationalism came about from the standard interpretation of quantum mechanics, according to which one could only speak in terms of the observations by different experimental arrangements, and not ask questions about what the underlying reality was.

B. On Animal Intelligence According to Descartes, animal behavior is a series of unthinking mechanical responses. Such behavior is an automatic response to stimuli that originate

268

SUBHASH C. KAK

in the animal's internal or external environments. Complex behavior can always be reduced to a configuration of reflexes where thought plays no role. Only humans are capable of thought, since only they have the capacity to learn language. Recent investigations of subhuman animal intelligence not only contradict Cartesian ideas but also present fascinating riddles. It had long been thought that the cognitive capacities of the humans were to be credited in part to the mediating role of the inner linguistic discourse. Terrace (1985) claims that animals do think but cannot master language, so the question arises as to how thinking can be done without language: Recent attempts to teach apes rudimentary grammatical skills have produced negative results. The basic obstacle appears to be at the level of the individual symbol which, for apes, functions only as a demand. Evidence is lacking that apes can use symbols as names, that is, as a means of simply transmitting information. Even though non-human animals lack linguistic competence, much evidence has recently accumulated that a variety of animals can represent particular features of their environment. What then is the non-verbal nature of animal representations?. . . [For example] learning to produce a particular sequence of four elements (colours), pigeons also acquire knowledge about a relation between non-adjacent elements and about the ordinal position of a particular element (Terrace, 1985, p. 113).

Clearly, the performance of animals points to representation of whole patterns that involves discrimination at a variety of levels. In an ingenious series of experiments, Herrnstein and Loveland (1964) were able to elicit responses about concept learning from pigeons. In another experiment, Herrnstein (1985) presented 80 photographic slides of natural scenes to pigeons that were accustomed to pecking at a switch for brief access to feed. The scenes were comparable, but half contained trees and the rest did not. The tree photographs had full views of single and multiple trees as well as obscure and distant views of a variety of types. The slides were shown in no particular order, and the pigeons were rewarded with food if they pecked at the switch in response to a tree slide; otherwise nothing was done. Even before all the slides had been shown, the pigeons were able to discriminate between the tree and the non-tree slides. To show that this ability, impossible for any machine to match, was not somehow learned through the long process of evolution and hardwired into the brain of the pigeons, another experiment was designed to check the discriminating ability of pigeons with respect to fish and non-fish scenes, and once again the birds had no problem doing so. Over the years it has been shown that pigeons can also distinguish: (1) oak leaves from leaves of other trees, (2) scenes with or without bodies of


269

water, (3) pictures showing a particular person from others with no people or different individuals. Herrnstein (1985) summarizes the evidence thus: Pigeons and other animals can categorize photographs or drawings as complex as those encountered in ordinary human experience. The fundamental riddle posed by natural categorization is how organisms devoid of language, and presumably also of the associated higher cognitive capacities, can rapidly extract abstract invariances for some (but not all) stimulus classes containing instances so variable that we cannot physically describe either the class rule or the instances, let alone account for the underlying capacity (p. 129).

Among other examples of animal intelligence are mynah birds that can recognize trees or people in pictures, and signal their identification by vocal utterances-words-instead of pecking at buttons (Turney, 1982), and a parrot that can answer, vocally, questions about shapes and colors of objects, even those not seen before (Pepperberg, 1983). The question of the relationship between intelligence and consciousness may be asked. Griffin infers animal consciousness from a variety of evidence: I. Versatile adaptability of behavior to novel challenges; 11. Physiological signals from the brain that may be correlated with conscious thinking; 111. Most promising of all, data concerning communicative behavior by which animals sometimes appear to convey to others at least some of their thoughts (Griffin, 1992, p. 27).

Consciousness implies using internal images and reconstructions of the world. Purposive behavior is contingent on these internal representations. These representations may be based on the stimulus from the environment, memories, or anticipation of future events. C . Gradations of Intelligence

The insight from experiments of animal intelligence, that one can attempt to define different gradations of cognitive function, is a useful one. The theory of evolution not only posits the evolution of human brains but also that of behavior. In this theory one would expect to see different levels of intelligence. It is obvious that animals are not as intelligent as humans; likewise, certain animals appear to be more intelligent than others. For example, pigeons did poorly at picking a pattern against two other identical ones, as in picking an A against two Bs. This is a very simple task for humans. Herrnstein (1985) describes how they seemed to do badly at certain tasks:

270

SUBHASH C. KAK

1. Pigeons did not do well at the categorization of certain man-made and three-dimensional objects. 2. Pigeons seem to require more information than humans for constructing a three-dimensional image from a plane representation. 3. Pigeons seem to have difficulty with dealing with problems involving classes of classes. Thus they do not do very well with the isolation of a relationship among variables, as againt a representation of a set of exemplars.

In a later experiment, Herrnstein et al. (1989) trained pigeons to follow an abstract relational rule by pecking at patterns in which one object was inside rather than outside a closed linear figure. It is to be noted that pigeons and other animals are made to respond in extremely unnatural conditions in Skinner boxes of various kinds. The abilities elicited in research must be taken to be merely suggestive of the intelligence of the animal, and not the limit of it. Animal intelligence experiments suggest that one can speak of different styles of solving A1 problems. Are the cognitive capabilities of pigeons limited because their style has fundamental limitations? Or is the cognitive style of all animals similar and the differences in their cognitive capabilities arise from the differences in the sizes of their mental hardware? And since current machines do not, and cannot, use inner representations, is it right to conclude that their performance can never match that of animals? Can one define a hierarchy of computational tasks that could be quantified as varying levels of intelligence? These tasks could be the goals defined in a sequence that could be set for AI research. If the simplest of these tasks proved intractable for the most powerful of computers, then the verdict would be clear that computers are designed based on principles that are deficient compared to the style at the basis of animal intelligence.

D . Animal and Machine Behavior 1. Linguistic Behavior

A classification of computer languages in the 1950s provided an impetus to examine the nature of animal communication. The focus of these studies on primates was not to decode this communication in their natural setting, but rather to see if animals could be taught to communicate with humans. Since the development of language, as part of behavior, is critically a component of social interaction, studies in unnatural laboratory settings are inherently limited,


27 1

Many projects have examined the grammatical competence in apes (e.g., Savage-Rumbaugh et al., 1985, Griffin, 1992). This research has clarified questions related to what we understand by language. Responding to signals in a consistent way is not sufficient to demonstrate linguisitic competence of the subject. It is essential for us to know that the subject uses different symbols for naming. Savage-Rumbaugh (1986) argues that, to qualify as a word, a signal must satisfy the following attributes: (1) It must be an arbitrary symbol that stands for some object, activity, or relationship; (2) it must be used intentionally to convey knowledge; and (3) the recipient must be able to decode and respond appropriately to the symbols. Many early ape-language projects did not satisfy all these attributes. Savage-Rumbaugh et al. (1985) report striking success in the acquisition of symbolic skills on a lexigram keyboard by an ape named Kanzi. How to grade competence in symbolic manipulation remains an interesting problem. If this were possible, one would have the picture of different levels of symbol manipulation leading finally to the development of language. Grammatical competence could likewise be classified in different categories. The relationship of these levels to the ability to solve A1 problems could be explored. A hierarchy of intelligence levels will be useful in the classification of animal behavior.

2. Recursive Behavior Animal behavior is recursive. Life can be seen at various levels, but consciousness is a characteristic of the individual alone. Considering this from the bottom up, animal societies may be viewed as “superorganisms.” For example, the ants in an ant colony may be compared to cells, their castes to tissues and organs, the queen and her drones to the generative system, and the exchange of liquid food among the colony members to the circulation of blood and lymph. Furthermore, corresponding to morphogenesis in organisms, the ant colony has sociogenesis, which consists of the processes by which the individuals undergo changes in caste and behavior. Such recursion has been viewed all the way up to the earth itself seen as a living entity. Parenthetically, it may be asked whether the earth itself, as a living but unconscious organism, may not be viewed like the unconscious brain. Paralleling this recursion is the individual, who can be viewed as a collection of several “agents,” where these agents have subagents, which are the sensory mechanisms, and so on. These agents are bound together and this binding defines consciousness.

272

SU3HASH C. KAK

Can an organism as well as a superorganism be simultaneously conscious? It appears that such nested consciousness will be impossible. In multiplepersonality disorder, the person’s consciousness remains a unity. Such a person might claim that “yesterday I was Alice and today I am Mary,” but each time the assertion would be in terms of an indivisible consciousness. In split-brain patients, the minor hemisphere does not have a conscious awareness. In other words, there is no nested consciousness.

3. Logical Tasks Logical tasks are easy for machines, whereas A1 tasks are hard. We have two perspectives. The first is: Why build machines in a hierarchy that mimics nature? After all, wheeled machines do better than animals at locomotion. The second is that there is something fundamental to be gained in building machines that have recursively defined behavior in the manner of life, and the same facility for solving A1 problems that subhumans possess. If the basis for animal competence at A1 tasks were understood, then new kinds of A1 machines could be designed. Logical tasks are difficult for animals to complete. Human language is more than reflexive behavior, and it has a logical component. A logical structure underpins human language. On the other hand, when apes have been taught associations of symbols with objects, they have found it hard to string these together into rule-based combinations that could be termed language. Although communication among animals seems to be very rich, it is clear that animal communication lacks many features that are present in human language. To see these differences in terms of less developed neural structures would connect neural hardware with specific function.

111. NEURAL NETWORK MODELS

A . The Brain

The central nervous system of vertebrates is organized in an ascending hierarchy of structures-spinal cord, brain stem, and brain. A human brain (Fig. 1) contains about 10’ neurons. The structure, composition, and functioning of central nervous systems in all vertebrates have general similarity. The peripheral nervous system consists of peripheral nerves and the ganglia of the autonomic nervous system. At the gross level, the brain is bilaterally symmetric and the two hemispheres are connected together by a large tract of about half a billion axons

’

273

QUANTUM NEURAL COMPUTING Corpus callosurn

LL

ond dliculi

ourth ventricle

Pons

FIGURE1. A midsagittal section of the human brain.

called the corpus callosum. At the base are structures such as the medulla, which regulates autonomic functions, and the cerebellum, which coordinates movement. Within lies the limbic system, a collection of structures involved in emotional behavior, long-term memory, and other functions. The convoluted surface of the cerebral hemispheres is called the cerebral cortex. The cerebral cortex is a part of the brain that is more developed in humans than in other species. Approximately 70% of all the neurons in the central nervous systems of primates are found in the cortex. This is where the most complex processing and coding of sensory information occurs. The most evolutionary ancient part of the cortex is part of the limbic system. The larger, younger neocortex is divided into frontal, temporal, parietal, and occipital lobes, which are separated by deep folds. Each of these lobes contains yet further subdivisions. The parietal, temporal, and occipital lobes receive sensory information such as hearing and vision. These sensoryreceiving areas occupy a small portion of each lobe, the remainder being termed the association cortex. Each sensory area of the neocortex receives projections primarily from a single sensory organ. Thus, the visual-receiving area in the occipital lobe processes sensations received by the retina; the auditory-receiving area in the temporal lobe processes sensations received by the cochlea; and the body sense-receiving area in the parietal lobe processes sensations received by the body surface. Within each sensory-receiving area, the sense-organ projections form a map of the sensory organ on the neocortical surface.

B. Neurons and Synapses The nervous tissue is made up of two types of cells, neurons and satellite cells. In the central nervous system the satellite cells are called neuroglia, and

274

SUBHASH C. KAK

I

Axon

Axon terminals

FIGURE2. An idealized neuron.

in the periphery they are called Schwann cells. The satellite cells provide insulating myelin sheathing around the neurons. This insulation increases the propagation velocity of the electrical signals. A neuron consists of treelike networks of nerve fiber called dendrites, the cell body (or soma), and a single long fiber called the axon, which branches into strands and substrands (Fig. 2). At the end of these substrands are the axon terminals, the transmitting ends of the synaptic junctions, or synapses (Fig. 3). The receiving ends of these junctions can be found both on the dendrites and on the cell bodies themselves. The axon of a typical neuron makes about a thousand synapses with other neurons. It may be argued that brain computations must really be seen to take place in the synapses rather than in neurons. This change in point of view has important philosophical implications. Not only are there more synapses than neurons-say, about 1015-but this changed perspective can be taken to mean a view of the brain as a chemical computer rather than an electrical digital system. This is an issue that will not be explored in this chapter. Two major types of synapse are electrical and chemical. In electrical synapses the channels formed by proteins in the presynaptic membrane are physically continuous with the similar channels in the postsynaptic membrane, or the synaptic gap is only about 2 nm, allowing the two cells to share some cytoplasm constituents and electrical currents. At chemical synapses, the gap of about 20-30 nm between the presynaptic and postsynaptic membranes prevents a direct flow of current. The signal is now transmitted to the next neuron through an intermediary chemical process in which neurotransmitter substances are released from spherical vesi-

275


\I

Glial cell

I Postsynaptic membrane

\Synaptic

gap

FIGURE3. The synapse.

cles in the synaptic bouton. This raises or lowers the electrical potential inside the body of the receiving cell. When this potential reaches a threshold, a pulse or action potential of fixed strength and duration is sent down the axon. This constitutes the firing of the cell. After firing, the cell cannot fire again until the end of the refractory period. Ionic currents flowing across surface membranes of nerve cells lead to electrical potential changes. Ions such as sodium, potassium, calcium, and chloride participate in this process. The nerve maintains an electrical polarization across its membrane by actively pumping sodium ions out and potassium ions in. When ion channels are opened by voltage changes, neurotransmitters, drugs, and so on, sodium and potassium flow rapidly through the channel, creating a depolarization. Action potentials occur on an all-ornone basis from integration of dendiritic input at the cell body region of the neuron. The frequency of firing is related to the intensity of the stimulus. Action potential velocity is fixed for given axons depending on axon diameter, degree of myelinization, and distribution of ion channels. A typical value of action potential velocity is 100 m/s.

276

SUBHASH C. KAK

Neurons come in a great variety of structures. The diversity is even greater if molecular differences are considered. Although all cells contain the same set of genes, individual cells exhibit, or activate, only a small subset of these genes. Selective gene expression has been found even in seemingly homogeneous populations of neurons. Evidence does not support the thesis that each neuron is unique; and certainly the brain cannot be viewed as a collection of identical elements. An order is imposed on the enormous diversity of the neurons and their connection patterns by the nested arrangement of certain circuits and subcircuits (Mountcastle, 1978; Sutton et al., 1988). For example, neurons of similar functions are grouped together in columns that extend through the thickness of the cortex. A typical module is the visual cortex, which could contain more than 100,000 cells, the great majority of which form local circuits devoted to a particular function. 1. The McCulloch-Pitts Model

McCulloch and Pitts (1943) proposed a binary threshold model for the neuron. The model neuron computes a weighted sum of its inputs from other inputs and outputs a 1 or a 0 according to whether this sum is above or below a certain threshold. The updating of the states of a network of such neurons was done according to a clock. Although the McCulloch-Pitts (MP) neurons do not model the biological reality well, networks built up of such neurons can perform useful computations. Variants of the M P neurons have graded response; this is achieved if the neuron performs a sigmoidal transformation on the sum of the input data. However, networks of such graded neurons do not provide any fundamental computing advantage over binary neurons. Networks of MP-like neurons have been used extensively to model brain circuits, in the hope that these networks will somehow capture the essentials of the processing done in the brain. Various kinds of networks have been used in these models. C . Feedback and Feedforward Networks

A broad distinction may be made between feedback and feedforward neural networks. In feedback networks the computation is taken to be over when the network has reached a stable state. The updating for the network is represented by the rule where f represents the concatenation of the transformation by the synaptic


277

interconnection weight matrix and the sign or the sigmoidal function. This updating occurs either synchronously or asynchronously. The operation of such a network in the asynchronous mode may be viewed as a descent to minimum energy value in the state space. Such minima can thus represent stored memories in a feedback model. Learning shapes attraction areas around exemplars. The notion of the attraction basis associated with each energy minimum provides the model for generalization. The major limitation of the feedback model is that, along with the useful memory, a very large number of spurious memories are automatically generated. Feedforward networks perform mapping from an input to an output. Once the training input-output pairs have been learned to define certain classes, this network also has the capacity to generalize. This means that neighbors of the training inputs are automatically classified. Although feedforward networks are quite popular as far as signal processing applications are concerned, they suffer from inherent limitations. This is owing to the fact that such networks perform no more than mapping transformations or signal classification. Feedback networks, on the other hand, offer greater parallels with biological memory, and biological structures, such as the vision system, have aspects that include feedback. It is due to this reason that recurrent networks, which are feedforward networks with global feedback, are being increasingly studied. The neuron model that we examine in this section is the on-off or the bipolar MP model. This model and its variant, where the neuron output can be any analog value over a range, have generally been used for signalprocessing artificial neural networks. However, biological neurons exhibit extremely complex behavior, and on-off-type neurons are a gross simplification of the biological reality (Libet, 1986). For example, neuron output is in terms of spikes; this output cannot be taken to be held to any fixed value. There is, furthermore, the question of the refractory phase of a neuron’s response. If one considers waves of neural activity, then a continuum model may be called for (Milton et al., 1994). Neuron outputs may also be considered to arise from the partial response of other neurons. In addition, there exist neurons that remain quiescent in a computation. We look at some of these issues and investigate how the bipolar model can be made more realistic. In many artificial intelligence problems, such as those of speech or image understanding, local operators can be properly designed only if some understanding of the global information is available. Networks of MP neurons do not provide any choice as far as resolution of the data is concerned. A generalization of the MP model of neurons appears essential in order to account for certain aspects of distributed information processing in biologi-

278

SUBHASH C. KAK

cal systems. One particular generalization allows one to deal with some recent findings of Optican and Richmond (1987), which indicate that in neuron activity the spike rate as well as spike distribution carry information. This may be represented by complex-valued neurons with relationship between the real and imaginary activations; this can form the first step in the storage of patterns together with their corresponding context clues. D . Iterative Transformations

Consider a fully connected neural network, composed of n neurons, with a symmetric synaptic interconnection matrix Tj,where Ti = 0. The matrix T is obtained by Hebbian learning, which is an outer product learning rule, or its generalization called the delta rule. In Hebbian learning,

T.=

c x:x; 1

k=l

where 1 is the number of memories, and X k represents the kth memory. Let xi be the output of neuron i. The updated value of this neuron is

where f is a sigmoid or a step function. Without loss of generality, we will now confine ourselves to the sign function (sgn) for f,and we will use the convention that sgn(0) = 1. Also note that the updating of the neurons may be done in any random order-in other words, asynchronously. The characteristics of such a feedback network have parallels with that of one-variable iterative systems. Consider the equation g(x) = 0, the solutions for x defining the roots of the equation. Let xo be a point near a root x. Then one may expand g(x) in a Taylor series so that

For values of x near xo, (x - xo) will be small. Assuming that g’(xo) is large compared to (x - xo), and that g”(xo) and the higher derivatives are not unduly large, the above equation may be rewritten as a first approximation: g(x) = d x o )

+ (x - xo)s’(xo)


279

Since we seek x such that g(x) = 0, we can simplify this equation as

The new x will be closer to the root than x,,, and this forms the basis of the Newton-Raphson iteration formula:

If the equation g ( x ) = 0 has several solutions, then the variable x will be divided into the same number of ranges. Starting in a specific range would take one to the corresponding solution. Each range can therefore be seen as an attraction basin. This may be generalized by the consideration of the transformation on the complex plane. For an analytic function one can use the Taylor series to give

For g(z), an nth-degree polynomial, one would see n attraction basins. A good example is g ( z ) = z 3 - 1 = 0, which leads to the iterative mapping 2z3 z,+1

+1

= ___

3z2

This has three attraction basins. The boundaries of the basins are fractals. Das (1994) has examined how neural networks that are a generalization of the above idea can be designed. We see that for an iterative transformation the concepts of attractors and attraction basins are defined similar to those for the feedback neural network. However, we do not obtain a regular structure of connected points defining an attraction basin. Moreover, the nature of the attraction basins is dependent on the iterative transformation, although the underlying polynomial may be the same. 1. Neuron Models and Complex Neurons

Biological neurons are characterized by extremely complex behavior expressed in a distribution of voltage and time dependence of current. Different types of membrane currents may be involved. Perhaps the simplest model is the classic Hodgkin-Huxley model of the squid giant axon. It has three main characteristics: (1) an action potential, (2) a refractory period after an action potential during which the slow recovery of the potassium and sodium con-

280

SUBHASH C. KAK

ductance has a great impact on electrical properties, and (3) the ability to capacitively integrate incoming current pulses. This model is approximated by two first-order differential equations in terms of four time-dependent dynamical variables. It has been suggested that a further approximation in terms of two dimensions is reasonably good. This description shows that the MP binary or bipolar neurons do not capture the intricacies of even the simple Hodgkin-Huxley model. Considering the problem at a gross level, we mentioned that neurons carry information in the distribution as well as the number of spikes in a response (Optican and Richmond, 1987). The nature of the relationship between these two types of information is not clear. In the most general case one may take these two channels of information to provide us with independent information, although they may actually be connected through some transformation. On the other hand, from an analytical perspective related to information processing, it is useful to see local/global information linkage whereby local information features are linked into global concepts. This may occur through stages of transformation so that information is delivered to neurons that deal directly with global concepts. One may use a straightforward generalization of the standard neuron model to include complex sequences where the real and the imaginary activations are related to each other. It may be assumed that usual observations, being based on time-averaged behavior, ignore the information in the distribution of the spikes. One might further assume that the real activation is related to the spike rate and the imaginary activation expresses the information in the spike distribution. In the general case the real and the imaginary quantities are connected to each other through a transformation. Note that each neuron in the standard network model does receive information from all other neurons and, therefore, there exists the potential of the delivery of global information at each neuron. However, this information is presented in a mixed fashion, from where it may not be retrievable. It is for this reason that it is useful to postulate an explicit transformation. Let the stored memories be represented by zj, where and The transformation g would in general be nonlinear. As a starting point, this transformation may be taken to be Fourier followed by a saturation function. Transformations such as Fourier decorrelate redundant patterns. This means that far fewer components of the pattern in the transformed domain are necessary to represent most of the pattern information.


28 1

For ease of illustration, one may consider bipolar neurons. The Hebbian learning rule for the interconnection weight matrix T can be stated to have the form

The neural network operates in the usual mode, Z,,, = sgm { TZ,,,}. The only difference with the standard Hebbian rule is that the column memory vector is multiplied with its complex conjugate transposed form. Also, if the staggering is done correctly for the patterns in a set, Hebbian learning will automatically associate the imaginary components of these patterns in a concatenated form. 2. Chaotic Maps on the Complex Plane The dynamics of a synchronous feedback network is represented by X,,, = J(Xold),where f is a linear map followed by a sigmoidal function and X is an n-component vector. In a synchronous update one can encounter oscillatory behavior. When the components of the X vector are continuous, the behavior can be chaotic. This will be seen in the context of a specific nonlinear transformation below. Consider z,,+~ = f(z,,), an iterative transformation on the complex plane. Such an iterative transformation must be considered for points on the unit circle because for ( z (< 1 the asymptotic value would end-up at 0, whereas for IzI > 1 it would end up at 03. The set of points on the unit circle is the Julia set. Consider now Zn+1

2

= zn

for the unit circle. This corresponds to z,+1

= x,+1

+

iY,+l

= xf

-yf

+ i2x,yn

Since x f + y i = 1, we obtain a pair of transformations for the x and y variables: x,+1

= 2x; -

1

- 1 I x, I 1

and 2

yn+l= 4~,2(1- y i )

Let us define another variable r, = y f ; then the above equation can be rewritten as I , + ~ = 4rn(l - r,,)

0 I r,, I 1

282

SUBHASH C. KAK

This represents the well-known logistic map. If r, is represented by an irrational point then the iterative transformation will never return to its starting point. In other words, the logistic map will generate an infinite period sequence, or a chaotic sequence. The same will also be true of the other maps listed above. To see this, consider z, = Iz,I exp(i8). It is clear that this map implies multiplying the angle 8 by 2 at each iteration. Now, if the starting angle 0, is represented in a binary form, then each step implies shifting the expansion one bit to the left. Therefore, if the original angle is represented by an infinite expansion (as for an irrational number), the iterative transformation will lead to chaotic behavior. It is also clear from the above that the two-dimensional structure of the Julia sets merely expresses the randomness properties in different scales of the underlying irrational number. One may consider different generalizations of the map on the unit circle. Thus one may consider

(

2 xn+l = Ax; 1 - GI 2 Y,+l

When GI = B we have

=

i2)

+ -y,

= B - I.,’Y,’

1 and 1 = 4, we obtain the logistic map. When only GI

=

fl = 1,

r,+1 = Irfl(1 - rfl) To determine the stability of a fixed point, one looks at the slope If’(r)l evaluated at the fixed point. The fixed point is unstable if I f ’ [ > 1. For this logistic function, when 1 < 1 < 3 there are two fixed points r = 0, (1- 1)/1, of which r = 0 is unstable and the other point is stable. For 1> 3 the slope at r = (1- 1)/1,which is f ’ = 2 - 1, becomes greater than 1 and both equilibrium points become unstable. At I = 3 the steady solution becomes unstable and a two-cycle orbit becomes stable. For 3 < A < 4 the map shows many multiple-period and chaotic motions. The first chaotic motion is seen for 1 = 3.56994... . Before the onset of chaos at this value, if the nth period doubling is obtained for I,, we get the wellknown result that in the limit

” -,4.6692016...

An

- In-1

which is the Feigenbaum number (Feigenbaum, 1978). This represents an example of the period-doubling route to chaos. There are other routes to chaos, such as quasi-periodicity and intermittency, that do not concern us in the present discussion.


283

Considering that chaos in the brain arises from a similar mechanism, one might ask whether the universality of the route to chaos provides a certain normative basis. 3. Implication The study of complex-valued neurons and iterative transformations shows that these alone are not capable of defining a rich enough framework in which cognitive functions can be understood. Thus a complex network may be replaced by four real ones, and so nothing fundamental is gained by such a generalization. Likewise, the transition from classical to quantum mechanics is much more than generalization from scalar to complex representations.

4. Performance of Interacting Units We now consider the question of a computation being carried out by a neural system of which a feedback network is a part. A network that deals with nontrivial computing tasks will be composed of several decision structures. If the neural network computation can be expressed in terms of a straightforward sequential algorithm, then it is inappropriate to call the computation style neural. We may call neurons that perform decision computations cognitive neurons. These cognitive neurons may take advantage of information stored in specialized networks, some of which perform logical processing. There would be other neurons that simply compute without cognitive function. Such cognitive neurons could lie at the end of the brain’s many convergence regions. It is essential to assume that many cognitive neurons compute in parallel for a computation to be neural. It was shown by Kak (1993b) that independent cognitive centers will lead to competing behavior. In the cortex, certain cells seem to be responsive to specific inputs. In other words, certain neurons process global information. Thus Hubel and Wiesel (1962) have shown that certain cells in the visual cortex of a cat are responsive to lines in the visual field which have a particular angle of slope. Other cells respond to specific colors, and others respond to the differences between what each eye has received, leading to depth perception. One might assume that neurons are organized hierarchically, with the higher-level neurons dealing with more abstract information. Can it be assumed that the two different kinds of neurons do interact with each other? Does the information processing of such a network have special characteristics o r limitations? Another interesting phenomenon is where damage to parts of the visual cortex makes a person blind in the corresponding part of the visual field. However, experiments have shown that often such subjects are able to

284

SUBHASH C. KAK

“guess” objects placed in this region with accuracy, even though they are blind in that region. The information from the retina is also processed by obscure regions lying in the lower temporal lobe. Clearly, these independent cognitive centers supply information to the subject’s consciousness, but this is done without direct awareness. Consider two nonlinearly interacting cognitive centers A and B, with three courses of action labeled 1, 2, and 3 with energies (- 1,2, 1) and (3, 5, - l), respectively. If the nonlinear interaction is represented by multiplication at the systems level, then the system states will be shown as in the following table: Energy

A,

A,

B, B2

-3 -5

6 10 -2

B3

1

A, 3 5 -1

The optimal system performance is represented by A , B,, which corresponds to the minimum energy of - 5 . However, the two centers would choose A , and B, considering individual energy distributions. This is indication that an energy function cannot be associated with a neural network that performs logical processing. If there is no energy function, how do the neurons operate cooperatively? Neural network models do not deal with or describe inner representations, so from our experience with animal intelligence (Section II.B), it is clear their behavior cannot be purposive. Current theory does not show how such inner representations could be developed.

Iv. THEBINDINGPROBLEM AND LEARNING A . The Binding Problem

How is a stimulus recognized if it belongs to a class that has been learned before? If the stimulus triggers the firing of certain neurons, how is the firing information “gathered” together to lead to the particular perception? Considering visual perception, there exists an unlimited number of objects that one is capable of seeing. Early theories postulated “grandmother neurons,” one for each image; this concept leads to a homunculus-the person’s representation inside the brain which observes and controls, and also represents the self. The postulation of grandmother neurons leads to logical problems:


285

How can we have a number that would exhaust all possible objects, and how does the homunculus organize all the sensory input that is received? Thus grandmother neurons cannot exist, although there might exist some specialized structures for certain features. If a large number of neurons fire in response to an image, the problem of how this set tires in unison, defining the perception of the image, is called the binding problem. No satisfactory solution to the binding problem is known at this time. In reality, the problem is much worse than the binding problem as stated above. The bound sets are in turn bound at a higher level of abstraction to define deeper relationships. The reductionist approach seeks explanations of brain behavior in terms of a sum of the behaviors of lower-level elements. Brain behavior has traditionally been examined at three different levels of hierarchy: (1) synaptic level, or in terms of biochemical changes; (2) neuron level, as in maps of the retina in the visual cortex; (3) that of neural networks, where the information is distributed over entire neuron cell assemblies. However, the explanation of brain function in terms of behavior at these levels has proved to be inadequate. Skarda and Freeman (1990) have argued that while reductionist models have an important function, they suffer from severe limitations. They claim that “perceptual processing is not a passive process of reaction, like a reflex, in which whatever hits the receptors is registered inside the brain. Perception does not begin with causal impact on receptors; it begins within the organism with internally generated (self-organized) neural activity that, by reafference, lays the ground for processing of future receptor input.. . . Perception is a self-organized dynamic process of interchange inaugurated by the brain in which the brain fails to respond to irrelevant input, opens itself to the input it accepts, reorganizes itself, and then reaches out to change its input (p. 279)” In other words, neural networks perform a perceptual processing function within a holistic framework that defines the selforganizing principle. B. Learning 1. The Biological Basis

Two types of learning or memories have been described by researchers. Memories that require a conscious record have been termed declarative or explicit. Memories where conscious participation is not needed are called nondeclarative or implicit. It is not known whether other types of memories exist.

286

SUBHASH C. KAK

Explicit learning is fast and may take place after only one training trial. It involves associations of simultaneous stimuli and defines memory of single events. In contrast, implicit learning is slow and requires repetition over many trials. It often connects sequential stimuli in a temporal sequence. Implicit learning is expressed by improved performance on certain tasks without the subject being able to describe what he or she has learned. Experiments indicate that the storage of both these memory types proceeds in stages. “Storage of the initial information, a type of short-term memory, lasts minutes to hours and involves changes in the strength of existing synaptic connections. The long-term changes (those that persist for weeks and months) are stored at the same site, but they require something entirely new: the activation of genes, the expression of new proteins and the growth of new connections” (Kandel and Hawkins, 1992, p. 86). Since gene activation at neuron sites, and the reorganization of the brain, are two of the consequences of learning, the question arises: Do we perceive as we do as a consequence of the genetic potential within us via its embodiment in the organization of the brain? In other words, is learning a part of the response to the universe? If the ape, or the pigeon, does not have the appropriate genes to activate specific development and reorganization of the brain, it will not respond to certain patterns in its environment. Hebb’s rule that coincident activity in the presynaptic and postsynaptic neurons strengthens the connection between them is one learning mechanism. According to another rule, called the premodulatory associative mechanism, the synaptic connection can be strengthened without activity of the postsynaptic cell when a third neuron, called a modulatory neuron, acts on the presynaptic neuron. The associations are formed when the action potentials on the presynaptic cell are coincident with the action potentials in the modulatory neuron. 2. Prescriptive Learning Although neural networks and other A1 techniques provide good generalization in many signal and image recognition applications, these methods are nowhere close to matching the performance of biological neural systems. As explained in Section 11, pigeons, parrots, and other animals can quite easily perform abstract generalization, recognizing abstract classes such as those of a tree, a human, and so on, which lies completely beyond the capability of any A1 machine. Are current learning strategies being employed fundamentally deficient? One would expect that biological systems, in their evolution through millions of years, would have evolved learning mechanisms that are optimal in some sense. This section examines biological learning and shows that artificial neural networks do not learn in the same manner


287

as animals. We claim that a two-step learning scheme, where the first step is a prescriptive design, offers a more versatile model for machine learning. Our starting point is developmental neurobiology. During embryogenesis, several kinds of cell death occur. One is histogenetic cell death, or the death of neurons that have ceased proliferation and have begun to differentiate; this is what is of relevance for our discussion. One may also speak of phylogenetic cell death, responsible for the regression of vestigal organs or of larval organs during metamorphosis (such as the tadpole tail); and morphogenetic cell death, which involves degeneration of cells during massive changes in shape, as in bending and folding of tissues and organs during early embryogenesis. According to Oppenheim (1981): “The neurons that exhibit cell death include such a wide variety of cells., .that it is impossible at this time to argue that any particular feature is characteristic of all neurons exhibiting cell death. Although it has been suggested that cell death may only occur among neurons that have greatly restricted or limited numbers of synaptic targets, the variety of cell types [that exhibit cell death] would appear to contradict such a suggestion” (p. 114). Soon after the issuance of Darwin’s The Origin of Species, it was suggested by T. H. Huxley, G. S. Lewes, and Wilhelm Roux that Darwinian ideas of natural selection might apply to the cells and tissues of the developing organism. The Spanish neuroanatomist Ramon y Cajal also suggested in 1919 that “during neurogenesis there is a kind of competitive struggle among the outgrowth (and perhaps even among the nerve cells) for space and nutrition. The victorious neurons.. . would have the privilege of attaining the adult phase and of developing stable dynamic connections” (Rambn y Cajal, 1929, p. 400). None of these early pioneers, however, anticipated the possiblity of massive neuronal death during normal development which was reported first by Hamburger and Levi-Montalcini (1949). We suggest that cell death is a result of many neurons being rendered redundant during a stage of reorganization that follows the first phase of development of the system. During the first phase all the neurons fulfil a specific function; this is followed by a structural reorganization. This twostep strategy can be used to devise a new approach to machine learning. 3. Reorganization

The two-step learning scheme allows us to view the developmental process in a different light. The first step consists of prescriptive learning which defines the layout and the synaptic weights of the interconnections. This learning may be called the biologically programmed component of the learning.

288

SUBHASH C. KAK

The nature of the prescriptive learning requires many more neurons than are necessary in a more efficiently organized network. During the second stage of learning a reorganization of the network takes place that renders a large number of neurons redundant. These are the neurons that suffer death as they do not receive appropriate signals from target neurons (Purves, 1988). In a more general setting this new learning strategy may be termed a structured approach. One may begin with a general kind of architecture for the intelligent machine which is refined during a subsequent phase of training. 4. The Example of Feedforward Neural Networks Backpropagation is the commonly used method for training feedforward networks, but it does not explicitly use any specific architecture related to the nature of the mapping. It has recently been shown that a prescriptive learning scheme can be devised (Kak, 1993a, 1994a). In its basic form this does not require any training whatsoever, but this form does not generalize. The weights are now changed slightly by adding random noise, which allows the network to generalize. Characteristics of this model are summarized in the work of Madineni (1994) and Raina (1994). This variant scheme provides significant learning generalization and a speed-up advantage of 100 over fastest versions of backpropagation. The penalty to be paid for this speed-up is the much larger number of hidden neurons that are required. The second step in this training would be to reorganize from the architecture obtained using prescriptive learning and reduce the number of hidden neurons. The speed of learning, and the simplicity of this procedure, make the new model a plausible mechanism for biological learning in certain structures. Interesting issues that remain to be investigated include further development of this approach so that one can obtain continuous-valued outputs and the development of competitive learning. However, we cannot expect such models to provide any insight regarding holistic behavior. C . Chaotic Dynamics

Freeman and his associates (e.g., Freeman, 1992, 1993 and Freeman and Jakubith, 1993) have argued that neuron populations are predisposed to instability and bifurcations that depend on external input and internal parameters. Freeman (1992) claims that chaotic dynamics makes it “possible


289

for microscopic sensory input that is received by the cortex to control the macroscopic activity that constitutes cortical output, owing to the the selective sensitivity of chaotic systems to small fluctuations and their capacity for rapid state transitions” (p. 452). The significance of this work is to point out the gulf that separates simple input-output mapping networks used in engineering research from the complexity of biological reality. But the claim that chaotic dynamics in themselves somehow carry the potential to explain macroscopic cortical activity relating to the binding problem seems to be without foundation. The nonlinearities at the basis of chaos have been seen as the basis of a self-organizing principle. Considering these ideas for the olfaction in a rabbit, Freeman (1993) says: “The olfactory system maintains a global chaotic attractor with multiple wings or side lobes, one for each odor that a subject has learned to discriminate. Each lobe is formed by a bifurcation during learning, which changes the entire structure of the attractor, including the pre-existing lobes and their modes of access through basins of attraction. During an act of perception the act of sampling a stimulus destabilizes the olfactory bulbar mechanism, drives it from the core of its basal chaotic attractor by a state transition, and constrains it into a lobe that is selected by the stimulus for as long as the stimulus lasts, on the order of a tenth of a second... . In this way the cortical response to a stimulus is ‘newly constructed’. . . rather than retrieved from a dead store” (p. 507). The above theory is very attractive for the olfactory system, but it remains very limited in its scope. It is hard to see how it could be generalized for more complex information processing.

D . Attention, Awareness, Consciousness Milner (1974) speculated that neurons responding to a “figure” fire synchronously in time, whereas neurons responding to the background fire randomly. More recently, von der Malsburg and Schneider (1986) have proposed a correlation theory to explain a temporal segregation of patterns. Crick and Koch (1990) have considered the problem of visual awareness. They distinguish between two kinds of memory: “very-short-term” or “iconic,” and a slower “short-term” or working memory. Iconic memory appears to involve visual primitives, such as orientation and movement, and it appears to last half a second or less. On the other hand, short-term or working memory lasts for a few seconds, and it deals with more abstract representations. This memory also has a limited capacity, and it has been claimed that this capacity is about seven items. For the “short-term” memory, Crick and Koch postulate an attentional mechanism that transiently binds together all those neurons whose activity

290

SUBHASH C. KAK

relates to the different features of the visual object. They argue that semisynchronous coherent oscillations in the 40- to 70-Hz range are an expression of this mechanism. These oscillations are sensory-evoked in response to auditory or visual stimuli. Postulating such a mechanism, however, merely trades one problem for another: How the neurons that participate in these oscillations are bound is still not explained. This theory is an extension of the earlier “searchlight” hypothesis relating to awareness. It shifts the basis of awareness to a more abstract mechanism of attention (Koch, 1993). This work has led to a reexamination of the question of what may be taken to define the loss of consciousness. In the standard view, anesthesia leads to four distinct effects: motionlessness in the face of surgery; attenuation or abolition of the autonomic responses such as tachycardia, hypertension, and so on, that would normally accompany surgery; lack of pain; and lack of recall. Use of electroencephalograms (EEGs) shows little difference between natural sleep and the anesthetized state. On the other hand, Kulli and Koch (1991) argue that the loss of consciousness, as defined by these four effects, is best represented by the loss of the 40-Hz sensory-evoked oscillations. Deepening anesthesia has also been seen as progressive loss of complexity of the EEG phase plot (Hameroff, 1987). Hameroff claims that anesthetics retard mobility of electrons, and doing so may disrupt hydrophobic links among proteins which interconnect microtubules which are filamentous substructures of the neuron (see next section).From a functional point of view one may see a prevention of memory consolidation from input to long-term memory storage as one of the consequences of anesthesia; other cognitive and motor functions may be similarly inhibited. 1. Scripts Yarbus (1967) recorded the eye movements when a picture is viewed. He found that there is a continual scanning of the scene, with a predominance of the fixation points on parts which carry important and complex features. From this one may infer that the brain constructs the regular image from fragments, obtained through the scanning process, that may be termed scripts. Such scripts may be defined with respect to both space as well as time. The structuring of events in dreams gives us important clues regarding time scripts. My father explained to me in 1955 that certain dreams run as scripts. Thus a foot slipping off another in sleep may be accompanied by a dream about falling off a precipice. That such a script includes an appropriate sequence of events preceding the climax indicates that the mind rearranges the events so that the dream appears to have started before the slipping of the foot. Similar scripts should be a part of normal awake cogni-


29 1

tion. For example, in delirious speech, ideas and words are strung together in an illogical manner. It is as if different brief scripts have been jumbled together without comparing the stream to the inner model of reality. Scripts might be seen as chunks of visualization or linguistic behavior, just as neural subsystems might be viewed as chunks of structure. 2. Time Effects Experiments focusing on time delays and rearrangement of events by consciousness have been performed by H. H. Kornhuber and his associates and by Benjamin Libet. Kornhuber and his associates (Deeke et al., 1976) found that the readiness potential, the averaged EEG trace from the precentral and parietal cortex, of a subject who was asked to flex his finger built up gradually for a second to a second and a half before the finger was actually flexed. This slow buildup of the readiness potential may be viewed as a response of the unconscious mind before it passes into the conscious mind and is expressed as a voluntary movement. In Libet’s work (Libet et al., 1979), the subjects were undergoing brain surgery for some reason unconnected to the experiment and they agreed to electrodes being placed at points in the brain, in the somatosensory cortex. It was found that when a stimulus was applied to the skin, the subject became aware of this half a second later, although the brain would have received the signal of the stimulus in barely a hundredth of a second. Furthermore, the subjects themselves believed that no delay had taken place in their becoming aware of the stimulus! In further experimentation the somatosensory cortex was electrically stimulated within half a second of skin stimulus. A backward masking phenomenon occurred, and the subject did not become aware of the earlier skin sensation. Now Libet initiated a persistent cortical stimulation first and then within half a second he also touched the skin. The subject now was aware of both stimulations, but he believed that the skin stimulation preceded that of the cortex. In other words, the subject extrapolated the skin-touching sensation backward in time by about half a second. These experiments demonstrate how the brain works as an active agent, reorganizing itself as well as the information. E . Cytoskeletal Networks

Recursive definition is one of the fascinating characteristics of life. For example, natural selection not only works at the level of species but also at the level of the individual and that of the nerve cells of the developing organism (Cowan, 198 1). Purposive behavior characterizes human societies as well as the societies of other animals such as ants. Recursion may be seen regarding

292

SUBHASH C. KAK

information processing as well, from animal societies to the neural structures of the individual or perhaps to the cytoskeleton of the cell. C. S. Sherrington (1951) argued that even single cells possess what might be called minds: “Many forms of motile single cells lead their own independent lives. They swim and crawl, they secure food, they conjugate, they multiply. The observer at once says ‘they are alive’; the amoeba, paramaecium, vorticella, and so on. They have specialized parts for movement, hair-like, whip-like, spiral, and spring-like.. . . [Of] sense organs.. .and nerves there is no trace. But the cell framework, the cyto-skeleton, might serve” (p. 208). Hameroff and his associates have argued that microtubules, hollow cylinders 25 nm across, which are the cytoskeletal filamentous polymers to be found in most cells, perform information processing. Hameroff (1987), Hameroff et al. (1992), and Rasmussen et al. (1990) propose that the cytoskeleton may be viewed as the cell’s nervous system, and that it may be involved in molecular-level information processing that subserves higher, collective neuronal functions ultimately relating to cognition. They further propose that microtubule automata may have a function in the guidance and movement of cells and cell processes during the morphogenesis of the brain’s network architecture, and that in learning they could regulate synaptic plasticity. In Hameroff (1994), interactions between the electric dipole field of water molecules confined within the hollow core of microtubules and the quantized electromagnetic radiation field were considered. These and BoseEinstein condensates in hydrophobic pockets of microtubule subunits were taken to be responsible for microtubule quantum coherence. It was suggested that optical signaling in microtubules would be free from both thermal noise and loss, and that this may provide a basis for biomolecular cognition and a substrate for consciousness. Regardless of the precise function of the microtubule information transmission, it could carry additional features that may be useful in biological information processing. The notion of a connectionist network within the neuron, which in turn is an element of the connectionist neural network, defines a recursive relationship that has many desirable features from the point of view of ability to model biological behavior.

F . Invariants and Wholeness The brain’s task can also be viewed as one that involves the extraction of invariants. For example, an image cannot be recognized based on the visual stimulus that is compared to a stored exemplar. The reason is that the details of the visual stimulus depend on illumination, the distance and orientation of the object, motion, and many other factors. In order to extract the in-


293

variant features of the image, the brain must, from the constantly changing cascade of photonic data, construct an internal visual world. Zeki (1992) has shown that color, form, motion, and possibly other attributes are processed separately in specialized parts of the visual cortex. These parts function as parallel processing modules, although there exist considerable connections among them. How the processing of these parallel modules is put together remains a puzzle. Zeki summarizes: “The entire network of connections within the visual cortex must function healthily for the brain to gain complete knowledge of the external world. Yet as patients with blindsight have shown, knowledge cannot be acquired without consciousness, which seems to be a crucial feature of a properly functioning visual apparatus. Consequently, no one will be able to understand the visual brain in any profound sense without tackling the problem of consciousness as well” (p. 76). It is the notion that awareness possesses a unity and that many aspects of consciousness are distributed over wide areas of the brain that has driven the search for quantum neural models. These issues are summarized in Kak (1993c, 1995) and Penrose (1989), but the presence of noise and dissipation inside the brain makes the development of such models a daunting task. Artificial neural network research has stressed pattern recognition and input-output maps. The development of machines that can match biological information processing requires much more than just pattern recognition. Corresponding to an input not only are many subpatterns bound together, there is also generated other relevant information defining the background and the context. This is the analog of the binding problem for biological neurons, and therefore further progress in the design of artificial neural networks appears to depend on the advances in understanding brain behavior. From the perspective of wholeness, it appears that a drastic change of perspective is necessary to make further advances. Consciousness is a recursive phenomenon: Not only is the subject aware, he or she is also aware of this awareness. If one were to postulate a certain region inside the brain from where the searchlight is shown on the rest of the brain and which provides the unity and wholeness to the human experience, the question of what would happen if this searchlight were to be turned on itself arises. V. ON INDIVISIBLE PHENOMENA

To understand the marvelous information processing ability of animals, researchers have investigated the neural structure of the brain and related it to a hypothesized nature of the mind. If brain structure is neuronal, then cognitive capabilities should be found for networks of neurons. Note, however, the argument (Sacks, 1990) that neuropsychology itself is flawed since it

294

SUBHASH C . KAK

does not take into account the notion of self, which is why it is hard put to explain phenomena such as that of phantom limbs (Melzack, 1989). The study of neural computers was inspired by possible parallels with the information processing of the brain. It was proposed that artificial neural networks constituted a new computing paradigm. However, our experience with these networks has shown that such a characterization is incorrect. When they are simulated, they represent sequential computing. In hardware, they may be viewed as a particular style of parallel processing, but as we know, parallel computing is not a departure from the basic Turing model. Neither does the use of continuous values provide us any real advantage, because such continuous values can always be quantized and processed on a digital machine. Artificial neural networks were assumed to offer a real advantage in the solution of optimization problems. However, the energy minimization technique, while sound in theory, fails in practice since the network gets stuck in local minima. Neural networks have provided useful insights, but they have failed at explaining the unity that characterizes biological information processing. Cognitive abilities may be seen to arise from a continuing reflection on the perceived world. This question of reflection is central to the brain-mind problem and the problem of determinism and free will (see, e.g., Kak, 1986; Penrose, 1989). A dualist hypothesis (e.g., Eccles, 1986) to explain brainmind interaction or the process of reflection meets with the criticism that this violates the conservation laws of physics. On the other hand, a brain-mind identity hypothesis, with a mechanistic or electronic representation of the brain processes, does not explain how self-awareness could arise. At the level of ordinary perception there exists a duality and complementarity between an autonomous (and reflexive) brain and a mind with intentionality. The notion of self seems to hinge on an indivisibility akin to that found in quantum mechanics. This was argued most forcefully by Bohr, Heisenberg, and Schrodinger (e.g., Moore, 1989). A . Uncertainty

Uncertainty is a fundamental limitation in a quantum description arising out of indivisibility. Since a quantum system is characterized by the wavefunction $, which is a superposition of states, and the measurement leads to the collapse of the wavefunction, one can never completely infer the state before the measurement. The wavefunction evolves according to the Schrodinger equation:


295

Here fi is the Hamiltonian. According to the probability interpretation of the wavefunction, the probability of finding a quantum object in the volume element dt is given by

*** Thus $$* is a probability density. For a particle the Heisenberg’s uncertainty relation,

limits the simultaneous measurements of the coordinate x and momentum p , where 6x and 6p are the uncertainties in the values of the coordinate and momentum. This relation also implies that one cannot speak of a trajectory of a particle. Since the initial values of x and p are inherently uncertain, so also is the future trajectory of the particle undetermined. One consequence of this picture is that we cannot have the concept of the velocity of a particle in the classical sense of the word. There are many scholars [see Jammer (1974) for a bibliography] who champion a statistical interpretation of the uncertainty relations, according to which the product of the standard deviations of two canonically conjugate variables has a lower bound given by h/2. Such an interpretation is based on the premise that quantum mechanics is a theory of ensembles rather than individual particles. But there is considerable evidence, including that related to the EPR experiment, described in the next section, that goes against the ensemble view. It appears, therefore, that the correct interpretation is the nonstatistical one, according to which one cannot simultaneously specify the precise values of conjugate variables that describe the behavior of a single object or system.

B. Complementarity The principle of complementarity, as a commonly used approach to the study of the individuality of quantum phenomena, goes beyond wave-particle duality. In the words of Bohr: The crucial point [is] the impossibility of any sharp separation between the behavior of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle can-

296

SUBHASH C. KAK

not be controlled. Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects (Bohr, 1951, p. 39).

Observe that complementarity is required at different levels of description. Just as one might use a probabilistic interpretation instead of complementarity for atomic descriptions, a probabilistic description may also be used for cognitive behavior. However, such a probabilistic behavior is inadequate to describe the behavior of individual agents, just as notions of probability break down for individual objects. According to complementarity, one can only speak of observations in relation to different experimental arrangements, and not an underlying reality. If such an underlying reality is sought, then it is seen that the framework of quantum mechanics suffers from paradoxical characteristics. One of these is nonlocal correlations that appear in the manner of action at a distance (Bell, 1987). However, quantum mechanics remains a very successful theory in its predictive power. Consider again the similarity between the thought process and the classical limit of the quantum theory. The logical process corresponds to the most general type of thought process as the classical limit corresponds to the most general quantum process. In the logical process, we deal with classifications. These classifications are conceived as being completely separate but related by the rules of logic, which may be regarded as the analog of the causal laws of classical physics. In any thought process, the component ideas are not separate but flow steadily and indivisibly. An attempt to analyze them into separate parts destroys or changes their meanings, yet there are certain types of concepts, among which are those involving the classification of objects, in which we can, without producing any essential changes, neglect the indivisible and incompletely controllable connection with other ideas. C. What Is a Quantum Model?

Wave-particle duality is not a characteristic only of light. Interference experiments have been conducted with neutrons. In one experiment, the interference pattern formed by neutrons, diffracted along two paths by silicon crystal, could be altered by changing the orientation of the interferometer with respect to the earth’s gravitational field. This demonstrated that the Schrodinger equation holds true under gravity. Interference experiments are also being designed for whole atoms. If we use complementarity in its widest sense as applying to all “indivisible” phenomena, reality is seen to be such that it cannot be captured by a single view.


297

This indicates that one should be able to define complementary variables in terms of higher-level attributes for “complex,” indivisible objects. From another perspective, a quantum system is characterized by probabilistic behavior. In Born’s probability interpretation, the wavefunction is not a material wave but rather a probability wave. Nevertheless, in relativistic quantum theory, the wavefunction cannot be used for the definition of a probability of a single particle. The orthodox Copenhagen interpretation of quantum mechanics is based on the fundamental notion of uncertainty and that of complementarity. According to this interpretation, we cannot speak of a reality without relating it to the nature of measurement. If one seeks a unified picture, one is compelled to accept the existence of effects propagating instantaneously. One example of paradoxical results arising from an insistence on a description that independent of the nature of observation is the delayed-choice variant of the double-slit experiment of the well-known physicist John Archibald Wheeler. 1. Delayed Choice

In the delayed-choice experiment, considering a specific reality (picture) before observations leads to the inference that we can influence the past. In the words of Wheeler (1980): A choice made in the here and now has irretrievable consequences for what one has the right to say about what has already happened in the very earliest days of the universe, long before there was any life on earth.

Wheeler considers a source that emits light of extremely low intensity, one photon at a time, with a long time interval between the photons. The light is incident on a semitransparent mirror MI and divides into two parts r and t ; after reflections by the totally reflecting mirrors A and B, it reaches the photomultipliers PI and Pz (Fig. 4).Since a photon will travel one of the two trajectories rAr or tBt, it will be detected either by PI or by P,. This experiment shows the corpuscular nature of photons. Now a semitransparent mirror M , is inserted at DCR (the delayed-choice region). The thickness of the mirror is so chosen that if light is considered wavelike, then the superposition of the waves toward P, combines in a destructive manner and the waves toward P, interfere in a constructive manner. With the insertion of M,, only the detector P, will register any reading, and it should be assumed that each photon follows both trajectories. Wheeler now wishes for us to imagine a situation where the mirror M 2 has been inserted at the last moment, when the photon has already passed through M , and is along the way either down rAr or tBt. If M , is inserted,

298

SUBHASH C. KAK

A

0

FIGURE4. Wheeler’s delayed-choice experiment; DCR is the delayed-choice region.

the photon will behave as a wave, as if it had followed both the paths. On the other hand, if M, is not inserted, then the photon chooses only one of the two paths. Since the insertion of M, is done at the last moment, it means that this choice modifies the past with regard to the behavior of that photon. Wheeler points out that astronomers could perform such a delayed-choice experiment on light from quasars that has passed through a galaxy or other massive object that acts as a gravitational lens. Such a lens can split the light from the quasar and refocus it in the direction of the distant observer. The astronomer’s choice of how to observe photons from the quasar appears to determine whether each photon took both paths or just one path around the gravitational lens in its journey commenced billions of years ago. When they approached the galactic beam splitter, did the photons make a choice that would satisfy the conditions of an experiment to be performed by unborn beings on a still-nonexistent planet? Clearly, this fallacy arises out of the view that a photon has a physical form before the astronomer’s observation.

VI. ON QUANTUM AND NEURAL COMPUTATION A . Parallels with a Neural System

By a quantum neural computer we mean a (strongly) connectionist network which is nevertheless characterized by a wavefunction. Let us assume that we are interested in considering only the computational potential of such a supposition. In parallel to the operational workings of a quantum model. we can sketch the basic elements of the working of such a computer. Such a computer will start out with a wavefunction, reflecting the state of the self-


299

organization of the connectionist structure, that is a sum of several different problem functions. After the evolution of the wavefunction, the measurement operator will force the wavefunction to reduce to the correct eigenfunction with the corresponding measurement that represents the computation. States of quantum systems are associated with unit vectors in an abstract vector space V , and observables are associated with self-adjoint linear operators on V. Consider the self-adjoint operator f.We can write

where $i are the eigenvectors and fi are the eigenvalues. The t+hi are the wavefunctions and fi represent the corresponding measurements. When $i are a complete orthonormal set of vectors in V, we have +it#‘ =S ,. Also, any wavefunction $ in V can be written as a linear combination of the $i. Thus

where the complex coeficients are given by ci

=

I

$$? dq

and q represents the configuration space. The sum of the probabilities of all possible values fi equals unity:

When an observation is made on an object characterized by a wavefunction $, the measurement process causes the wavefunction to collapse to the eigenfunction t,bi that corresponds to the measurement fi. This measurement itself is obtained with the probability Ici12. Since the wavefunction evolves regardless of whether any observations are being made, this will endow the system to compute several problems simultaneously. Analogously, animals can simultaneously perform several tasks, although such performance has traditionally been explained as being reflexive. From a functional point of view, this has parallels with the workings of a feedback neural computer. In such a computer the final measurement is one of the stored “eigenvectors” Xi,where the neural computer is itself characterized by the synaptic interconnection weight matrix T, so that sgm(TXi} = Xi where sgm is a nonlinear sigmoid function, so as to define a X with discrete component values.

300

SUBHASH C . KAK

One difference between the quantum mechanical framework and the above equation is the nonlinearity introduced by the sigmoidal function. This nonlinearity may be seen to be at the end of a chain of transformations, where the first step is a linear transformation as in quantum mechanics. On the other hand, the matrix T is like one of the many measurement operators of a quantum system. In other words, a neural network is associated with a single measurement operator, whereas a quantum computer is associated with a variety of measurement operators. Equivalently, we may view a quantum computer as a collection of many neural computers that are “bound” together into a unity. The view of the cognitive process representing self-organized neural activity implies that each such process does set up what may be considered a different neural computer. The set of the self-organized states may then be viewed as representing a collection of neural structures that are bound together. The search for a local mechanism for this binding is as difficult as for the binding problem mentioned in Section IV. If, on the other hand, we postulate a global function defining such a binding, then we speak of a quantum neural computer. If the wavefunction is associated with several operators, then the neural hardware for a specific problem will secure its solution. The measurement process will appear instantaneous after the decision to choose a specific measurement has been made. How this choice might be made constitutes another problem. Some of the characteristics of such a model are: 1. It explains intuition, or the spontaneous computation of the kind performed in a creative moment, as has been reported by Poincark., Hadamard, and Penrose (1989). 2. A wavefunction that is a sum of several component functions explains why the free-running mind is a succession of unconnected images or episodes. The classical neural model does not explain this behavior. 3. One can admit the possibility of tunneling through potential barriers. Such a computer can then compute global minima, which cannot be done by a classical neural computer, unless by the artifice of simulated annealing. (The simulated annealing algorithms may not converge.) 4. Being a linear sum of a large (or infinite) number of terms, the individual can shift the focus to any desired context by the application of an appropriate measurement hardware that has been designed through previous exposure (reinforcement) or through inheritance. Such a shifting focus is necessary in speech or image understanding.


301

B. Measurements

Let us consider a pure state $ to be a superposition of the eigenfuctions, or $=

1

ci$i

i

The overlapping realities collapse into one of the alternative worlds of a specific eigenfunction when a measurement by a nonquantum device is made. However, not all measurement devices are completely nonquantum; thus superconductivity represents a quantum phenomenon at the macroscopic level. If the measuring system is also a quantum system, it too should be described by a wavefunction. The measurement state would then arise out of the interference of the two wavefunctions and, in itself, would represent a superposition of several states. As shown by von Neumann, this requires that further measurements be made on the measuring apparatus to resolve the wavefunction, and so on (Schommers, 1989). 1. The Example of Vision

It was argued by Gibson (1979) and Schumacher (1986) that it is not necessary to view vision as resulting from the passage of signals through the optic nerve, but rather as a reorganization of the brain as a response to an environment. According to Gibson: It is not necessary to assume that anything whatever is transmitted along the optic nerve in the activity of perception. We can think of vision as a perceptual system, the brain being simply part of the system. The eye is also part of the system, since retinal inputs lead to ocular adjustments and then to altered retinal inputs, and so on. The process is circular, not a one-way transmission. The eye-head-brain-body system registers the invariants in the structure of ambient light. The eye is not a camera that forms and delivers an image, nor is the retina simply a keyboard that can be struck by fingers of light (p. 61).

Consider the parallel between the tracks in the photosensitive emulsion used to detect deflected particles and the eye. Bohm (1980) argued that the track in the emulsion is to be viewed as resolving the relevant wavefunction; likewise, the track in the eye may be viewed as resolving the wavefunction associated with the incident beam. However, we have seen in Section LA that the eye cannot be seen to be entirely classical. It appears, then, that the vision system itself should be decomposed into a concatenation of subsystems where this resolution proceeds. From this perspective the vision system may be viewed as quantum and classical measuring instruments associated with a quantum process.

302

SUBHASH C. KAK

C . More on Information

That a quantum neural computer will have characteristics different from that of a collection of classical computers is seen from an examination of information. In contrast to classical systems, information in a quantum mechanical situation is affected by the process of measurement. If a linearly polarized photon strikes a polarizer oriented at 90” from its plane of polarization, the probability is zero that the photon will pass to the other side. If another polarizer tilted at 45” is placed before the first one, there is a 25% chance that a photon will pass both polarizers. By the process of measurement, the 45” polarizer transforms the photons so that half of the initial photons can pass through it. The collapse of the wavefunction also causes nonlocal effects. These characteristics can be looked for in determining whether biological information systems should be considered quantum. 1. The E P R Experiment

We consider the thought experiment described by Einstein, Podolsky, and Rosen (1935) (Bell, 1987) now known as the EPR experiment. Einstein et al. assume the following condition for an element of a mathematical model to represent an element of reality: If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical reality (P. 777)

They also assert that “every element of the physical reality must have a counterpart in the physical theory.” They argue that if a pair of particles has strongly interacted in the past, then, after they have separated, a measurement on one will yield the corresponding value for the other. Since the position and momentum values are supposed to be undefined before the measurement is made and, nevertheless, the value is revealed after the measurement on the remote particle is made, Einstein et al. argue that the measurements should correspond to aspects of reality. They conclude that “the quantum-mechanical description of physical reality given by wave functions is not complete.” Since after separation each particle is to be considered as physically independent of the other (locality condition), this can also be taken to imply that effects propagate instantaneously. Bell (1987) has shown that the EPR reality criterion is incompatible with the predictions of quantum mechanics. Experiments have confirmed the predictions of the latter. We now consider the EPR experiment from an information theoretic viewpoint. In its Bohm variant, a pair of spin-f particles, A and B, have


303

formed somehow in the singlet spin state and are moving freely in opposite directions. The wavefunction of this pair may be represented by 1 -(v+ w-+ v- W')

Jz

where I/+ and I/- represent the measurements of spin + f and -f for particle A and W + and W - represent the measurement of + fand -f for particle B. The EPR argument considers the particles A and B to have separated. Now if a measurement is made on A along a certain direction, it guarantees that a measurement made on B along the same direction will give the opposite spin. The important point here is that the spin is determined as soon as, but not before, one of the particles is measured. This has been interpreted to mean that knowledge about A somehow reduces the wavefunction for B in a specific sense. In other words, the EPR correlation has been taken to imply a nonlocal character for quantum mechanics, or instantaneous action at a distance. To reexamine these questions in an informationtheoretic perspective, it is essential to determine the extent of information obtained in each measurement. Given a spin3 particle, an observation on it produces log, 2 = 1 bit of information. A further measurement made along a direction at an angle of 8 to the previous measurement leads to a probability cos' 8/2 that the measurement would give the same sign of spin, and a probability sin' 8/2 that it will give spin of the opposite sign. The information associated with the second measurement is 8 8 e 8 H ( 0 ) = -cos2 - log, cos2 - - sin2 - log, sin' 2 2 2 2

The average information considering all angles is Ha, =

1

:j

H ( 8 ) d8 = 1 - log,

& bits = 0.27865 bits

In other words, the information obtained from the second measurement is somewhat less than a quarter of a bit. These sequential experiments are correlated. It is also important to consider that all further measurements provide exactly 0.27865 bit of information. This indicates that information in a quantum description is not a locally additive variable. 2. Information in the Experimental Setup Now consider the information to be obtained from the measurement of spin of two spin-f particles that are correlated in the EPR sense. If information is

304

SUBHASK C. KAK

additive, then the first measurement provides 1 bit of information and after the end of the second measurement we have a total of 1.27865 bits. Since the EPR correlation reveals the spin of the particle B, as soon as the measurement of A has been made, one might infer that the information that the two arms of the experimental setup provide equals 0.27865 bit. In reality we cannot do so, as the calculus for information, in a quantum description, is unknown. Not being locally additive, the information in the experimental setup cannot be used in subsequent repetitions of the experiment. D . A Generalized Correspondence Principle

In a study several years ago (Kak, 1976, 1984) it was argued that the fundamental Heisenberg uncertainty was compensated by information in terms of new symmetries of the quantum description. In other words, one can generalize the correspondence principle to include uncertainty:

Iclassica, = Ism+ uncertainty A calculation of this information yields plausible results regarding the number of such symmetries. The analysis given here provides further elaboration of that idea. Although we appear to be unable to use the information in the various measurements, it is clear that the experimental setup itself plays a fundamental role in our knowledge. From another perspective, information can be associated with the angular (as well as spatial) position of an object. Nevertheless, this information cannot be considered in a local fashion. In other words, this information cannot be considered separately for the components of the system. In the context of a quantum neural computer, this means that behavioral characteristics of such a machine can be defined only in terms of the manner in which measurements on it are made. If such measurements are stored in the machine, then this experience can be regarded only as an interaction between the machine and the environment. Berger (1977, p. xiv) argues that one needs to use a similar relativism in the study of biological systems. “A necessary condition of experience is interaction between the organism and the environment, so that the structure of experience reflects the structure of both the organism and the environment with which it interacts. It follows that an organism cannot experience its own structure or that of the environment independently of the other.” Assume that an emergent phenomenon arises from interactions that are noncomputable in terms of the descriptions at the lower level. Assume further that the higher-level description is associated with a fundamental uncertainty. Will the generalized correspondence principle apply to this situation?


305

And if it does, will the emergent phenomenon be associated with new attributes that provide information equal in magnitude to this uncertainty? If information in biological information processing exhibits other nonlocal characteristics as well, such behavior may represent further parallels between the quantum mechanical paradigm and biological reality. VII. STRUCTURE AND INFORMATION A . A Subneuron Field

In earlier sections we critiqued materialist models, where an identity is assumed between mental events and neural events in the higher centers of the brain. Our main criticism was that these models did not deal with the binding problem. We presented evidence in support of the view that biological systems perform quantum computing. We now raise the question of the location of the quantum field that associates experience with its unity. Two proposals are reviewed in this section. 1. A Dualist Hypothesis

Popper and Eccles (1977) presented the view in which the world of mental events (they call it World 2) is sharply separated from the world of brain processes (World 1). Both these worlds are supposed to have autonomous existence. These ideas were further developed in Eccles (1986, 1990). In his theory, Eccles (1990) assumes that the entire set of the inner and outer senses is composed of elementary mental events, which he calls psychons. He further proposes that each psychon is linked to a corresponding functional aggregate of dendrites, which he names a dendron. A dendron is the basic receptive unit of the cerebral cortex, and it is a bundle or cluster of apical dendrites of certain pyramidal cells. Each dendron could involve about 200 neurons and as many as 100,000 synapses. There are about 40 million dendrites in the human cerebral cortex. He assumes that the mind exerts a minimal additional excitation on the dendronic vesicles, which are at a state just below the threshold for excitation. In order to meet the objection that immaterial mental events such as thoughts cannot act on material structures, such as the neurons in the cerebral cortex, Eccles has recourse to quantum mechanics. He says that “a calculation on the basis of the Heisenberg uncertainty principle shows that a vesicle of the presynaptic vesicular grid could conceivably be selected for exocytosis by a psychon acting analogously to a quanta1 probability field. The energy required to initiate the exocytosis by a particle displacement

306

SUBHASH C . KAK

could be paid back at the same time and place by the escaping transmitter molecules from a high to a low concentration. In quantum physics at microsites energy can be borrowed provided it is paid back at once. So the transaction of exocytosis need involve no violation of the conservation laws of physics” (Eccles, 1990, p. 447) Although the central idea of this theory is the linkage between the psychons and the microsite dendrons, Eccles’ theory is a field theory of psychons. A dendron represents the measurement hardware with which each unitary or elementary mental event is associated. Psychons are autonomous entities that may exist apart from dendrons and be related among themselves. In the brain-to-mind transaction, Eccles proposes that each time a psychon selects a vesicle for exocytosis, this “microsuccess” is registered in the psychon for transmission through the mental world. This signal carries into the mental world the special experiential character of that psychon. Eccles has only sketched the outline of his theory. He sees the psychons and the brain processes to be interlinked entities, which is why he labels his theory dualistic. He does not explain what is the physical support of psychons. Seen from this perspective, psychons are nothing but an artifice to circumvent the brain-mind problem. If they need the dendrons to emerge as mental events, then we are again confronted with the problem of the processes that lead to such an emergent phenomenon. Eccles does see the need for a quantum theoretic basis to explain psychon -dendron interaction of mental events. It appears, therefore, that the psychon field ought to be a quantum field. He does not explain how the world of mental events, as a separate, autonomous reality, functions. 2. Microtubule Field As a field one cannot take awareness to be localized. Considering the parallel of the wavefunction, it is a unity and it cannot be taken to be identical to the physical body although it is associated with it. Hameroff and his collaborators (Hameroff, 1994; Jibu et al., 1994) have suggested that the subneural structures called microtubules provide coherent information transmission that leads to the development of a quantized field that solves the “binding problem” and explains the unitary sense of self. These cytoskeletal microtubules, which provide structural support to cells, are hollow cylinders 25 nm in diameter whose walls are made of subunit proteins known as tubulin. Each tubulin subunit is an 8-nm dimer that is made up of two slightly different classes of 4-nm dimers known as a and fl tubulin. Hameroff has argued that the quantum dynamical system of water molecules and the quantized electromagnetic field confined inside the hollow microtubule core manifests a collective dynamics by which coherent photons are created inside


307

the microtubule. The question of why consciousness characterizes only certain cells, although microtubules are to be found in all cells, is not clearly answered. Hameroff (1994) suggests that consciousness might be an attribute of all quantum phenomena. Commenting on the results of the EPR experiment, he says that “quantum entities are ‘aware’ of the states of their spatially separated relatives!” It is possible also to see the microtubule information as providing the “imaginary” component in the complex neural information that was described in an earlier section. If this is the mechanism that provides the simultaneous global and local information which is essential for artificial intelligence tasks, then it is clearly not feasible to build such quantum neural computers at this time because microtubule information processing is still imperfectly understood.

3. A Universal Field If one did not wish for a reductionist explanation as in inherent in the cytoskeletal model, one might postulate a different origin for the quantum field. Just as the unified theories explain the emergence of electromagnetic and weak forces from a mechanism of symmetry breaking, one might postulate a unified field of consciousness-unified force-gravity from where the individual fields emerge. The notion of a universal field still requires for one to admit the emergence of the individual’s I-ness at specialized areas of the brain. This I-ness is intimately related to memories, both short term and long term. The recall of these memories may be seen to result from operations by neural networks. Lesions to different brain centers effect the ability to recall or store memories. For example, lesions to area V1 of the primary visual cortex lead to blindsight (Weiskrantz, 1986). These people can “see,” but they are unaware that they have seen. Although such visual information is processed, and it can be recalled through a guessing-game protocol, it is not passed to the conscious self. B. Uncertainty Relations

By structure we mean a stable organization of the neural system. The notion of stability may be understood from the perspective of energy of the neural system. Each stable state is an energy minimum. For example, the following energy expression (Hopfield, 1982) is suitable for a feedback neural network:

308

SUBHASH C. KAK

Any structure may be represented by a number, or a binary sequence. Thus, in one dimension, the sequences

abc, ba, cab represent three structures that can be coded into numbers by a binary code. Assume that a neural structure has been represented by a sequence. Since this representation can be done in a variety of ways, the question of a unique representation becomes relevant. Definition 1. Let the shortest binary program that generates the sequence representing the structure be called p .

The idea of the shortest program gives us a measure for the structure that is independent of the coding scheme used for the representation. The length of this program may be taken to be a measure of the information to be associated with the organization of the system. This length will depend on the class of sequences being generated by the program. In other words, this reflects the properties of the class of structures being considered. Evidence from biology requires that the brain be viewed as an active system which reorganizes itself in response to external stimuli. This means that the structure p is a variable with respect to time. Assuming, by generalized complementarity, that the structure itself is not defined prior to measurement, then for each state of an energy value E, we may, in analogy with Heisenberg’s uncertainty principle, say that

where k l is a constant based on the nature of the organizational principle of the neural system. The external environment changes when the neural system is observed, due to the interference of the observer. This means that as the measurement is made, the structure of the system changes. This also means that at such a fundamental level, a system cannot be associated with a single structure, but rather with a superposition of several structures. Might this be a reason behind pleomorphism, the multiplicity of forms of microbes? The representation described above may also be employed for the external environment. Definition 2. Let the shortest binary program that generates the external environment be called x.

If the external environment is a eigenstate of the system, then the system organization will not change; otherwise, it will.


309

We now propose an uncertainty principle for neural system structure: Sx S p 2 k ,

This relation says that the environment and the structure cannot be simultaneously fixed. If one of the variables is precisely defined, the other becomes uncontrollably large. Either of these two conditions implies the death of the system. In other words, such a system will operate only within a narrow range of values of the environment and structure. We conjecture that k , = k2 = k. One may pose the following questions: Are all living systems characterized by the same value of k? Can one devise stable self-organizing systems that are characterized by a different value of k? Would artificial life have a value of k different from that of natural life? What is the minimum energy required to change the value of p by one unit? Does a Schrodinger-type equation define the evolution of structure? It is also clear that, before a measurement is made, one cannot speak of a definite state of the machine, nor of a definite state of the environment. VIII. CONCLUDING REMARKS In the previous sections we have argued that humans and other animals perform what might be called quantum neural computing. If we accept the reverse of this claim, then a quantum neural computer would be characterized with life and consequently consciousness. On the other hand, ordinary machines cannot be conscious, since they do not come with a set of potentialities that consciousness provides. Machines are therefore like the neural hardware that provides extension. A machine that is so designed that it has infinite set of potentialities would be alive, but such an alive machine need not be based on the organic molecules of normal life. Our review has highlighted the following points. Conventional computing has been unable to devise schemes for holistic processing. This is why computers cannot recognize faces or understand text. Conventional computing is based on a reductionist algorithmic approach, but such an approach presupposes that the particles, or objects of the computation, have been selected. The selection of these objects is normally left to a human in the solution of any real-world problem. Conventional computing has failed at deriving methods for such a selection.

310

*

SUBHASH C. KAK

Biological computing may be viewed as a potentially infinite collection of conventional computing devices. This versatility arises from the brain reorganizing its structure in response to an association, presented by the environment or by a self-generated state, to become a special-purpose computing machine suited to the solution of that problem. We postulate a wavefunction associated with brain behavior that allows the biological system to get into a definite structural state corresponding to the stimulus. We can simulate a quantum neural computer by a collection of a large number of conventional computers. To build a true quantum neural computer one will have to develop a framework for organization. In other words, one would then know how the different conventional computers that together constitute an approximation to the quantum neural computer could have a plasticity built into them so that the same machine would be able to reorganize itself. We speculate that self-awareness is associated with the wavefunction that allows for the selectivity in the biological organism. If this is true, then a quantum neural computer will be self-aware.

This chapter presents arguments for a holistic computing paradigm that has parallels with biological information processing and with quantum computing. In this paradigm, inputs trigger an internal reorganization of the connectionist computer that makes it selective to the input. Such a holistic paradigm suffers from several paradoxical aspects. No wonder the determination of the phenomenological correlates of the holistic function, and therefore the design of a computer that operates in this paradigm, remain perplexing problems. A quantum neural computer represents an underlying quantum system that interacts with classical measurement structures composed of neural networks. This parallels the basic quantum theory framework, where measurement is to be performed using macroscopic apparatus. Learning by a biological system then represents the development of the measuring instruments, which are the neural structures in the brain. This picture still needs to address other questions, such as how, in response to a stimulus, does the brain reorganize itself so as to pick the appropriate neural network for measurement; and how the unity of the wavefunction leads to self-awareness. Unless we consider all matter to be conscious, we must take consciousness to be an emergent property of a quantum system that requires a certain structural basis supporting specific neuronal activity.

REFERENCES Aurobindo, S. (1970). “The Synthesis of Yoga.” Aurobindo Ashram, Pondicherry, India. Baylor, D. A., Lamb, T. D., and Yau, K.-W. (1979). J. Physiol. 288,613.


311

Bell, J. S. (1987). “Speakable and Unspeakable in Quantum Mechanics.” Cambridge University Press, Cambridge. Beniof, P. (1982). J . Stat. Phys. 29, 515. Berger, R. (1977). “Psyclosis: The Circularity of Experience.” Freeman, San Francisco. Bohm, D. (1980). “Wholeness and the Implicate Order.” Routledge & Kegan Paul, London. Bohr, N. (1951). “Atomic Physics and Human Knowledge.” Science Editions, New York. Cowan, W. M., ed. (198 I). “Studies in Developmental Neurobiology.” Oxford University Press, New York. Crick, F., and Koch, C. (1990). Sem. Neurosci. 2, 263. Das, S. (1994). Second order neural networks. Ph.D. dissertation, Louisiana State University, Baton Rouge. Deeke, L., Grotzinger, B., and Kornhuber, H. H. (1976). Biol. Cybernet. 23,99. Deutsch, D. (1985). Proc. R. SOC.Lond. A 400,97. Deutsch, D. (1989). Proc. R . Soc. Lond. A 425, 73. Dyczkowski, M. S. G. (1987). “The Doctrine of Vibration.” State University of New York Press, Albany. Eccles, J. C. (1986). Proc. R. Soc. Lond. B 227,411. Eccles, J. C. (1990). Proc. R. SOC.Lond. B 240,433. Einstein, A., Podolsky, B., and Rosen, N. (1935). Phys. Rev. 47,777. Feigenbaum, M. J. (1978). J . Stat. Phys. 19,25. Feynman, R. P. (1986). Found. Phys 16,507. Freeman, W. A. (1992). Int. J . Bijircarion and Chaos 2,451. Freeman, W. A. (1993). In “Rethinking Neural Networks: Quantum Fields and Biological Data” (Karl H. Pribram, ed.). Lawrence Erlbaum Associates, Hillsdale, NJ. Freeman, W. A,, and Jakubith, S. (1993). In “Brain Theory” (A. Aertsen, ed.). Elsevier, New York. Gibson, J. J. (1979). “The Ecological Approach to Visual Perception.” Houghton & Mimin, Boston. Gramss, T. (1994). On the speed of quantum computation. Tech. Rep. 94-04-017, Santa Fe Institute, Santa Fe, NM. Grifin, D. R. (1992). “Animal Minds.” University of Chicago Press, Chicago. Hamburger, V., and Levi-Montalcini, R. (1949). J . Exp. 2001.111,457. Hameroff, S. R. (1987). “ Ultimate Computing: Biomolecular Consciousness and NanoTechnology.” North-Holland, Amsterdam. Hameroff, S. R. (1994). J . Consciousness Stud. 1,91. Hameroff, S . R., Dayhoff, J. E., Lahoz-Beltra, R., Samsonovich, A. V., and Rasmussen, S. (1992). IEEE Comput. 25(11), 33. Hermstein, R. J. (1985). Phil. Trans. R . Soc. Lond. B 308, 129. Herrnstein, R. J., and Loveland, D. H. (1964). Science 146, 549. Herrnstein, R. J., Vaughan, Jr., W., Mumford, D. B., and Kosslyn, S. M. (1989). Perception and Psychophysics 46, 56. Hopfield, J. J. (1982). Proc. Natl. Acad. Sci. USA 81, 2554. Hubel, D. H., and Wiesel, T. N. (1962). J . Physiology (London) 160, 106. Jammer, M. (1974). “The Philosophy of Quantum Mechanics.” Wiley, New York. Jibu, M., Hagan, S., Hameroff, S. R., Pribram, K. H., and Yasue, K. (1994). BioSystems 32, 195. Kak, S. C. (1976). Nuouo Cimento 33B, 530. Kak, S. C. (1984). Proc. Indian Natl. Sci. Acad. 50, 386. Kak, S. C. (1986). “The Nature of Physical Reality.” Peter Lang, New York. Kak, S. C. (1988). Proc. 22nd Annu. Conf. on Information Science and Systems (Princeton, NJ), p. 131. Kak, S. C. (1993a). Pramana J . Phys. 40, 35. Kak, S. C. (1993b). Circuits, Systems, and Signal Processing 12,263.

312

SUBHASH C. KAK

Kak, S. C. (1993~).Reflections in clouded mirrors: Selfhood in animals and machines. Symposium on Aliens, Apes, and AI: Who Is a Person in the Postmodern World. Southern Humanities Council Annual Conference, Huntsville, AL, February 1993. Kak, S. C. (1994a). Pattern Recognition Lett. 15, 295. Kak, S. C. (1994b). From Vedic science to Vedanta. Fifth Int. Congress of Vedanta, Miami University, Oxford, Ohio, August 1994. Kak, S. C. (1994~).“The Astronomical Code of the Rgveda.” Aditya, New Delhi. Kak, S. C. (1995). Inform. Sci. 83, 143. Kandel, E. R., and Hawkins, R. D. (1992). Sci. Am. 267(3), 79. Koch, C. (1993). Curr. Opin. Neurobiol. 3, 203. Kulli, J., and Koch, C. (1991). Trends Neurosci. 14,6. Landauer, R. (1991). Phys. Today 44,23. Libet, 9. (1986). Fed. Proc. 45, 2678. Libet, B., Wright, E. W., Feinstein, B., and Pearl, D. K. (1979). Brain 102, 193. Madineni, K. B. (1994). Inform. Sci. 81,229. McCulloch, J. L., and Pitts, W. (1943). Bull. Math. Biophys. 5, 115. Melzack, R. (1989). Can. Psychol. 30, 1. Milner, P. M. (1974). Psych. Rev. 81, 521. Milton, J., Mundel, T., an der Heiden, U., Spire, J.-P., and Cowan, J. (1994). Neural activity waves and the relative refractory states of neurons. Tech. Rep. 94-09-052, Santa Fe Institute, Santa Fe, NM. Moore, W. (1989). “Schrodinger: Life and Thought.” Cambridge University Press, Cambridge. Mountcastle, V. 9. (1989). In “The Mindful Brain” (G. M. Edelman and V. 9. Mountcastle, eds.), p. 7. MIT Press, Cambridge. Oppenheim, R. W. (1981). In “Studies in Developmental Neurobiology” (W. M. Cowan, ed.). Oxford University Press, New York. Optican, L. M., and Richmond, B. J. (1987). J. Neurophys. 57, 162. Penrose, R. (1989). “The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics.” Oxford University Press, Oxford. Pepperberg, I. M. (1983). Anim. Learn. Behau. 11, 179. Popper, K. R., and Eccles, J. C. (1977). “The Self and Its Brain.” Springer International, Berlin. Purves, D. (1988). “Body and Brain: A Trophic Theory of Neural Connections.” Harvard University Press, Cambridge, MA. Raina, P. (1994). Inform. Sci. 81, 261. Ramon y Cajal, S. (1929). “Studies on Vertebrate Neurogenesis.” Thomas, Springfield. Rasmussen, S., Karampurwala, H., Vaidyanath, R., Jensen, K. S., and Hameroff, S. (1990). Physica D 42, 428. Sacks, 0. (1990). “Awakenings, A Leg to Stand On, The Man Who Mistook His Wife for a Hat.” Book of the Month Club, New York. Savage-Rumbaugh, E. S. (1986). “Ape Language: From Conditioned Response to Symbol.” Columbia University Press, New York. Savage-Rumbaugh, E. S., Sevcik, R. A,, Rumbaugh, D. M., and Rubert, E. (1985). Phil. Trans. R. SOC.Lond. B 308,177. Schnapf, J. L., and Baylor, D. A. (1987). Sci. Am. 256(4), 40. Schommers, W. (1989) “Quantum Theory and Pictures of Reality.” Springer-Verlag, Berlin. Schrodinger, E. (1961). “Meine Weltansicht.” Paul Zsolnay, Vienna. Schrodinger, E. (1965). “What Is Life?” Macmillan, New York. Schumacher, J. A. (1986). In “Fundamental Questions in Quantum Mechanics” (L. M. Roth and A. Inomata, eds.), p. 93. Gordon & Breach, New York. Sherrington, C. S. (1951). “Man on His Nature.” Cambridge University Press, Cambridge.


313

Sinha, J. (1958). “Indian Psychology.” University of Calcutta Press, Calcutta. Skarda, C. A., and Freeman, W. J. (1990). Concepts in Neurosci. 1,275. Sutton, J. P., Beis, J. S., and Trainor, L. E. H. (1988). J . Phys. A: Math. Gen. 21,4443. Terrace, H. S. (1985). Phil. Trans. R. Soc. Lo&. B 308, 113. Turney, T. H. (1982). Bull. Psychon. Soc. 19, 56. Turing, A. M. (1950). Mind 59,433. von der Malsburg, C., and Schneider, W. (1986). Biol. Cybernet. 54,29. Weiskrantz, L. (1986). “Blindsight.” Oxford University Press, London. Wheeler, J. A. (1980). Delayed choice experiments and the Bohr-Einstein dialog. Paper presented at the Joint Meeting of the American Philosophical Society and the Royal Society, London, June 5. Wigner, E., (1967). “Symmetries and Reflections.” Indiana University Press, Bloomington. Yarbus, A. L. (1967). “Eye Movements and Vision.” Plenum Press, New York. Zeki, S. (1992). Sci. Am. 267(3), 69.



Signal Description: New Approaches. New Results ALEXANDER M . ZAYEZDNY Department of Electrical Engineering and Computers. Ben Gurion University of the Negev. Beer Sheva. Israel

ILAN DRUCKMANN Lasers Group. Nuclear Research Center Negev. Beer Sheva. Israel

I . Introduction . . . . . . . . . . . . . . . . . . . A. On the Meaning of Signal Description . . . . . . . . . . B. Review of Signal Description Methods . . . . . . . . . C. Criteria for Comparing Different Methods of Signal Description . I1. Theoretical Basis of the Structural Properties Method . . . . . . A . Basic Definitions and Terminology . . . . . . . . . . . B. Elementary State Functions . . . . . . . . . . . . . C. Nonelementary State Functions . . . . . . . . . . . . D . Generating Differential Equations . . . . . . . . . . . E. A Second-Order Linear Generating Differential Equation Based on the Amplitude-Phase (A-Psi) Representation of a Signal . . . . . F. The Linear Second-Order Generating Differential Equation with . . . . . . . ThreeCoeficients, and Some Generalizations I11. Generalized Phase Planes . . . . . . . . . . . . . . . A. Definitions . . . . . . . . . . . . . . . . . . B. Construction of Generalized Phase Planes for Given Signals . . . C. Reconstruction of the Original Signal from Given Phase Trajectories D . Signal Processing Using Phase Trajectories . . . . . . . . IV. Applications . . . . . . . . . . . . . . . . . . . A. Parameter Estimation . . . . . . . . . . . . . . . B. Signal Separation . . . . . . . . . . . . . . . . C . Filtering and Rejection of Signals . . . . . . . . . . . D . Signal Identification . . . . . . . . . . . . . . . E. Data Compression . . . . . . . . . . . . . . . . F. Extrapolation of Time Series and Real-Time Processes . . . . G . Malfunction Diagnosis . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

315 316 318 325 321 327 329 333 336 339

. . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . . . . I

343 341 341 349 353 358 365 365 316 319 383 386 391 392 394

I. INTRODUCTION Signal theory has developed during the last decades as an independent branch of science which incorporates the general mathematical theory and tools for dealing with signals originating from a variety of fields. The main 315

Copyright 0 1995 by Academic Press. Inc. All rights of reproduction in any form reserved.

316

ALEXANDER M. ZAYEZDNY A N D ILAN D R U C K M A N N

part of signal theory is dedicated to mathematical methods of signal representation and classification, i.e., signal description methods. There is an abundant literature dedicated to this subject, e.g., Franks (1969), Trachtman (1972), Papoulis (1977), Urkowitz (1983), and Marple (1987). The primary attention in this literature is given to the analysis of signal properties, to their relationships with systems, and also to the synthesis of signals and systems. There are many methods of signal description; some of them can be viewed as classical (Fourier analysis and its many branches and related transforms, envelope-phase representation, series expansions, and linear system models) and others as new methods which have gained much popularity recently (wavelet analysis, higher-order spectra). We shall review these methods briefly, with emphasis on the newer methods. However, this chapter is dedicated mainly to the new approach called the method of “structural properties” of signals. The main idea of this approach is to make use of the relationships between the signal and its derivatives, and possibly also other transforms (e.g., Hilbert). These relationships are in general nonlinear, and are expressed by combinations of derivatives called state functions. Three tools are used in this method: the state function, the generating differential equation (GDE), and the generalized phase plane. In our view, the structural-properties approach has two main advantages: It makes use of more information (information about the relationships between the derivatives is information about the nature of the signal); and the description by this method is transparent to system synthesis. The chapter is organized as follows. In Sections LA and 1.C are presented some basic ideas about the meaning of signal description and some criteria for comparing description methods. In Section 1.B we briefly review the methods of signal description. Section I1 is dedicated to two of the tools of the structural properties method: the state function (Sections 1I.A-I1.C) and the GDE (Sections 1I.D-1I.F). In Section I11 we present the third tool, the generalized phase plane. Finally, Section IV is dedicated to the applications of the method to signal processing. Some of the sections of Section IV include brief reviews of existent methods. We cite here a few early works on the method of structural properties: Zayezdny (1967), Zayezdny and Zaytsev (1971), Zayezdny et al. (1971), Plotkin and Zayezdny (1979). A . On the Meaning of Signal Description

First we shall define some basic notions. A signal is a function of time (or space) which contains information. Usually a signal implies a combination of

SIGNAL DESCRIPTION: NEW APPROACHES, NEW RESULTS

317

a “pure” signal, containing the information, and a disturbance. The disturbance may be a noise or an interference, i.e., an unwanted signal. We shall consider in this chapter only one-dimensional real functions of time. Of course, the generalization is readily available. Signal description is defined as an operation of imaging, in which the signal is represented as a convenient analytical expression. This can be processed for the extraction of the information, and also for the synthesis of systems. A general representation of unidimensional signals is x ( t ) = p(t)s(t;a, b)

+ n(t),

(1)

where all functions are time-dependent. However, in the following, we shall drop the argument t except when we want to emphasize it. The vector a contains the informative parameters; the vector b contains the noninformative ones. p(t) denotes a multiplicative disturbance, and n(t) an additive one. n(t) can also include other signals, unrelated by nature with the functions s(t) and p ( t ) . We shall use the term “signal” to denote both s ( t ) itself and the mixture s ( t ) + disturbance. Signal theory deals with different problems: measurement of parameters a; suppressing the disturbances p ( t ) and n(t) and diminishing the influence of the noninfonnative parameters b on the parameters a; separation of signals (by their parameters or by their different functional form), problems of transforming the signals in order to increase the efficiency of processing systems, etc. From the point of view of practical applications, all these problems can be divided into the following groups: 1. Estimation or measuring of signal parameters: in communication, radar, etc. 2. Signal identification and signal classification: pattern recognition, speech identification, malfunction diagnosis, spectrum monitoring, medical diagnosis, etc. 3. Signal separation, filtering, and signal rejection 4. Data compression 5. Additional problems: extrapolation, etc. The purpose of signal description methods is to facilitate the solution of one or several problems. Solving the problem means obtaining a mathematical algorithm which can be implemented as a real device or system. From the great diversity of signal processing problems arises a diversity of description methods, and a method which is adequate for a certain problem may not be satisfactory for another type of problem. In the following sections we shall briefly review existing methods of signal description, and then we shall propose and discuss some criteria for the comparison of their efficiency.

318

ALEXANDER M. ZAYEZDNY AND ILAN DRUCKMANN

B. Review of Signal Description Methods

All signal description methods can be classified into two main groups: description by contour and description by structural properties. Contour properties are those properties which can be revealed by using instantaneous values of the signal, and also of its derivatives and integrals. By structural properties we mean those properties by which we can reveal the relationships between the signal and its derivatives, or between the derivatives themselves, and which in some sense can generate the signal. In the following we give a brief review of the methods of signal description. 1. The Direct (Immediate) Description

The signal is described by means of an analytical expression, as in (1). This method is used for dealing with only the most simple problems. For the solution of other problems it is used as a preliminary step for the transition to some other description. 2. Representation by Envelope and Phase

The signal is represented by means of the instantaneous values of two functions: the envelope A ( t ) and the phase d(t): x ( t ) = A(t) cos q(t),

(2)

where A(t),q ( t ) are defined as

and A(t) is the Hilbert integral transform:

It is convenient to express the phase in the form d t ) = mot

+ W),

(5)

where w,t is the linear term of the expansion of q ( t )about the frequency w,,. The theory of envelope-phase representation of signals is widely treated in the literature and is used in applications connected with oscillatory processes (Franks, 1969; Rice, 1982; Urkowitz, 1983; Picinbono and Martin, 1983; Papoulis, 1984).


3 19

3. Representation by Expansion in a Series of Orthonormal Functions The signal is expanded in a series of basis functions,

where, in most cases, the basis functions are orthonormal on some interval ( T may be finite or infinite),

and the coefficients cn are given by

The coefficients serve as a representation of x ( t ) in the signal vector space spanned by the family of basis functions. The equality in (6a) becomes an approximation if only N terms are used. In general, the coefficients are computed according to some criterion of minimum error for N-terms approximations, and also of independence of the set {c,} on N . For example, the specific form of coefficients (6c) results from the minimization of the mean-square error, for given basis functions and given N . An example of such an expansion is the Fourier series expansion in the interval (- T/2, T/2), T = 2n/w,, with $, = exp jnw,t. Other popular basis functions are families of orthogonal polynomials, the sine function, Walsh functions, etc. A related topic is the expansion of random processes in a series of orthonormal functions. In this case the coefficients are random variables. For a given family of basis functions, the optimal coefficients under the same minimum error criterion are as in (6c). An additional problem appears: What is the optimal family of functions, in the sense of decorrelating the coefficients? The solution to this problem is the Karhunen-Lohe expansion, in which the basis functions are the eigenfunctions of the integral equation involving the autocorrelation of x ( t ) (Van Trees, 1968; Proakis, 1983). The theory of representation of a signal x ( t ) by a linear combination of a family of basis functions &,(t) is called “the generalized spectral theory” (Trachtman, 1972). This topic is well developed and finds numerous applications-for example, in signal classification and signal detection (Van Trees, 1968; Franks, 1969; Proakis, 1983; Proakis and Manolakis, 1989).

320


4. Description by Means of integral Transforms The signal is represented in another domain, of the variable s, by a new function X(s),related to the original function by (74 ~ ( t=)

b

X ( s ) Y ( t ,S ) ds

where O(s, t), Y(t,s) are the kernels of the direct and inverse transforms, respectively. These transforms can be considered as continuous analogs of the discrete representations (6). The result of the direct transform is a function X ( s ) of a new variable s. This function is the image of the original signal x ( t ) . This new image is the object of processing in this method. The integral transforms which are most commonly used in signal processing are the Laplace and Fourier transforms (Proakis and Manolakis, 1989; Bracewell, 1986; Elliot and Rao, 1982).

5 . Mixed Time-Frequency Transforms The Fourier transform and its relatives are global, ie., are based on integrating the entire time axis, and as such, are inadequate for describing nonstationary signals. For this kind of signals one uses, instead of global transforms, various types of localized transforms, called mixed timefrequency transforms, such as the short-time Fourier transform (STFT), the Gabor transform, and the Wigner-Ville distribution. All these transforms map the signal into a time-frequency plane, where certain properties of the signal are made visible. The STFT or windowed Fourier transform is defined as X(w, t) =

s t

t+r/z

-r/z

x(a)e-j"("-') da.

(8)

This transform adequately describes the characteristics of a varying spectrum signal. In this definition the signal has been windowed by a rectangular window. In a more general definition, the window shape is arbitrary: m

X ( o , t) =

[ J

x(a)w(a - t)e-iocu-t)da.

(9)

-m

An important member of this class is the Gabor transform, in which the window is Gaussian shaped (Bastiaans, 1980; Wexler and Raz, 1990), and which has applications in image analysis and compression (Porat and Zeevi, 1988).


321

The Wigner-Ville distribution (WD) is defined as WD,(w, t ) =

x (t

+

i)

x* ( t -

i)

e-jw7dz.

Its interest is connected with some geometric properties in the time-frequency plane, which enables localization of various signals, particularly of linearly frequency-modulated signals (Mecklenbrauker, 1993). There also exist generalizations of the WD. One of these is the Cohen class of distributions (Cohen, 1989). These distributions can be used to classify signals from the point of view of complexity and measure of information (Orr, 1991; Baraniuk et al., 1993). 6. Wavelet Transforms Another localizing analysis, which has gained much attention recently, is wavelet analysis. This is basically a generalization of the Fourier mixed timefrequency analysis described above. The main differences are replacing the frequency by a scale variable, and the basic sinusoid by another suitably chosen signal g(t), called a wavelet. The signal is projected on a family of functions that are derived from g(t) by time scaling and shifting. The wavelet transform (WT) is defined as

Wavelet analysis was first introduced by Grossmann and Morlet (1984), and developed by Daubechies et al. (1986), Daubechies (1988, 1990), and Mallat (1989a, 1989b). It is now recognized as a powerful tool in signal processing and has a rapidly growing literature, from among which we mention Combes et al. (1990); Meyer (1991); Special issue (1992); Schumaker and Webb (1993). In STFT, Gabor, and similar transforms there is a fundamental problem: The fact that the window length is constant makes it impossible to represent both low- and high-frequency components equally well. We have an uncertainty-principle issue. Short windows contain enough high-frequency cycles to localize their spectrum well, but spectra of low-frequency components are not accurately determined. Larger windows can do this, but then the highfrequency components are not well localized in time. What we need is a window whose size adapts itself to the frequency contents of the signal. This is precisely what the wavelet transform does: The scaling variable allows dilating or shrinking the effective width according to the frequency component being analyzed.

322

ALEXANDER M. ZAYEZDNY A N D ILAN DRUCKMANN

7. Description b y Equivalent Linear Time-Invariant Discrete Systems

A very fruitful concept in signal theory is the description of a signal by an equivalent system (mostly linear). The signal is viewed as the output of a linear time-invariant system, and therefore, instead of the signal, we describe the system and the input signal. This method is useful if the system and the input are more simple to characterize than the output signal. The linear system can be described by its impulse response, or by the corresponding transfer function. The most widely used models of this kind are discrete linear systems of the forms all-poles, called autoregressive (AR); all-zeros, called moving-average (MA); and poles-zeros, called ARMA. The input to these systems is in general considered to be a white stochastic process or a delta sequence. Both these inputs have a flat power spectrum. In fact, the procedures for determining the parameters of these models are based on flattening (whitening) the spectrum of the assumed input, and minimizing its energy. These models, especially AR and ARMA, have been very popular for 20 years, and have applications in many areas: speech processing, recognition, and synthesis; seismic data processing; biomedical signals; spectral estimation; data compression; etc. They have been thoroughly investigated and have become part of many textbooks on signal processing. We shall emphasize two conceptual aspects of the models. (a) They may reflect an actual physical mechanism generating the signal, as in the case of speech signals. (b) They offer a possibility of separating the signal into two parts: a known part, or deterministic, described as the system, whose parameters can be determined; and an unknown part, described as a stochastic process-the input to the system. The interaction between the two parts forms an approximation to the real signal. These remarks can be applied to any model consisting of a deterministic system and a stochastic input. There are cases where the linear time-invariant system is not a good enough approximation. Two directions of generalization exist: time-varying AR models and nonlinear systems. 8. Higher-Order Spectra

One of the most-used tools in signal description is the power spectral density (PSD) or power spectrum (Papoulis, 1977; Kay, 1988). It is defined as the Fourier transform (FT) of the autocorrelation of the signal (instead of the FT of the signal), and in practice is used especially for discrete signals. The motivation for this definition and also for the popularity of the tool is the


323

fact that it can be defined for both deterministic and random processes, and therefore Fourier analysis can be extended to random processes. Given a stationary random process x and a deterministic one y , we define autocorrelation, respectively, as r,[k] = E { x [ n ] x [ n

+ k]},

rJk]

=

lim

N

+

1

2N ~

~

c YCnlYCn + kl9 N

+1

n = - ~

(12) and the power spectrum, for both cases, as m

P(w) =

1

k=-m

r [ k ] exp(-jwk),

(wI In.

(13)

The description of signals and their processing using these tools (autocorrelation, PSD) have some limitations (e.g., loss of information about phase, or about harmonic relationship between spectral components). These have caused the evolution, in recent years, of more powerful extensions to the PSD: these are the higher-order spectra (HOS) or polyspectra (Brillinger and Rosenblatt, 1967; Nikias and Raghuveer, 1987; Mendel, 1991). These are based on the cumulants of the process, in the same manner that PSD is based on the autocorrelation. The joint cumulants of a set of random variables (r.v.’s) are defined in terms of derivatives of the logarithmic characteristic function of the r.v.3 (Papoulis, 1984). They can be expressed in terms of the moments. In the case of stationary processes, a cumulant is a function of several delays between the samples. For example, the cumulants of order 2 and 3 are identical, respectively, with the moments of the same order. Given a stationary random process with zero mean, the second-order cumulant is the autocorrelation, and the third-order cumulant is c 3 [ k , , k z l = m3[k1, k z l = E { x [ n l x C n + k l l x C n

+4 1 } .

(14)

The fourth-order cumulant is not identical with the fourth moment m4, but is given by

The spectrum of order N is defined (Nikias and Raghuveer, 1987) as

CN(%,%, ...l w N - l )

324


For example, the third-order cumulant spectrum or bispectrum is defined as

c F m

Ho,,WZ) = k,=-m

k2=-m

c,Ck,, k z l exp{-j(o,k,

+ wzkz)).

(17)

HOS have several properties that make them useful: The higher-order cumulants (HOC, N 2) are zero for Gaussian processes, therefore higherorder spectra can serve as a measure of gaussianity; if a group of random variables can be divided into mutually independent subgroups, HOC are zero, therefore HOS provide a criterion of statistical independence; if a component frequency of an output process is harmonically related to other frequencies, HOS can distinguish if this is due to a nonlinearly coupling or these are independent components, etc. HOS techniques have been used for identification of nonlinear and non-minimum-phase processes, speech and biomedical signals processing, passive sonar, source separation, blind deconvolution, etc (Matsuoka and Ulrych, 1984; Raghuveer and Nikias, 1986; Nikias and Raghuveer, 1987; Giannakis and Mendel, 1989; Mendel, 1991; Lacoume, 1991; Lacoume et al., 1992). The following two methods are structural properties methods.

=-

9. Representation by State Functions and Generating Di#erential Equations In this method the given signal x ( t ) is transformed into a generating differential equation (GDE), which has one partial solution identically equal to x(t). It is important to emphasize that for all twice-differentiable, signals, there exists a GDE (see Section 11.C): x

+ a , ( t ) i + a,(t)x = 0,

(18)

which can be termed the canonical linear form GDE. Equation (18), together with appropriate initial conditions, is the image of x ( t ) . For the method of differential transforms we created a special mathematical apparatus by using nonlinear relationships between derivatives in the form of new variables (state functions). The basis of this method, which is called the method of structural properties, (Zayezdny and Druckmann, 1991) is presented in Section 11. 10. Description by Generalized Phase Planes

According to this method, the signal x ( t ) is represented by the image u(u), where u and u are any two of the above-mentioned state functions, viewed as operators applied on the signal x. As a special case, when the operators are u = 1, u = d/dt, the image is k(x)-the Poincare phase plane. In the general case, the curves u(u), where u = u [ x ( t ) ] , u = u [ x ( t ) ] , are called generating phase trajectories. Signal processing is performed on these curves. The theory


325

and practice of using generating phase trajectories are presented in Section 111.

Now we shall turn to criteria for comparing different methods of signal descriptions, from the point of view of their efficiency for signal processing. C . Criteria for Comparing Different Methods of Signal Description

The problem of signal description is an essentially complex problem, and the advantages of different methods can be estimated only by reference to a number of qualitative criteria, or characteristics. Therefore, we cannot give an exhaustive and definite answer to the question of which method is the better one in the context of a real problem of analysis, and, what is more important, of synthesis. At the same time, however, we present below a number of criteria which are helpful for selecting the most preferable method from those listed in Section I.B. We discuss here only the main criteria; other criteria, such as physical size of devices, computational complexity, and cost, are not addressed here. We propose the following criteria for comparing signal description methods. 1. Constructability Criterion

The meaning of the constructability criterion is the possibility to realize (to construct) a device directly from formulas of description. For example, a description by infinite series or polynomials is less preferable than a description by closed-form formulas, which are accessible for realization. By accessibility for realization we mean the possibility to obtain a working algorithm or block diagram. 2. Technologicality and Universality of Realization By technologicality we mean the possibility of constructing the block diagram of the algorithm or system into elementary blocks of as small a number of types as possible. Universality means that these elementary blocks or elements can be used in constructing systems for a wide spectrum of applications. For example, the malfunction diagnosis system described in Section 1V.G is made up of a great number of only one type of universal elementdissipantors (see Section II.C, list 3). 3. “Znformation per Symbol” as a Characteristic of Description

The most powerful methods of description are those which make use of complex concepts, i.e., concepts which combine simpler ones. For example, the “vector” concept is a more powerful tool than a “scalar,” and the

326


“matrix” concept, as a generalization of “vector,” is still more powerful. We say that the matrix and vector symbols are more informative. Another example is complex variables used as the “phasor” symbol to represent sinusoidal signals.

4. Transparency of Description High transparency of description means that the parameters associated with the description have an evident physical meaning, and give insight into the essence of the process. For example, according to this criterion, description by polynomials is less transparent than description by closed-form formulas, which reflect clear physical meaning.

5. Universality of Description Universality of description tools means that these tools can be easily adapted to general classes of signals or systems. For example, autoregressive (AR) models are suited for multisinusoidal-multiexponential signals, or equivalently, for linear systems. Related to these are the large class of Prony methods for estimating parameters of multiple sinusoids. Superior to these from the point of view of universality is the method of wavelet analysis, which is suited to describe and analyze arbitrary signals. Also more universal is the method of structural properties proposed in this chapter, which provides the means (generating differential equations) to describe arbitrary signals. 6. Additional Characteristics of Description Methods

Different description methods or different models have various influences on the characteristics of the real systems or devices that were implemented on the basis of these methods. There are even situations when models which are mathematically equivalent produce real systems that are widely different from the point of view of stability or immunity to noise, for example. Therefore, the choice of the description method may have a decisive importance on the characteristics of the real, synthesized system. We give some examples of these real system characteristics, which may serve as criteria for comparison among description methods. 1. Computational precision of the signal processing operations. Related to this is the sensitivity to the quantization of system parameters. 2. Sensitivity of output parameters with respect to stability of system elements, external conditions, or irrelevant system parameters. Is it possible to minimize the influence of some parameters by creating a system which is invariant to them? See Section 1I.D.


327

3. Resolution of measurement (of parameters): Does the description method have the possibility of increasing the resolution? See Section IV.B.2. 4. The frequency range and the dynamic range of the real system.

It is not possible to assess the relative importance of each criterion exactly, and moreover, this depends on the specific application. However, the list of criteria we have presented may serve as a guideline for comparing the description methods and for selecting the most advantageous.

11. THEORETICAL BASISOF THE STRUCTURAL PROPERTIES METHOD A . Basic Definitions and Terminology

When in Section 1.B we reviewed the signal description methods, we mentioned the “method of structural properties.” The main idea of this method is that, for describing a signal, we use a transformation from the original x ( t ) to a generating differential equation (GDE). This GDE is the object on which signal processing is performed, in accordance with the requirements of the specific problem. We mentioned there that, to this end, a special mathematical apparatus was created; the basis of this apparatus is described in this section. The main feature of this special apparatus consists of changing the variable x into a related variable which is more informative. This approach enables us, by essentially increasing the information per symbol, to modify the canonical form of differential equations, in the interests of signal processing. For this description it is necessary to introduce some new definitions and concepts, to which this section is dedicated (Zayezdny, 1967; Zayezdny and Druckmann, 1991). In addition to the function x(t),we shall consider the current variation of x(t),

+

AX(^) = ~ ( tAt) - ~ ( t ) ,

and the relative current variation,

This function is called the relative function or the chi function. By means of this function we obtain an increase of information per symbol, relative to signal x(t).

328


Further, we introduce the velocity of the chi function, defined as follows:

or S,(t) =

dx/dt i -x(t) x

[s-’1.

~

We shall call this function the dissipunt of the function (signal) x(t). The reason for this terminology will be given below. The new function 6,(t) can be used to express a linear homogeneous differential equation of first order and with variable coefficient in a more concise form: dx dt

_-

6,(t)x = 0,

x(0) = xg.

Equation (21) represents a differential image of x ( t ) and can be used as an object of signal processing instead of the signal x itself. With respect to the velocity of function y ( t ) = i ( t ) , we can introduce the dissipant of velocity,

j x 6,(t) = - = y Y

cs-’1,

X

which corresponds to a homogeneous linear differential equation of second order: x - 6,(t)2 = 0,

x ( 0 ) = xo. (23) We introduce also the product of the dissipants 6,6, = K,, termed the conservant of x , and their ratio 6,/6, = I,, termed the condiunt of x : i(0) = lo,

i x

x

K , ( t ) = 6,6, = - -

=-

x - x,(t)x

= 0,

x l

I&)

6 6,

Cs-’]

x

xx il

7

(25)

xx g2’

=2=-- =-

2 - I&)-

l2 = 0. X

Each of the new functions A, S,, x, I , contains in one symbol not only information about one point as in the signal x(t), but also information about


329

current variation of x , its relationship to the value of the signal, about its velocity and acceleration, i.e., information about the state of a point. For this reason we call these functions state functions. These four state functions are called elementary state functions because the solutions of their corresponding differential equations with constant coefficients are functions which are commonly called “elementary functions”: the exponential function e”’, trigonometric and hyperbolic functions sin at, sinh at, . ..,power functions ( t + a)”. We once more draw attention to the fact that, in the structural properties method, differential equations are modified by using new variables, nonlinear combinations of derivatives, which are themselves functions of time. In the following sections we shall develop this subject. B. Elementary State Functions Definition.

A state function (SF) of a signal x is an expression of the form u(t) = u ( L , x , L 2 x , ...) L,x),

(28)

where L , , L 2 , . . . are arbitrary operators. The operators commonly used in the structural properties method are the differentiator and integrator operators. Other operators may also be used occasionally, e.g., Hilbert transform, raising to power, logarithmic, etc. In the last section we introduced four elementary state functionsformulas (20), (22), (24), and (26); these functions play a central role in the structural properties method. Each elementary function is characterized by its physical meaning and some properties. Nonelementary state functions will be considered in the next section. 1. The Dissipant of Function x ( t ) For the interpretation of the physical meaning of S,, and the reason for its name, we must turn to the canonical GDE (18), rewriting it after dividing by x in the following form: K,

+ al(t)Sx+ a&)

= 0.

(29)

The second term of this expression is connected with energy dissipation, and in the theory of differential equations is called the dissipative term [or resistance term (e.g., Boyce and diPrima, 1969, p. 417)]. Consequently, S, will be called the dissipant. Another way to see this is the following. Let us define x 2 ( t )as the instantaneous power of signal x, and let us compute the relative variation rate of this

330


power:

We see that S, expresses the relative rate of variation of P . Namely, if 6, < 0, then PIP decreases, i.e., energy dissipation occurs; if 8, = 0, then there is no instantaneous change of PIP; and if 6, > 0, then PIP increases. We note that when x > 0, the dissipant equals the logarithmic derivative, i d 6, = - = - In x = in' x , x dt

x > 0.

Clearly, the dissipant generalizes the definition of the logarithmic derivative, because the dissipant is not subject to the limitation x > 0. The physical meaning of the dissipant follows directly from its definition: dxlx &(t) = dt

Cs-'l,

and is read as: the dissipant equals the velocity of relative variations of the function x(t). Another interpretation follows from the fact that 1,the velocity of the current variations of x(t), equals the product of x itself and the dissipant (velocity of relative variations), 1 = xs,,

(30)

and by analogy,

(3 1) In the Poincare phase plane 1 = f ( x ) , the dissipant equals the slope of the tangent to the phase trajectory. In the conventional methods of signal description we use primary notions: the point of function x ( t ) and its velocity 1(t).In the method of structural properties we introduce new notions: the point of relative function (AxIx)( t ) and the velocity of its relative variations S,(t). This approach requires a new form of thinking, because the velocity of relative variations is a notion of deeper insight-contains more information-than the velocity of the process itself. This reorientation of thought needs some training, similar to that required for the transition from time representation to spectral representation of signals. f = xSxSy.

2. The Dissipant of the Velocity of Function x ( t )

The physical meaning of the dissipant of velocity S,, y = 1,is identical to the meaning of S,, but replacing x with y. Here we must warn against confusion


33 1

with the concept of acceleration, when considering relative variations. The dissipant 6, is not acceleration of relative variations, and it has the dimension [s-'1. It expresses velocity of variations of i ,but not relative variations. The velocity of velocity of relative variations, i.e., velocity of S,, equals 8,:

or, by Eq. (241,

8,

= SX(6, - 6),

cs-21. (33) We can see that the acceleration of relative variations is defined by (32) and has dimension [s-']. 3. The Conservant of Function x ( t )

The values of the conservant can serve as a measure of the closeness of a given process to a conservative process. According to (32), the following relationship exists between K, and 6,: K, = 6:

+ 8,.

(34) This equation gives the possibility to reduce the order of some differential equations. Another interpretation of the conservant is as a basic operator for the solution of a general linear differential equations of order I I ; that is, using this operator and its inverse (in analogy to the differentiator and integrator operators) enables us to express the solution of such an equation analytically. Let us consider the conservant as an operator K , and define its inverse K-', by analogy with the differentiation and its inverse, the integration operator: x

K [ x ( t ) ] = - = KX(t) X

(354

K-': K ( t ) + x ( t ) (35W By introducing these definitions, it is possible to express the solution of a general second-order linear differential equation, as explained in the following theorem.

Theorem. Given a linear homogeneous DE of order IZ,

R

+ p(t)i + q(t)x = 0,

(36)

one of its general solutions is (one solution is sufficient to find both general solutions)

332


To solve a linear DE of order I1 we need a table of anti-kappa, in addition to a table of integrals (anti-derivatives).

4. The Condiant of Function x ( t ) The condiant has the physical meaning of expressing in a succint way the behavior of the power function. By definition, the condiant is the ratio of the velocity of relative variations of the function’s derivative to the velocity of the relative variation of the function itself. When I, is a constant q, then the corresponding process is a power function: x = ( t a)”(’-@, q z 1.

+

5 . Inversion Formulas We close this section by giving the formulas for computing the original signal x ( t ) when one of the elementary state functions is known. The derivation of these formulas follows from solving the appropriate differential equation, associated with each elementary state function (21), (23), (25), (27). Dissipant 6, i= 8,(t)x,

dx

-=

X

6,(t) dt,

In x =

s

6,(t) dt

+ c, (38)

x = c exp[jd,(t) d t ] .

Dissipant 6,

3 = 6,(t)Y,

dy Y

-=

dy(t)dt,

y

= c1 e x p [

j 6 J t ) dt],

x = s y ( z ) dz,

(39) x ( t ) = c1 sexp[

/‘s,,(e)

de] d s

+ c2.

Conservant K,. Given x = K,(t)x, we make the transition from K , to 8, by (20) and obtain the Riccati equation:

8, + 6: - KX(t)= 0.

(40) Solving this equation numerically gives d,(t), and from this we can obtain X(t) by (38). Condiant I ,

SIGNAL DESCRIPTION NEW APPROACHES, NEW RESULTS

333

C . Nonelementary State Functions

In addition to elementary state functions, which were described in detail in the preceding subsection, nonelementary state functions are also widely used in the method of structural properties. A great variety of nonelementary state functions may be defined, but in the following we shall present only the most widespread, expressed in the form of basic combinations of elementary state functions (SF) of the signal x and some other functions (e.g., i). The symbols of SFs can be viewed as operators applied on signals. We give below formulas for nonelementary SF, ordered in groups according to the operation involved. The derivation of these formulas is straightforward and consequently is not given here. These formulas are useful for the design of block diagrams and transformations between state functions. In the following the symbols u, u, w designate arbitrary functions, and x designates a combination of u, u, .. . , e.g., x ( t ) = u(t)u(t).The notation &(u, u) is read as follows: the dissipant of the signal (process) x, which is the product of functions u and u. 1. Separation of product (division) of functions under the application of the elementary SF operators:

6,(uu) = (UUY ~

uu

6,

(:>

- = 6, -

K,(UU) =

(uu)" ~

uu

= 6(u)

+ 6(u) = 6, + 6,,

a,, = K,

+ K , + 26,6,,

334


3. Formulas for expressing different state functions by means of only the dissipant S, and the original function x:

s,

= 6, = S(XS,),

Sj

= 6, = 6 i: = S ( i )

( :)

sxsy = 6,6(X6,), = K y = s,sj = [s(Xs,)]2

(444

+ S(6,) = S(x6,) + S[S(xS,)],

(444

K, =

Ki

(44b)

+ s(Xsx)d[6(X6,)],

(444

The formulas of this group give the possibility to generate all state functions by using only one element, the dissipantor, whose output is the dissipant of its input (see also Section I.C.2).


335

4. Formulas for expressing the dissipant of elementary state functions and of their derivatives: 6(6,)

(:>

=6 -

6(6,) = 6

= s, - 6, = d(X6,) - ,s,

(3

- s,

- = 6,

b(K,) = 6(6,6,) = 6(6,)

6(A,)

(45d

(3

(4W

+ 6(6,) = 6, - ,s,

= 6 - = 6(6,) - 6(6,) = 6,

+ 6,

(454 - 24,

C ~ , W x )=l 6(4) + SC~(6,)1, 6@,) = m,w,)l= W,) + W~,)l> W,) = = 6 ( K , ) + 6C6(K,)l, S ( L ) = “,W,)l = W,) + sCW,)l, S(&)

=~

4

~

X

W

X

)

I

(454 (454 (450 (45d (434

5. Formulas for expressing differences of elementary state functions and dissipants of their differences:

We have given above the most common nonelementary state functions; there are many ways to generate others. For example, we can add to the list l/u, where u is an elementary function, or different functional transformations. We mention here that a similar apparatus of state functions for representation of signals has been developed by Maizus, who calls them differential operators or S(D)operators (Maizus, 1989, 1992, 1993; Grinberg et al., 1989).

336


D . Generating Differential Equations

We have already seen that one of the basic tools of the method of signal description by structural properties is the state functions, which are nonlinear combinations of the signal and its derivatives. These state functions serve as images of the original signal for the purpose of signal processing. Another basic tool of the method of structural properties is the generating differential eguation (GDE). Definition I : Generating Diflerential Equation (GDE). Let x ( t ) be a signal. A differential equation, of which x is the solution (with given appropriate initial conditions), is called a generating differential equation (GDE) of x.

A general form of a GDE is F ( x , 2,X,...; a, b, C, ...; t ) = 0,

GDE:

(47)

where a, b, c, . . . are parameters contained in the signal x(t). In addition to this usual form, we can rewrite any GDE in a new form, using state functions instead of the variable x. Such equations are called differential equations of states. Definition 2: Differential equation of states (DES). A differential equation of state (DES) is an equation of the form

DES: F[dl(t), d2(t)7 where dl(t), dz(t),. . . are state functions.

* . . I=

(48)

O7

Examples. Differential equations (DE) versus DES:

DE x

+ q ( t ) 2 + a,(t)x = 0

... x+

DES Ic,

+ a,(t)6, + a&)

Ky

+ 26; + K , = 0

x2

55 27+-=0 x

x

=0

We should pay attention to the difference between the use of differential equations (DE) here and in the classical context. In the theory of differential equations the problem consists of finding the solution of a given DE, i.e., an analysis problem. In the signal processing context the signal is given, and the problem consists of constructing a DE corresponding to this signal; i.e., our problem is the synthesis of a DE. Methods for the synthesis of GDEs will be considered below. A GDE of the form (47) contains parameters a, b, . .. . There are instances when it is useful to consider GDEs containing one parameter only, or no parameters at all. Therefore we define the following.


337

Definition 3: Ensemble GDE (EGDE). Let { x ( t ; a, b, c, . . .)> be a family of signals dependent on the parameters a, b, c, . .. . If there exists a GDE of the form

GDE:

F ( x ,i, I , ...;t ) = 0,

(49)

i.e., independent of a, b, c, .. .,then it is called an ensemble GDE of the signal family (signal ensemble) x.

A GDE which is not an ensemble GDE is called a partial GDE. A partial GDE may contain all the parameters a, b, c, . . ., or some of them, e.g., only parameter a: GDE:

F,(x, i, -it,

. . . ; a; t ) = 0.

(50)

Now we shall present a procedure by which we can decrease the number of parameters on which a partial GDE is dependent until, ultimately, an ensemble GDE is obtained. Let us assume a GDE of the form (47). By differentiatingwith respect to t, we obtain a new equation, GDE:

G(x, i, X,

. . .; a, b, C , . . .; t ) = 0.

(51)

Then we can eliminate one of the parameters between the two equations. Let us assume (without loss of generality)that the eliminated parameter is a. The result will be a GDE independent of parameter a, GDE:

F1(x,i,X , . . .; b, C, . . .; t ) = 0.

(52) By performing this step a sufficient number of times and eliminating all the parameters one by one, we shall ultimately reach a stage in which the resulting equation will be an ensemble GDE.

Example. For a sinusoidal signal x = b cos(wt

+ O), we have

+ wZx = 0, GDE(x, X; b) = X(b2 - x’) + i Z x = 0,

GDE(x, 2 ; W ) = X

EGDE(x, i, R, X) = X i - XX = 0.

(53) (54) (55)

Obtaining a GDE which contains only one parameter can be of interest in the following situation. Suppose we want to use a GDE to measure a certain parameter a. This is done by solving the GDE expression (viewed as an algebraic equation) for a. Then the measuring algorithm is u = F,(x, 1, X , ...; b, C, ...; t).

(56)

This expression depends in general on other parameters b, c, . . . . If we make use of a GDE which contains only a, we obtain a measuring algorithm which depends only on x and its derivatives, i.e., is invariant with respect to other parameters.

338


In the procedure for parameter elimination that we presented above, we used differentiation of the original GDE in order to generate new GDEs. In general, we can also use other transformations, e.g., Hilbert transforms (HT), to produce new GDEs. In this case, the resulting GDE will include x, its HT, and their derivatives. In the following we give a list of examples of signals, their GDEs, and DESs, including ensemble equations. Note the fact that the formulation by means of state functions, i.e., in the form of DES, is more concise. Example 1. x = eat

DES

GDE i- ax = 0

Partial:

6, = u

Ensemble: f x - k2 = 0, Example 2. x = cos(ot

(;)

=o,

=o

+ cp)

GDE Partial:

x

6(6,) = 0

DES

+ 02x=0

Ensemble: xi - Yx

K, = --o

=0

2

?I, = 0,

6,

K y - K, = 0 - 6, = 0, 6(K,) = 0

Example 3. x = t P ,p # 0, 1

GDE

DES

x2

Ensemble: Y - 2T x Example 4. x

= exp

+ xx i = 0 -

1, = 0,

(- $)

GDE Partial

6(6,,) - S(S,) = 0

DES

+

a,=

i Ptx = 0 x i

Ensemble: W - 3 - + 2 - = 0 X

-Pt

23

X2

&=O,

(K,--;),=o


Example 5. x = e-”‘ cos(ot

+ q) DES

GDE Partial Ensemble:

339

x + 2ai

+ (w’ + a 2 ) x = 0 xx2 + X P X . . . xi” + X x2ix + i3

Ic,

+ 2 4 + w2 + a2 = 0

8 ( K y - Ic,) - S(S, - 8,) = 0

Given a signal x, there exists an infinite number of corresponding GDEs. Of course, most of them are more complicated in structure than x itself, and hence of no use to us. Our aim is to construct GDEs that have simple coefficients (constant, polynomial, etc.), and in any case, coefficients that have a simpler structure than x itself. These GDEs may be linear or nonlinear. We shall present in the following sections some methods to construct GDEs. E . A Second-Order Linear Generating Dgerential Equation Based on the Amplitude-Phase (A-Psi) Representation of a Signal 1. The A-Psi Type of GDE

It is well known that any analytical signal can be represented in the following form (Papoulis, 1984): x ( t ) = A(t) cos $(t),

(57)

where A ( t ) is the envelope (or the instantaneous amplitude) and $ ( t ) is the phase, and they are given by A(t) =

Jm,

(58)

where 2(t) denotes the Hilbert transform (or conjugate process) of x(t). We also define the instantaneous angular frequency w(t): w(t)= $(t).

(60)

We shall set out from this representation to synthesize a second-order linear GDE and a corresponding DES.

340


Theorem 1. Given a twice-differentiable signal x(t),for which an amplitudephase representation (57)-(59) exists, then it has a GDE of the form

x - (26, + 6,)1 +

[w2

+ 6,(26, + 6,)

- KA]X =0

(61)

or K, - (26,

+ 6,)6, + [a2+ 6,(26, + S,)

=0

(DES), (62) where 6, = A / A is the dissipant of amplitude, 6, = ci)/w is the dissipant of frequency, and K , = A / A is the conservant of amplitude. - KA]

Proof: We differentiate twice, then we eliminate cos $ and sin $ between the three equations for x, 1, x. We obtain (61) (Zayezdny and Druckmann, 0 1991). This type of GDE is a special case of the canonical form 2

+ a l(t)i + a,(t)x = 0.

(63) We shall call it the A-Psi equation type. This DE is a generating equation for any oscillatory signal modulated by both amplitude and frequency. It is possible to simplify the A-Psi GDE (61) by using, instead of the variable x, its normalized counterpart x/A; after performing this substitution we get the following equivalent equation:

The equivalence between (64) and (61) is easily proven by expanding the expressions rc(x/A),6 ( x / A )by means of Eq. (42),Section 1I.B. Equation (64) can be written in yet another equivalent form, by using the identity K , = 6,,6,. We obtain

2. Special Cases of the A-Psi GDE Type: A M and FM Signals

We shall now derive from (61) special forms of GDE for two cases: an amplitude-modulated (AM) signal and a frequency-modulated(FM) one. For an AM signal we have w = w, = const., hence 6, = 0, and therefore (61) becomes K, - 26A6,

+ (mi + 26;

- KA) = 0

(AM).

(66)

For example, if A = Aoe-", then 8, = -a, K , = -a2, and we obtain the well-known equation of a dissipative oscillatory circuit: K,

+ 2a6, + (w," + a') = 0.


34 1

In the case of an FM signal, we have A = const., hence 6, = 0, IC, = 0, and (61) takes the simple form: IC, - Sodx

+ w2 = 0

(FM).

(67)

This equation, if written in the usual form (not with 6, IC),has been used by Hess (1966). 3. Separate Equations for A(t) and o ( t )

In the following we shall derive GDEs for A ( t ) only, and GDEs for $ ( t ) (or w ) only, under the condition of conservativeness, and also in the general case. The resulting equations are encountered in some areas, of circuit synthesis. The Case of a Conservative Process. For a conservative process, the dissipative coefficient a1 in (61) is zero, a l ( t ) = 26,

+ 6,

= 0,

(68)

and Eq. (61) therefore becomes x

+ a,(t)x = 0,

IC,

+ a,(t) = 0.

(69)

Equation (68) is called the condition of adiabatic invariance.

Theorem 2. Given a twice-dgerentiable, conservative process x(t), its instantaneous amplitude A ( t ) satisfies the GDE

.. c A - 7- ao(t)A = 0, A and its instantaneous Pequency satisfies the GDE 1

ZK,

-

-t(3' = ao(t).

In Eq. (70), c is a constant which equals c = const. = x 2 - i 2 .

(72)

Proof: Substituting the condition (68) in (62), we get

x, - KA t'm2 = 0.

(73)

In this equation we identify the coefficient a,(t): a, =

-KA

+0 . 2

(74)

Now we shall express w in terms of A, by using the relationship provided by (68):

342

26,


+ 6,

dA 2A

= 0,

+ dww = const.,

In A'

-

w=-

+ In w = const.,

A 2 w = c,

C

(75)

A2'

Substituting (75) in (74) gives the desired Eq. (70). Now we shall prove (71), by expressing (74) in terms of w. First we make the following substitution in (74): KA

=8A

+ 6i.

Then we express 6, in terms of d, [from (68)]: dA -- _ - id,.

(77)

By performing these substitutions in (74), we obtain (71). Finally, to find the constant c in (75), we express w by differentiating the phase in (59), and we have

The Case of a General (Nonconseruative)Process

Theorem 3. Given a general process satisfying the second-order linear GDE (63), then its instantaneous amplitude A ( t ) satisfies the GDE K,

+ a,6,

-c

exp[-2S(al

+ 26,)

dt] = -ao,

(78)

and its instantaneous frequency satisfies the GDE

+ 02

= a, - +a: - "4 2 1'

(79) Proof; This follows the same lines as the preceding proof. We have two equations, in which a , ( t ) and ao(t) are given input functions, and A, w are unknown variables: + K ~

First we use the equation for a l ( t )from above to express w in terms of A. We have

dm = -26,

- a,,

dw

-=

dt

(-26,

-

a,)w,

w2 = c exp[-2J(al

+ 26,)

dt],

and then, substituting this in the equation for ao(t),we obtain (78). Alternatively, we can use the equation for a l ( t )to express A in terms of w, and then substitute this in the equation for ao(t) to obtain (78).


343

4. The A-Psi G D E with Coefficients Expressed in Terms of x , 2 , i , . . . Until now we have expressed the coefficients of the A-Psi GDE and of all related equations in terms of A , w. We shall give here the same equations expressed in terms of x, 2 and their derivatives. Note that these are not merely equivalent equations, but exactly the same ones, only with coenicients expressed differently. The new forms are derived from the old ones simply by making use of Eqs. (58) and (59). These equations enable us to express A, w , and their derivatives in terms of x, i?, and their derivatives. Carrying out the calculations, we obtain the A-Psi equation (61) expressed as (Zayezdny and Druckmann, 1991) i j - xi XZ - x i a, = a , =(81) Xi? - i i ? . X i - 22’ Similarly, the condition of adiabatic invariance, i.e., the condition for a process being conservative, is expressed as [see (68)] x2 - X Z = 0,

or

K , - K~

= 0.

(82)

F . The Linear Second-Order Generating Differential Equation with Three Coefficients, and Some Generalizations

In the preceding section we discussed a special kind of linear GDE, the A-Psi GDE, which is one type of the more general class (which we repeat here for convenience)

x + a l ( t ) i + a,(t)x = 0.

(63) It is indeed customary in the field of differential equations to describe second-order linear homogeneous DEs in this form (the canonical form), which includes two Coefficients. Let us consider a more general form, containing three coefficients: b,(t)x

+ b l ( t ) i + b,(t)x = 0.

(83) This equation can be readily converted to the canonical (two-coefficients) form by dividing by the leading coefficient:

It is customary to view (63) and (83) as the same equation. By this is meant that both equations have the same set of solutions. And indeed, in the field of DEs, the question of the solutions is the only matter of interest. In contrast, we have here other interests. In fact, we may say that the solution of the DE

344


is not a problem at all for us, because we know the solution a priori: This is our original signal x(t).Instead, we are concerned with the form of the GDE, or, more explicitly, with the form of the coefficients. We are interested in certain properties of these coefficients, for example, their bandwidth. We shall show in this section that the coefficients of (63) and (83) have in general different bandwidths, although those equations are equivalent from the point of view of their solutions. Therefore, in the following, we shall view (63) and (83) as distinct GDEs. 1 . The A-Psi GDE in the Three-Coeficients Form

A three-coefficientsGDE equivalent to form (61) is A2wX - A(2kw

+ A & ) i + [ A 2 w 3 + k(2ko + A h ) - AAo]x

= 0. (85)

We can rewrite the equation by expressing its coefficients in terms of x, A, that is, b,(t)X b, = x i - i A ,

+ b l ( t ) l + bo(t)x= 0, bl = XA - x i ,

bo =

- XL

(86)

This GDE is the counterpart of (81). We emphasize that (85) and (86) are the same equation-that is, their respective coefficients are exactly the same functions, only expressed differently. We shall call this form the general (ternary)A-psi GDE, to distinguish it from the equivalent two-coefficients form (61) and (81), which will be termed the reduced A-Psi GDE. It is easily seen that the coefficients b,, b, of (86) [identical to those of ( 8 9 1 are related by the following relationship: b, = -b2.

(87)

The significance of this relationship is that the GDE (86) can be entirely specified by only the two coefficients (b,, bo). The remaining coefficient b,(t) can be easily reconstructed by (87). 2. Generalization of the A-Psi Type of GDE Theorem 4. Given a signal x ( t ) and an arbitrary transform y ( t ) of x(t), then the following is a GDE of x(t),for which the relationship b, = -6, holds:

+

b2(t)% b l ( t ) i b2(t) = x j - i y ,

+ bo(t)x = 0,

b,(t) = Xy - xjj,

b,(t) = ijj - Xj.

(88)


345

Proof. We construct the following determinant:

The determinant is identically zero because it has two equal rows. Expanding the determinant with respect to the first row, we obtain the GDE (88). The relationship b, = -8, is proven simply by carrying out the differentiation of b,, as given by (88). 0 For the special case when y = 3, we get (86).As an additional example, we take y = 1.We have then the GDE

The Condition for Conservativeness. From the condition b, = 0 and from (87) we have b, = const. (conservativeness) (91) This is true in general, for any GDE obtained by a procedure as described in the preceding theorem. When y is the Hilbert transform of x, we get (75)as a special case. 3. Equivalent GDEs May Have Coefficients with DiSferent Bandwidths

Now we shall show that the coefficients of (63) and (83) may have very different bandwidths. In fact, we shall show that there are cases when we have two equivalent GDEs, one with two coefficients and one with three, and the three coefficients have a total bandwidth that is less than the total bandwidth of the two coefficients.

Theorem 5. Given two independent low-pass processes A@), $(t), having bandwidths, respectively, of W,, W,, we form the modulated process x ( t ) given by (57).The three coefficients b,, b,, b,, of the GDE (of x) given by (86) are low-pass, and they have bandwidths of W, = W, = 2wA + W,, Wo = 2wA + 3w,. Proof. We shall consider the coefficients {b,} in the form given by (85) [as already mentioned, these are the same coefficients as in (86)].First we shall note, that if A is low-pass, with bandwidth W,, so are k,k: Similarly, if $ ( t ) is low-pass, with bandwidth W,, so are o,6.Then we recall that the Fourier spectrum of a product of functions is the convolution of their spectra, and if the functions have bounded spectra, their product also has a bounded

346


spectrum of bandwidth equal to the sum of their individual bandwidths (Bracewell, 1986). We therefore have b, = AZw

its bandwidth W, = 2 W,

W, = W,; b, = A2w3+ k(2kw + A h ) - AAw b, = -bz

+ W,;

-

W, = max(2W,

+ 3W,

= 2w* -k 3w,.

2 W,

+

(92) W,)

0

There is no such bandwidth-restricting theorem for the coefficients a, , a, of (63). Moreover, if A has zeros, then b, has zeros too, and hence a, and a,, having b, at the denominator according to (84), are unbounded. Therefore the bandwidth of a, and a, can even be infinite. As another simple example, consider the following differential equations, which are both GDEs of the Bessel function of order zero Jo(ot):

+ 1 + wZtx = 0,

(934

1 ++iw2x = 0. t

(93b)

tj;. f

The bandwidths of the coefficients of (93a) are zero (being polynomials). In contrast, the bandwidth of the coefficient al(t) of (93b) is infinite. 3. Other Classes of Linear Homogeneous Second-Order GDEs Equation (88) represents a quite general class of linear, homogeneous, second-order GDEs, of which the A-Psi GDE is a special case. However, there are many other classes of linear GDEs for the same signal x(t), and even more classes of nonlinear GDEs. We shall present now a new type of GDE, which illustrates two points: (a) This is a GDE which is not included in the class of Eq. (88). (b) The coefficients of different GDEs have different spectra, and therefore one kind of GDE is useful for a certain class of signals, while another kind of GDE is useful for another class of signals. Assume two low-pass processes O(t), $(t) of bandwidth W, W,, respectively, and define a(t) = 4,

w ( t ) = I).

(94)

Ce-O(')cos $ ( t )

(95)

Then the signal x(t) =

has the following GDE:

wx

+ (2ao - h ) i + (w3+ 20 - a h + iw)x = 0.

(96)


347

Note that this equation is not of the class of (88). For example, the relationship 6, = -b2, which was a property of the GDEs of class (88), does not hold here. Of course, the signal has also a GDE of the type A-Psi, whose coefficients may be computed according to (85), but these coefficients are less simple (in the sense that the have larger bandwidth) than the coefficients of Eq. (96). As an illustration, let us compare the leading coefficients of both equations. In Eq. (96) the coefficient is w and has bandwidth W,. On the other hand, the coefficient of the A-Psi GDE is b2(t)= A2w = C2e-2e(‘’w,

(97)

and it has in general an infinite bandwidth, because of the exponential term. In conclusion, there are signals for which the GDE (96) is more efficient, and others for which the A-Psi GDE is more efficient. “Eficient” means here “having simpler coefficients.” The signals of the first class are essentially of low-pass nature, and those of the second class are essentially bandpass. 111. GENERALIZED PHASE PLANES A. Definitions

In Section I1 we have discussed two kinds of tools for signal description: state functions and GDEs. These signal description tools have this in common, that they use instantaneous values of the signals, of their derivatives and combinations thereof. In addition to this approach, it is possible to introduce new functions, which express relationships between two state functions, so that the time t does not appear in explicit form. These functional relationships are represented as phase trajectories in the plane defined by the two state functions, which is called a phase plane. The simplest example of a phase plane is the well-known PoincarC phase plane x i , in which differential equations are represented as trajectories of the form i ( x ) (Andronov et al., 1966), and which has been in use for a century in the qualitative theory of differential equations. Generalizing, we define the following. Definition. Given a signal x(t),let u(t).v ( t ) be two state functions (as defined in Section 1I.B). Then the curve described by the relationship v = F(u) is called a trajectory in the generalized phase plane uv. In general, in order to obtain the expression of the trajectory, we must find the inverse function t(u), and substitute it in the expression for v(t). That is,

348


the trajectory is constructed as a composite function u[t(u)] = u(u).

(98)

The classical phase plane x i is widely used in the general oscillation theory and the theory of nonlinear differential equations, in control theory, and in other applications (Andronov et al., 1966), and we shall not discuss it in this chapter. Instead, we shall focus on the method of the generalized phase plane (GPP). We shall consider phase trajectories that correspond, for instance, to the following state functions of the signal x(t): 1. With argument x: dx(x),

d,(X),

K,(X),

&Ax), &(x);

2. With argument 6,: ~,(~,>,K A & J Y nX(d,),

8,(&),

46,) [Wx)I;

3. More complicated GPPs, for example:

w,- a),

Cd(KY - KAI,

etc.

We can expand this list of GPPs. We have therefore many different GPPs. We shall explain now why this is useful for signal processing. Consider the following problem, which we call the linearization of functions: Given a function x(t), defined on a bounded time interval, how can we transform it into a new function which is a straight line, without loss of information? The meaning of this is that if the given function has N parameters, and since a straight line has only two parameters, then the remaining N - 2 parameters must be expressed by initial conditions or boundary conditions of the straight line. In other words, the variables which are related by the linear function (straight line) contain in their expressions some derivatives of the original function, and hence, initial conditions. To date we do not have a solution in closed form to this problem. Therefore the only way toward a practical solution is to compose a dictionary (list) of the transformations which transform different functions to straight lines by using generalized phase coordinates. In Section 1II.B we shall present some examples of such transformations. Having found a phase plane in which the trajectory of our signal has a suitable form (linear or other), we can perform the signal processing operations on this trajectory. We shall show in the following sections that this method has its own advantages and properties, and therefore will receive progressive attention in the course of time.


349

Now we state the problems appearing when processing signals by means of phase trajectories. Let x(t, a, b) be a signal. Our task is to estimate its parameters or to use some other properties for signal identification.We must answer the following problems. 1. We must choose the phase coordinates u(t),v(t) in accordance with the conditions of the specific problem. In general, several possibilities are available, and we must choose the optimal. 2. For the given x ( t ) , we must compose the expression for the phase trajectory u(u). This can be done analytically or numerically. 3. ?he inverse problem: Given the phase trajectory v(u), we must recover the original signal x ( t ) . 4. We must create metrics for phase trajectories processing and measuring, that is, define measures and criteria appropriate for the specific problem. For example, we may define a decision law or an identification algorithm that is based on the staying of the trajectory in a definite region of the plane. We shall consider these matters in the following sections. B. Construction of Generalized Phase Planes for Given Signals

The first problem of signal processing by using generalized phase planes is to choose the phase plane according to the signals we must process, and according to the kind of problem we want to solve. We are interested in a plane, in which the trajectory will be simple, will emphasize features that are important for our problem, and will be, if possible, invariant to parameters that are not relevant for the problem. There is no general method for finding such a phase plane. We can get help from illustrated tables of phase planes and trajectories, computed for various families of signals. We shall show now some examples of how choosing an appropriate phase plane can simplify the trajectory and make it invariant to sume parameters. Additional examples can be found in Zayezdny and Shcholkunov (1976). Example 1. We are given the signal x = a sin@ + 6) and want to construct a phase plane in which the signal has a simple trajectory and it is convenient to measure the frequency w. It is well known that, in the classical phase plane x i , the trajectory of this signal is an ellipse (Andronov et al., 1966): x-+-2 i2 - 1, a’ a202

350


and the frequency can be computed by the algorithm w = Axis on i/Axis on x. However, by choosing another phase plane, we can obtain a simpler image. Consider the plane with coordinates v=x.

u=x,

The trajectory of the signal in this plane is a straight line:

u = -w%, (99) from whose slope the frequency can easily be computed. Note that this trajectory is invariant to a and 8. Example 2. Given the signal x = be-"' sin(wt + O), find a phase plane in which the trajectory is a straight line, for measuring the parameters u, w. As is well known, in the classical phase plane the trajectory is a spiral (Figs. la and lb). Now let us consider the state equation for this signal: K,

+ 2a6, + w2 + uz = 0.

We therefore choose the phase plane U = d,,

U = Kx,

and in this plane the trajectory of the signal is a straight line, namely, u = -2uu - m2 - u2.

( 100)

The slope and the intercept with the u axis provide us with the desired means of measuring the parameters (Fig. lc). Example 3. Given the signal x = b exp( -ut2/2), find a phase plane in which the trajectory is a straight line, for measuring the parameter u. This signal has the state equation K, - 62 + u = 0. We therefore choose the phase plane U

=a:,

V = K,,

and in this plane the trajectory of the signal is a straight line, namely, u=u-a.

The parameter is measured by means of the intercept with the v axis. Example 4. Consider the family of signals x = a sin w1t

+ b sin(w, t + 0).

Figures 2a-2d show the signals and the trajectories in the Poincark phase plane for o2= $ml and for two values of the phase difference 8. We see that the trajectories are quite complicated. Moreover, if the ratio between the frequencies is not a simple fraction, then the image is even more complicated.


a

1

1

I

I

1 0.5

'n -

351

-

'-

0 -

-0.5

-

1

0

I

I

I

5

10

15

20

'n

4r-------. --. \

2-

/-. o(

-2 2

(72)

)

1 ' 1 -0 5 -0.5

05

0 X

1

n

C

FIGURE1. 'The signal x = be-"' sin(ot + 0): (a) in the time domain; (b) in the Poincare phase plane x e (c) in the generalized phase plane S,, K,.

=

d 2

-1 -7

X2'

05

-

-0

-05

100

50

e

0~~

2

'\ 0

2

l

.... 5

0

X

1

5

0

5

KX

FIGURE 2. (a) A bisinusoidal signal in the time domain. o z / w , = 7/3.(b) The same signal in the Poincare phase plane x i . (c) A signal as in (a), but with another phase difference. (d). The same signal in the Poincark plane. (e) The two precedent signals in the generalized phase plane K,, Xiw/X.


353

Let us now consider the phase plane

In this plane the trajectory is the straight line (see Fig. 2e): u

+ (0:f 0l)u + 0:w;

= 0.

(102)

It remains a straight line for all signals of the family (but not the same straight line!). The position of the trajectory in the plane will change when varying the frequencies. Information about the frequencies can be recovered from the intersections with the axes. The trajectory is invariant to phase and amplitudes (however, knowledge of the time difference between two points on the trajectory allows the calculation of the amplitudes and phase). This image can serve to signal identification, or to malfunction diagnosis (for instance, appearance of higher frequencies would destroy the linearity of the trajectory). C. Reconstruction of the Original Signal from Given Phase Trajectories

During signal processing by using phase trajectories u(u), we may encounter the problem of reconstructing the original signal x(t) when the current phase coordinates u(t), u(t) are known, but the implicit time variable is unknown. In the particular case when u(t) = t and u(t) is one of the four elementary state functions a, S,,, K,, I,, this problem has already been considered in Section II.B, and there were obtained inversion formulas for the original signal x(t). Now we shall consider methods for solving this problem in the general case, and shall also give specific formulas for the case when the phase coordinates are the most simple nonelementary state functions. The solution of these problems is based on numerical methods, which involve piecewise linear approximation of a phase trajectory (Zayezdny, 1973, p. 434), and the method of isoclines. The latter is widely used in the theory of nonlinear oscillations (Andronov et al., 1966, p. 8). Given a segment of phase trajectory u = f(u), we want to find the function x(t). We do this in two main steps. 1. First we compute the inverse function t(u) [or t(u)],and consequently, u(t) Cml. 2. Then we compute x ( t ) from u(t). The first problem is special to the subject of this section, generalized phase planes. The second is the inverse problem of state functions, which has already been discussed in Section 11. In this section we shall focus on the first, and more dimcult, problem.

354


In the first problem we can distinguish two cases: (1) the function f(u) is known analytically, and (2) only the coordinates u,,, u, are known. The second case is more realistic, so it is this case that will be treated here. In this case we shall approximate the trajectory piecewise by N linear segments:

v 2 a,

+ b,u,

n

=

1, ..., N ,

tn-l I t I t,,,

to = 0,

(103)

each one connecting a point (u,,-~,u,,-~)to a point (u,,, u,). Each point corresponds to a time instant t,, not known. We have therefore to determine N time points (except to = 0). We shall do this by computing the interval At,, on each linear segment, and then t,,

= t,,-l

+ At,,,

n

=

1, ..., N .

We must also know that we have %Vn-1 - Un-lUn b,, = A v n - o n - on-1 a, = Au,, u,,- ~ , - 1 ufl - 4 - 1 In the following we shall consider specific phase trajectories. ~

1 . The Phase Trajectory y

9

(104)

= i = f ( x ) (Poincarb Plane)

Given a trajectory y = f (x), we approximate it piecewise by linear segments: dx y = - 2 a, dt

+ b,X,

t,,-l It It,,

to = 0.

(105)

For each segment we integrate the equation and obtain (the index of a, b is dropped for clarity):

Since from Eq. (104) we have

then

We can simplify the algorithm as follows:


355

and therefore

Axn At,, = __ F Yn-1

(2)'

In z F ( z ) = __ z - 1'

where

(108)

In this way we have obtained the desired function x ( t ) in the form of a set of pieces of the inverse function t = g(x). 2. The Phase Trajectory 6, = f ( x ) 1 dx 6, = - - E a, x dt

+ b,X,

t,-l 5 t 5 t ,

After using (109) and (104), we obtain

3. The Phase Trajectory 8, = f(6,)

The sequence of operations for deriving the algorithm for this case is similar to that of Subsection 1, except for the final stage: dd, dt

-= a i bh,,

...

We have obtained a sequence oft, corresponding to d,,, which enables us to find an approximation to &(t). Since 1 dx x dt

= - -,

then x(t) = c

exp[f 6&) d z ] .

356


4. The Phase Trajectory K, = f( x )

Since d2x dt2

x K = - = -

dy dt’

dx y=dt

and dY - d Y dx - dY

dt - zz - yz’ then a linear differential equation of second order,

2 + UI%

+ a,x

= 0,

can always be rewritten in a form which does not contain time t explicitly: dY --y dx

+ a , y + a,x

= 0,

or

dY

X

+ a, + a,- Y = 0. dx -

The solution of this equation gives the phase trajectory y = f(x), from which we can return to x = f ( t ) according to Subsection 1. We recommend to solve this equation by the method of isocline lines (Andronov et al., 1966, p. 8). (An isocline is the locus of points at which the tangent to all integral curves has the same dope.) Denoting dyldx = k, we obtain the expression of the isocline: X

k+a,+a,-=O, Y

a0 y = -____ x a, k

+

(Equation of isocline). (113)

As a rule, for approximate calculations, it is enough to use only four isoclines, which are called the main isoclines (Zayezdny, 1973, p. 382):

By marking on the isoclines the points corresponding to the given a,, a,, we obtain a field of vectors describing the phase trajectory i = f ( x ) , and from this curve we can make the transition to x ( t ) according to Subsection 1.


357

5 . The Phase Trajectory K , = f(6,) Using the equation

K, =

f(a,) we have

8,

+ 8,” and the piecewise linear approximation for 8,

+ 62 z a, + b,6,

and we get the differential equation with initial condition, in the variable 6,:

8,

+ 62 - b.6,

= a,,

6,(0)

(114)

=

Integrating this equation numerically, we obtain (z is the shifted time t - t,-l) 6,(t) = g(t.),

0 I5 IAt,.

We know the value of g(z) at the end of the segment: g(At,) = a,, and from this we calculate At,. Doing this for all the segments, we get the sequence { t , } corresponding to {d,), that is, we have an approximation of the function &.(t), and from this we can calculate x(t).

6. The Phase Trajectory 1, = f ( x ) We have

d y - f ( x )d x , Y

In Jyl =

X

i’e

dx

+ C.

The last expression is the equation of the trajectory in the plane x y , that is, we have obtained a function y = i= g(x), and now we can make transition to x ( t ) according to Subsection 1. 7 . The Phase Trajectory 6, = f(8,)

For a linear segment approximation, we have

We obtain the differentialequation with initial condition

8,

= a,&

+ (b, - l)S$

6,(0)

Solving numerically, we obtain the function 6,(z) know the value of g(z) at the end of the segment, g(Atn) = 4,

=

(116)

= g(z), 0 I z I At,. We

358


and from this we calculate At,,. Doing this for all the segments, we get the sequence ( t n } corresponding to {Sn}, that is, we have an approximation of the function S,(t), and from this we can calculate x(t). 8. The Phase Trajectory 1, = f(S,) For a linear segment approximation, we have

We obtain the differential equation with initial condition

8,

= (a,, -

1)s:

+ b&,

S,(O)

=

(1 17)

Solving numerically, we obtain the function S,(T) = g(T), 0 I T 5 At,,. We know the value of g(z) at the end of the segment, g(Atn) =

6fl9

and from this we calculate At,,. Doing this for all the segments, we get the sequence { t,,} corresponding to {dn} that is, we have an approximation of the function S,(t), and from this we can calculate x(t). 9. The Phase Trajectory 8, = f ( x )

Using the equation 6, = K, -,:S we have K,

- s:

=f ( x ) .

K, =

dY = f ( x ) + sxsy = -xY -dx

After some manipulations, K,

=f(x)

+

(2)' -

,

we obtain the equation of the isocline:

From this, transition to x ( t ) according to Subsection 1. D. Signal Processing Using Phase Trajectories

Signal processing is usually performed by functions of time-the signals themselves. We can readdress all the operations involved in signal processing to phase trajectories, which are images of signals. According to this statement of the problem, its solution consists of two parts: (1) to construct the


359

phase trajectory corresponding to the given signal; and (2) to apply to the phase trajectory special processing, according to the conditions of the concrete problem, for example signal separation, measurement of signal parameters, identification, pattern recognition, etc. We intend to consider here the second part of the problem, and to give some examples of phase trajectory processing for these applications. A phase trajectory is a composite function u(u), where u ( t ) and u(t) are implicit functions of time. Since there are always definite relationships between the parameters of the composite function and the parameters of x(t), we can estimate the properties of the original signal by using the coordinates of the trajectory in the u, u plane. We propose two methods for characterization of phase trajectories, based on the following parameters: 1. The number of intersections of the representative point with certain straight lines defined on the phase plane and, as a particular case, with the axes. 2. The time during which the representative point remains in a certain region of the phase plane. We shall call this the staying time. We give some illustrative examples of applying the two methods. In these examples we use only very simple trajectories, for the sake of clarity. However, the recommended procedures are valid for any phase trajectories. Example 1. Let us consider a well-known problem in detection theory, that of the reception of a binary set of signals sl,s2. The received signal is

+

x = s ~ , W ,~

0 < t < T,

where w is an interference (noise), and we must decide which one of the two signals was transmitted. The classical algorithm is

and is based on the following decision rule: If the upper inequality holds, then we infer that it was s1 that was transmitted, if the lower inequality holds, then s2 was transmitted. To be specific, we shall assume that sl,s2 are the two antipodal signals: s1 = u

sin wt,

s2 = --a sin wt,

0 < t < T.

( 120)

Now we shall propose a new algorithm, using a phase plane. In this algorithm, we replace the comparison of integrals as in (119) by comparison of the “staying time,” which is the amount of time that the representing point remains in a certain region of the plane. To see this, let us consider the case without interference, x = s ~ , The ~ . integrand functions are shown in Figs. 3a

360


-2

200

100

0

-2

100

0

200

10

0

-10

-1 5

0

15

SI2

s1 'S2

FIGURE3. Detection of two signals using phase planes. The received signal is multiplied by sl. (a) The output when receiving s, (no noise). (b) The output when receiving s2. (c, d) The outputs of (a) and (b) in the Poincare phase plane.

and 3b and their corresponding phase trajectories on the Poincare phase plane C(u) in Fig. 3c and 3d. We introduce four unit functions corresponding to the four quadrants of the phase plane:

i

1, x > 0, i> 0, "@) = 0, otherwise,

I

s3(t)=

1, x < 0 , i < 0, 0, otherwise

{ {0,

1, x < 0 , i > 0, rlz(t) = 0, otherwise. 1, x > 0,2< 0, r14(t) = otherwise.

(121)

We calculate the vector of staying time parameters: Oi = J

qi(t)dt,

i = 1, ..., 4.

0

This vector is a characteristic that can fully discriminate between the two


361

original signals. The algorithm can be written in the following form:

e, + 0,s e3 + e4,

S1

x=

.

s2

(123)

In the real case, when noise is present, the signals must be smoothed before applying the algorithm. Example 2. Besides the method of staying time, we can use the method of number of crossings: in this case, the crossings of the two semiaxes which form the boundaries between the quadrants 1-4, respectively, 3-2 (Figs. 3c and 3d). The algorithm for this method will be Ni-4

2 N3-2,

S1

X*

.

(124)

s2

Note that the method of staying time is to be preferred to the method of number of crossings, if there is noise. This because the number of crossings due to noise may be much larger than those due to the signal, and therefore the number of crossings loses significance. Example 3. Let us consider the problem of discriminating between the two antipodal signals shown in Figs. 4a and 4b (together with their phase trajectories in Figs. 4c and 4d). We will use the method of number of crossings. We shall designate by N, (right) the number of crossings of the semiaxis 1-4, and by Nf(left) the number of crossings of the semiaxis 3-2. The two kinds of crossings take place when

N,:

s > 0, S = 0,

N,:

s < 0, :: = 0.

In our example, for the first signal N, = 3, iVl = 1, and for the second signal N, = 1, Nf = 3. The decision law is N,

2 N,,

x*

S1

.

(125)

s2

Example 4. Consider the four signals (see Figs. 5a-5d) a nt s1 = - sinh -,

. nt s2 = - a sin -, T

s3 = -s1,

s4=-s2,

b

T

where b = / sinh T -2.nl .

O l t l T .

362


IKj ; r j

0

-/'1-1

0

100

200

-1

0

100

5E

200

5

0

0

-.

- -~

-5 1

0

I'

1

- -1

s1

0

1

s2

FIGURE 4. Detection of two signals using number of crossings in a phase plane (Example 3). (a, b) The signals in the time domain. (c, d) The signals in the Poincark plane. The number of crossings of the positive semiaxis x is different for each signal.

This specific value for b has been chosen, in order to have equienergetic signals. We must receive them and decide which one has been transmitteda quaternary decision problem. We shall make use of the following plane uv:

The trajectories of the four signals are shown in Fig. 5e. These are the four bisectrices of the quadrants. The decision procedure is: (1) For the received signal, compute the four staying times ei in each quadrant i, i = 1, 2, 3,4. (2) Decide according to the law: if max(Oi) = 0, => sj.

(127)

Example 5. This is another example of a quaternary decision problem. Consider the following set of four equienergetic signals, defined in the interval 0 5 t I T (Figs. 6a-6d):

:t

-2 0

1

2

tIT

=

O -2

t/T

d

s3

'

-

7

s4

I

0

1

-2

2

tIT

0

1

2

t/T

e 2

..

v=x

0

-2

-1

0

1

u=02x

FIGURE5. Detection of the four signals of Example 4. (a-d) The signals in the time domain. (e) The signals in the generalized phase plane uu, u = d x , u = f. Each signal stays in a different quadrant.

tlT

=

tlT s4

=3

2r-----

2r----l

1

00

:L

2

0

1

2

tIT e

tlT

30

0

I

D

a

FIGURE 6. Detection of the four signals of Example 5. (a-d) The signals in the time domain. (e) The signals in the generalized phase plane uu, u = 6(6,), u = b(6,). Each signal stays in a different quadrant.


s1 = b(t - T)’,

s2 = a sin wt,

s3 = bt2,

sq = a cos wt,

where b = a&(T2fi).

365

7L

w =2T’

(128)

We choose the following phase plane:

u = 6(6,),

u = 6(Sy),

where y = 1.

The four trajectories are along the bisectrices of the quadrants, and are shown in Fig. 6e. The decision law is as in the preceding example. Note that if we replace s2 by -s2, and s4 by -s4, the trajectories do not change. This replacement is recommended if we want to obtain a signal set with zero dc component.

IV. APPLICATIONS

A. Parameter Estimation (Note: In this section only, x’ denotes the Hilbert transform of x, and Z denotes the estimate of x.) We have already seen some particular cases of parameter estimation by means of GDEs in Section I1.D. In this section we shall consider this problem in more detail. The problem of parameter estimation in the general case is formulated as follows. Assume a signal, function of time t and of a vector of parameters a. We are given N samples of this signal, corrupted by noise w, x, = s(t,; a)

+ w,,

t,

= nT,

n = 0 , .. ., N - 1 ,

(129)

where T is the sampling interval. The problem consists of estimating the vector of parameters a from these samples. The most widely treated problem is the case of a multiple sinusoid, and the closely related case of the multiple exponential signal. For these problems a large amount of literature exists. In contrast, literature on nonsinusoidal signals is very scarce. In Subsection 1 we present the general approach of using structural properties to parameter estimation, and then deal with a test case of nonsinusoidal signals. We emphasize that it is especially in this specific field that the structural properties method has the greatest potential advantages, because of its generality. In the remaining subsections we focus on the multiple sinusoidal signals. We

366


shall give a brief survey of the existing methods for the multisinusoidal case in Subsection 2. In Subsection 3 we shall discuss the Cramer-Rao lower bounds (CRB) for the one-sinusoidal signal. This bound sets a theoretical limit to how small (good) the variance of an estimated parameter can be. We shall show that by making use of the second derivative of the signal we obtain a CRB that is smaller than the classical bound. Thus we may expect better estimates by using derivatives, i.e. structural properties methods. In Subsection 4 we present structural properties-based algorithms for estimating the parameters of multiple sinusoids, and an illustrative example of their performance. 1. Parameter Estimation b y Structural Properties: General Approach and a

Test Case for Nonsinusoidal Signals

The general approach to parameter estimation by structural properties consists of using generating differential equations (GDEs), and especially GDEs dependent on only the parameter of interest (and invariant to others). This topic was developed in Section 1I.D. We shall only briefly review it here. Given a signal x dependent on parameters a, b, c, for example, then we can construct GDEs of several forms: F(x, i, f, . . .; U , b, C; t ) = 0, F,,,(x,i,f

,... ; a , b ; t ) = O ,

F,(x, 1,f, ...; a; t ) = 0, and similarly for b, c, bc, etc. In practice, these equations must be viewed as algebraic (not differential) equations existing for each discrete sample t,, x,, etc. If we want to measure parameter a, we can use the GDE F,, which is invariant to the other two parameters b, c. From Fa we can derive an equation for a, of the form a = GJx, i, f,. . .; t). Moreover, we can eliminate the time t in the above equations, and obtain time-independent algorithms for measuring the parameter. If the signal is corrupted by noise, we have, instead of the aforementioned equations, equations of the form (el",e2, are errors): F(x,, in, x,, . . .; a, b, c; t.) = el,,, F,(x,, in,f,, ..., a; t,) = e2,, etc.

(131)

The parameter a is computed by minimizing the total mean squared error E e l : or z e 2 : , etc. In general, we obtain an algebraic equation of some degree K, of the form


361

with coefficients of the form A, = (Gk(xn, kn,

- ..; t,)>,

(133) where (. ) denotes discrete averaging over N samples. One of the solutions u of the equation is the desired parameter a. The selection of the correct solution can be made in one of two ways: (1) by careful checking the derivation of (132) in order to take in consideration various conditions which the solution must satisfy; (2) by substituting all the solutions back in (131) and choosing the one producing the minimum error. As a test case for nonsinusoidal signals, let us estimate the “frequency” w of a sinc signal in the presence of noise: X, = b

In,

sin w T ( n - N/2) +w,, w T ( n - N/2)

n = O ,..., N - 1 .

The following results are taken from Druckmann (1991, 1994). It is shown that, when there is no noise, o satisfies the following nonlinear GDE (for each sample xn): 04x2+ 2w2(2xf - i 2 )

+ 3 2 - 2 i x = 0.

(134) We view this equation as an algebraic equation in the unknown o.When there is noise, the zero on the right side is replaced by an error e. Minimizing the total squared error, we develop an algorithm which involves the solving of the following cubic equation, whose solution is 02: au3

+ bu2 + cu + d = 0, b = 3(x2(2x2 - a’)),

a = (x4),

+ 2 i 4 - 2x22f - 8xi2x),

c = (1 1x2x2

(135)

d = ((3x2 - 22i](2xx - i 2 ) )

This is an example of an equation of the type (132). The algorithm requires the computation of the first three derivatives of the signal. Two numerical differentiation algorithms were used: one based on a seven-point interpolating polynomial, the other on Fast Fourier transforms (FFTs). Both methods were tested at low and high signal-to-noise ratios, and for various sampling rates. For each case, a set of 400 noisy records was used for statistical averaging. Some results are shown in Fig. 7 the root-mean-square error as a function of the sampling rate, for the two differentiation methods, and at two SNR levels, - 5 dB and 40 dB. 2. Parameter Estimation for Multiple Sinusoids: Statement of the Problem and Existing Methods The problem consists of estimating the frequencies, amplitudes, and phases of an M-tuple sinusoid embedded in noise:

368

--


L

0,

b

0.01z

FFT der. 40 dB

I 5

0.0012

6

4

3

fs,

7

8

I

sampling freq. [sampl./timeunit]

FIGURE7. Frequency estimation of a sinc signal in noise: the rms error of the estimate vs. the sampling frequency. Frequencies are normalized to the true frequency. The results are for two differentiation methods (one using a 7-point interpolation polynomial, the other based on FFT), and for two SNRs (- 5 dB, 40 dB). (After Druckmann, 1994.) M Xn

=

bk k=t

COS(%Tn + 6,)

+ Wk,

n = 0,..., N - 1,

(136)

from the knowledge of N samples. The interference w k is a discrete Gaussian white noise, with zero mean and variance 02.It is usual to consider, instead of (136), the complex sinusoid, or cissoid:

where w k is a complex Gaussian white noise, crk = 20’. Note that having N samples of the complex signal (137) is equivalent to having N samples of the real signal x, (136) and, in addition, N samples of the Hilbert transform of x,. The three kinds of parameters are not equal from the point of view of difficulty of estimation. If the frequencies w, are known, then finding bk,6, is a simple linear problem; i.e., they can be computed by simply inverting a matrix. The difficult problem is computing the frequencies. Let us suppose that we want the optimal solution for the frequencies, amplitudes, and phases, i.e., that set of frequencies, amplitudes, and phases which minimizes the total squared error c k w k w k * . Because the parameters w, appear inside cos or sin functions, this is a nonlinear problem. Therefore it is customary to divide the problem of parameter estimation into two parts. First, a suboptimal algorithm is used to find the frequencies. Then, with the frequencies known, the


369

optimal bk, 6, are computed (a linear problem). The central issue in all algorithms for multisinusoidal signals is therefore the estimation of frequencies, and this is also the issue by which the various algorithms differ. Most algorithms are based on a difference equation for finding the frequencies. We shall give here a brief presentation of three of these methods. The Extended Prony Method. The extended proxy method is a family of methods, suited to exponential, decaying sinusoids and sinusoidal signals. The variant adapted to the multiple sinusoid case is known as line spectrum prony (Kay and Marple, 1981; Kay, 1988). The method is based on the fact that a multiple sinusoid (complex or real) satisfies the difference equation

where the vector (ak)is symmetric and is called the prediction vector. This vector is found by minimizing the total squared error l e Z . The resulting procedure for frequency estimation is: Compute the matrix and the vector,

N-M Yk

=

1

n=M+l

(%+k

-k

-k

xn+k)(xn+M

X,-M),

k = 0, .. ., M - 1,

(140)

and then the coefficient vector, Form the p-degree polynomial, p

= 2M,

M

H(2) =

1

Ck(ZM+k

+zM-k),

k=O

and compute the M pairs of conjugate roots. From the M arguments of the roots (the positive ones), compute the frequencies:

The Pisarenko Method. The Pisarenko method (Pisarenko, 1973) is based on a difference equation of the ARMA (p, p) type. The procedure is as follows. Compute the autocorrelation matrix:

370


Compute the minimum eigenvalue Amin and its associated eigenvector vmin. Then rs2 = Amin and the prediction vector a is vmin.As in the Prony method, construct the p-degree polynomial, compute its roots and the frequencies according to (143). In the original version of the method, all the eigenvalues are computed, and the minimal is selected. There are more recent algorithms which compute iteratively the minimum eigenvalue directly (Hayes and Clements, 1986; Pitarque et al., 1991), and are thus more computationally efficient.

Forward-Backward Linear Prediction Method. The forward-backward linear prediction method (FBLP) is more accurate than the other linear prediction methods, but more computationally complex. It uses a prediction vector of length L greater than M , and therefore obtains a number of eigenvectors and roots greater than M , the number of sinusoids. It was originally proposed by Ulrych and Clayton (1976) and by Nuttall (1976). We shall present here an improved version of the method, proposed by Tufts and Kumaresan (1982) and called the modified FBLP. The procedure is as follows. Compute the correlation matrix R, according to ( H denotes complex conjugate transpose): XL-1

XL-2

XL

XL-1

...

XO X1

... X N - L - 1 ...............................

xN-2

A=

...

xr x;

*

XN-L

xN-3

xf xf x;I;-L+i

..'

,

R=AHA.

x2* X2+1

...

*

xN-l

Compute the L eigenvalues and eigenvectors uiof R . From these, choose the M greatest eigenvalues and their associated eigenvectors. Now compute the prediction vector g as follows:

Compute the spectral estimator using g:

37 1


This spectrum has L maxima, attained at L frequencies. From among these, we choose the M frequencies whose corresponding poles are nearest the unit circle, and these are the best estimates. 3. Cramer-Rao Bounds for One Sinuisoid When Using the Second Derivative of the Signal A good estimate for a parameter means (1) small bias, and (2) small variance. A very small bias is attainable for high enough SNRs. For unbiased estimators there is a theoretical bound on how small a variance can be obtained. This is the Cramer-Rao lower bound (CRB) (Van Trees, 1968).In the case of a complex sinusoid, the CRB for the three parameters are (Rife and Boorstyn, 1974): 12a2 var{d} 2 = CRB, (146a) b2T2N(N2- 1) IT2

(146b)

var(6) 2 N = CRB, var{@ 2

2aZ(2N- 1) = CRB, bZN(N 1)

+

(146c)

We recall that complex sinusoid means that we assume having at our disposal not only N samples of the original real signal, but also N samples of its Hilbert transform. In analogy to this, let us suppose that we also have N samples of the second derivative of the complex signal. What is the CRB under these conditions? The answer is contained in the following theorem (Druckmann, 1994, Druckmann, in prep.). Theorem. Given N samples of each of the complex signals (T = sampling interval), 2, = x, +jy,,

Z , = s,

+ jz,,

where:

+ w,, y, = 1, = b sin(wTn + 0) + Gj,, = - b o 2 sin(oTn + 0) + &,, = 2, = - bo2 cos(o7'n + 0) + w,, z, =

x, = b cos(oTn + 0) s,

then the Cramer-Rao bounds for the frequency, amplitude, and phase are var{d} 2

16a2R = CRB2, b2T2N[3R2(NZ- 1) (128O/~~)(f/f,)~]

+

(147a)

372


var{b} 2

+

2aZ[R2(N- 1)(2N - 1) ( 6 4 0 / 3 ~ ~ ) ( f / f J ~ ] = CRB2,. b 2 N . $ R [ R Z ( N 2- 1) (1280/3~~)(f/f,)~]

+

(147~)

The proof can be found in the references, and will not be reproduced here. However we shall mention some basic facts. The samples of x, 2, x, 2, which we shall designate by x,, y,, s,, z, respectively, form together a 4N Gaussian random vector, having the following parameters: E{X,2} = E{y,2} = 2, p=-

E{s,Z} = E(z,2} =

azn4f,4 ~

5

’

Js

Cov{xnsn} _ - COV{y n z n > - ax 0 s ay=z 3 ’

and all other correlations are zero. Expressing the logarithmic probability density function of this vector and then computing from it the corresponding Fisher information matrix, produces, after some lengthy derivation, the results of the theorem. We define the gain of the new bounds with respect to the classical bounds as CRB,, classical = CRB,, with 2nd derivative and similarly for amplitude and phase. These are functions of the relative frequency w/o,, and of N. The gain for the frequency CRB, as a function of relative frequency, is shown in Fig. 8 (N = 64 samples). We see that it is always greater than 1, except at one point, where it is 1. Maximum gain is obtained at the extremal frequencies, 0 and 0.5 f,. The other two gains, of the amplitude and phase, have the same behavior (not shown). In conclusion, making use of the second derivative of a complex sinusoid allows us to expect to obtain estimators with lower variance. 4. Algorithms

One-Sinusoidal Case. For the one-sinusoidal case, when the noise is very small, the conservant can be used to measure the frequency (Zayezdny et al., 1992a): &

=--*

x,

(148)

Xn

The derivative is computed by a numerical algorithm. This simple estimation algorithm can be used to track the instantaneous frequency, if frequency variations are small compared to 2KT, where T is the sampling interval and 2K is the number of samples around n that are necessary to compute the

SIGNAL DESCRIPTION: NEW APPROACHES, NEW RESULTS N = 64 samples

Gain of CRB for Freq (DB)

0.1

0

02

373

0.3

0.4

0.5

Freq I Fs

FIGURE 8. The gain of the Cramer-Rao bound using second-derivative knowledge, with respect to the classical CRB, in the case of sinusoid frequency estimation. The gain is plotted vs. the frequency (normalized to sampling rate). The gain is never less than 0 dB, i.e., the CRB with derivative is smaller (better) than the classical one. (After Druckmann, 1994.)

second derivative at n. Smoothing before applying the algorithm can improve the result, if the noise is small. For the case of larger noise, a better algorithm is (Zayezdny et al., 1992a): lj)

”

(XX)

=---

(2)’

(149)

where ( ) denotes discrete averaging over N samples. This estimator is obtained by minimizing the total squared error for the differential equation:

x, + w’x,

= en.

( 150)

This algorithm, with suitable modifications, has also been used successfully to estimate waveforms of the form A cos(wt + 0) + C from samples of less than half a period (Druckmann et al., 1994). Multiple-Sinusoidal Case. Assume we are given an M-multiple sinusoid with M frequencies, and its derivatives of even order, up to order 2M. Then the samples of these signals satisfy the equation M-1

x“”)

+ i=O

aixi2’)= 0,

n = 0, ..., N - 1,

(151)

where dk)= d k x / d t k .From this equation we obtain, by using the Fourier

374


transform, the following algebraic equation, whose roots are the frequencies of the sinusoids:

We can obtain an equivalent equation of reduced degree, by setting u

uM +

M-1

= s2:

,

a i d = 0.

(153)

i=O

This equation has M negative real roots. The frequencies are given by

ok=t/-uk,

k = l , ..., M .

(154)

If the signal is corrupted by noise, then Eq. (151) becomes

where en is an error due to noise. We want to find those coefficients which minimize the total squared error. Differentiation of the error with respect to the coefficients gives the following linear equation in the coefficient vector: r

‘00

‘10

r20

r10

rll

121

120

‘21

r2 2

‘M-1,l

l;M-1.2

,‘M-l.O

... ... ... ...

...

>

rM-l,O

rM-l, 1

rM-1,2

r~-i,~-i,

where we have introduced the notation rk

= (X(”)X(~~)).

Note that the matrix is symmetrical. The algorithm for computing the frequencies is summarized as follows: 1. Compute M derivatives of even order, dZk), k = 1, ..., M. 2. Compute the M ( M + 3)/2 terms: rk,i= ( x ( ~ ~ ) x ( ~k ~=) 0, ), i = k , ..., M. 3. Compute the Coefficients a, . . ., u ~ - according ~ , to:

Matrix R

= (rk,j ) ,

Vector y, a = -R-l

yi= rM,i,

Y.

. .., M

- 1,


375

4. With the coefficient vector, construct the following equation and solve

it for M negative solutions uk: uM

+ a M - I U M - l + * . . a , u + a, = 0.

5. Compute the frequencies according to: o,= J- uk.

Note 1 . Step (1) requires a numerical differentiation algorithm. Therefore the given general frequency computation algorithm is in fact a family of algorithms, each one differing from the other by the specific differentiation algorithm used. Note 2. The differentiation algorithm computes the highest derivative at point k on the basis of 2L + 1 points around point k. (L depends on the algorithm). Therefore, the derivatives are available at only N - 2L points of the signal’s original N data points. Therefore the averaging required at step 2 is performed on N - 2L samples. Simulations. The performance of the presented methods has been tested by computer simulations: variance of one-sinusoid case (Zayezdny et al., 1992a), variance and root-mean-square (rms)error of one- and two-sinusoid case (Druckmann, 1994). We give two results from the last reference. Figure 9 shows the “performance” (defined as the inverse of the variance) of the frequency estimates by the structural properties method and by the 40

ONE SlNUSOlD Freq = 1 SNR= -5 DB 400 TRIALS Np = 4 PERIODS

’ 5

30-

r

-

r

-1

r

a: 5

+

fl d

+ - 4 -

+

+

--

a

4 Srnoothings

9 20

104

4

I

I

I

.

a

I

I

I

12

x + w:w;x.

= a,

sin(w,t

Lol,ol[X] = K , K i

(e) The signal x

+ wo'

exp( -pt2).

Ls[x] = x x - 1 2

(d) The signal x

L,,[x] = K,

=a

+ (U:+ Wi)K, +

0:Wi

sin(wot + 0) . The filter is derived from Eq (134). w,t 8

+

L,[x] = 2k-2 - 3x2 - 4w2xx + 2 w 2 i 2 - w4x2.

SIGNAL DESCRIPTION. NEW APPROACHES, NEW RESULTS

383

2. Pass Filters

We can synthesize filters that allow the passing of a given class of signals, and reject another class of signals. This problem is closely related to the separation problem of the preceding section. The procedure is as follows. We synthesize a GDE that corresponds to both classes of signals, dependent on parameters of both classes, or dependent only on the parameters of the first class (the passing class). Then we derive the equations that estimate (calculate) the parameters of the passing signal. Finally, from these calculated parameters we reconstruct the passing signal. Note that we have introduced here a new type of GDE: one that is dependent on the parameters of one class of signals, and independent on the parameters of another class of signals.

D. Signal Identification Signal identification is closely related both to parameter estimation and to signal separation. Let s(t; a, b) be a signal including parameters a, b, and let us construct the following state functions and GDEs. The state functions that produce constant output for input s are (we have assumed that the output is the parameter itself, e.g., a, but it may be a', ab, or some other combination of parameters):

u = FJs, S,

S, ...),

b = Fs,b(s,&i, ...).

The GDEs, including an ensemble GDE, are of the form

. . .; a, b) L,[s; a, b] = 0, L,,,[x] = EGDE(x, i, X, . . .) * L e , s [ ~=] 0.

L,[x;a, b] = GDE(x, i, X,

(164) (165)

The first GDE includes in its expression the parameters a, b; the second is independent of the parameters. Taking into account the presence of noise, we shall also construct estimators for the parameters, according to the procedure of (13 1)-( 132). The estimator for a is of the form 5 = Fest((Gi(x% % . . . )>, ( G ~ ( x ,& . ..I>,

.. .),

(166)

where F,,,contains averages of functions GI, C,, . .. over a specified running time interval. A.11 these functions, viewed as filters acting on a signal x, can be used as identification systems for the general signal s(t; ., .) or for the specific signal s(t; a, b) with specified parameters a, b. In the first case we have a signal x,

384


and must decide whether it is s(t; ., .) or not. We can use some of the abovedefined functions as filters and apply them to the input x. We have at the output: F,,,[x = s] = a = const., Fs,b[X

= s] = b = Const.,

L,,,[x = s] = 0 = const.,

F,,,[x # s] # const. Fs,b[X

# S] # Const.

Le,,[x # s] # 0.

For all these systems we check the outputs. If the output is essentially constant (or zero), then we conclude that the input is the signal s(t; ., .). When the output departs from the constant, we conclude that the input is no longer s. Of the three systems above, the first two not only identify s, but also produce some more information about it: They measure one of its parameters. The third system performs only identification. We shall call all these three systems or filters exact identification filters, because they are derived under the assumption that the signals are “clean,” without noise (Kazovsky et al., 1983). If the signal is corrupted by noise, then the output of the three identification systems is not constant any more (note that in general the systems are not linear, and therefore the noise at the output is not simply related to the input noise). If we allow the presence of a certain amount of noise and we still want to recognize the corrupted signal as ‘‘s(t),” then we can apply smoothing at the output, and check the smoothed output for “constantness.” The length of the smoothing time window is to be determined according to the amount of noise allowed: If we perform too much smoothing (large time window), then we can obtain a constant output even for a signal s2 other than s, and hence we shall wrongly identify it as s; on the other hand, too small a smoothing window will produce a noisy signal (not constant), and hence s will not be identified as such. Another possibility is, in the case of noisy signals, not to use an exact filter and to smooth its the output, but to use a filter that is synthesized according to a procedure which from the beginning takes into account the presence of noise. Such a filter is, for example, one designed as a parameter estimator under the condition of minimizing the output error, such as (166). The identification is carried out according to FeSt,,,,[x# s] = variable > threshold.

F,,,,,,,[x = s] w u = const.,

The criterion for constantness can be the variance of the output, computed by time averaging. If y , denotes the output (as discrete samples), then the time variance is defined as var

=

(Y: -

(Y.>2)9

(167)


385

and the “constantness” is defined according to var < threshold. A filter of this type will be called an aueraging identification filter. It includes averaging (smoothing) as an integral part of its design, and is optimal in the sense of minimizing the power of the output noise. This filter was designed not only for estimating a, but also b and all other parameters, simultaneously. Therefore a better identification system is one which tests simultaneously the constantness of the outputs of Fest,s,a[x], Fest,s,b[~], etc. We define a multioutput (or uector) filter,

I 1 Fest,s,aCxI

Fest,sCxI

=

Fest,s,bCxI

( 168)

-..

and the identity test will be for a constant vector at the output. Finally, we also define an exact filter that is a vector, i.e., a multiple-output exact filter. This is based on a new notion, the vector state function, which is a generalization of a state function. There are signals that are not fully identified by a state function which produces a constant, but are fully identified by a vector of state functions which produces a constant vector for this signal. For a two-parameter signal we have the vector state function

3 i i

F,,,(s, S, s, .. .) s, s, . . .)

= FJS,

= Fs(s,S, i, ...).

Example 1. Given a bisinusoidal signal s, we define the vector state function

Applying this vector filter on s, we obtain a constant output:

In a certain sense. this is a bidimensional conservant. Example 2: Identification of a Caussian-Shaped Pulse. We return to the example of the mixed signal of Section 1V.B. We showed there how to separate between a Gaussian pulse and an exponential one. But suppose that we have a broad Gaussian pulse, and the exponential pulse appears somewhere during the Gaussian pulse. We must use an identifier for the Gaussian pulse: as long as its output is constant, we know there is no exponential interfer-

386


ence. As soon as its output departs from constantness, we know the exponential pulse has appeared, and we must apply the separation algorithm. Here are three examples of identification filters for Gaussian pulses, of the types discussed above:

E. Data Compression 1. Definitions and Brief Survey of Existing Methods

Data compression is connected with the representation of an analog signal by a string of digital characters (e.g., binary digits). Given a band-limited signal of bandwidth W, the simplest and crudest way to do this is (1) to sample it at a rate f greater than the Nyquist rate 2W, and then (2) to quantize each sample to L uniform levels, representing it by M bits, L = 2'. The signal is thus encoded by a sequence of bits transmitted at a rate of R

=fM

(bits/s).

This is called the canonical representation. There are ways of decreasing both f and M ,and thus obtaining an average rate R* smaller than R. If this is the case, we say that we have achieved data compression. Other names are bandwidth reduction and waveform coding. Following the above description, we see that the transition from the analog signal (continuous both in time and value) to the digital representation (discrete in both) is done in two steps: discretization in time (sampling) and then discretization in value (quantization). Consequently, a data compression system consists of two main blocks: a generalized sampler and a generalized quantizer. A generalized sampler obtains from the samples a set of P parameters (paramete+), from which the signal can be reconstructed. These parameters are quantized. The generalized quantizer transforms the sequence of bits into another sequence, more economical, containing in average C bits/parameter. This is done by exploiting statistical properties of the original sequence (lossless or entropy encoding).For most practical signals, P < f and C < M , and therefore data compression can be achieved. In this chapter we are concerned only with the first block, that is, parametrization or generalized sampling. There are many methods to do this, but


387

most of them can be classified in three groups: linear predictive methods, transform methods, and syntactic methods. In methods of the the first group, the signal is viewed as the output of a linear discrete filter (AR or ARMA), which can be represented by one of several sets of parameters: coefficients of the difference equation, coefficients of the impulse response, reflection coeffcients, poles-zeros, etc. According to the second group, the signal is transformed by a linear transform, and the most significant of its coefficients are retained as the representation. Several transforms are used: cosine, Hadamard, Karhunen-Lobe. Walsh, Fourier, wavelet, etc. In the third kind of methods, the signal is segmented into components which have special significance for the signals involved, and these are compressed separately. Accordingly, the methods of this group are strongly dependent on the nature of the signals involved, and different from each other. For example, picture signals are decomposed in contour and texture signals, and ECG signals are segmented in a small number of special-type complexes (Q, R, S, etc.). Some basic references are Jayant and No11 (1984), Lynch (1985), and Kunt et al. (1985). 2. Statement of the Problem for Use of Generating Dgerential Equations For our aims, the problem of data compression can be formulated as follows (Druckmann, 1987; Zayezdny and Druckmann, 1991). Let x ( t ) be a signal, and T [ '1 a transformation that generates from x a set of K new signals:

Data compression is achieved if the following conditions are satisfied. (a) The set { s k } contains full information about x, that is, there exists an algorithm, which can reconstruct x from this set:

(b) The set { S k } can be represented by R, bits/s, which is less than R,, the number of bits per sample required to represent x with the same reconstruction error:

where Mk are the number of bits per sample of signal sk, and jkis the sampling rate of sk. If the spectra of sk and x have bandwidths W,, Wx, respectively, then a condition that may cause (171) to hold is that the

388


total bandwidth of

{sk}

be less than the bandwidth of x:

We deal here with a particular case of this general formulation: For us, { s k } are the time-varying coefficients of a GDE, and T - ’ [ . ] is the algorithm

of solving the differential equation. In the following we shall confine ourselves to second-order linear GDEs. Two types of GDEs have been used: the A-Psi type with three coefficients (see Section II.F), and an equivalent set of two first-order equations, in which A and o are transmitted directly. Some work on a fourth-order GDE has also been done (Druckmann and Zayezdny, 1991). 3. Use of a GDE of A-Psi Type with Three Coeflicients to Data Compression

Consider a band-limited signal x, whose instantaneous amplitude and frequency have bandwidths W,, W,, respectively, and let us construct its GDE of type A-Psi. As a consequence of Theorem 5, Section II.F, if the bandwidths W,, W,,satisfy

w,,+ 8

0

= 4(W*

+ W,) < w,,

( 172)

then we have a possibility of data compression. We say only “have a possibility,” because (172) is not sufficient: We must check the reconstruction error after quantization of the coefficients. Now we shall give a brief description of the compression system. This consists of a transmitter and a receiver. At the transmitter the following operations are performed. (a) The coefficients az(t),ao(t)are computed: a,(t) = x i - ii,

ao(t) =

-xi.

Actually, they are computed as discrete samples, not as continuous signals: - f,,i,,. a,[n] = x,,i,, - inin, ao[n] = inin

(b) The coefficients are sampled at a lower rate than the original signal, because they have a smaller bandwidth than it has. (c) The samples of the coefficients and of the initial conditions are quantized and transmitted. At the receiver, the reconstruction of x is performed according to the following steps:


389

(a) Interpolation is performed on the coefficient samples, in order to restore them to a sampling rate greater than the Nyquist rate of the original signal. (b) The coefficient al(t) is computed by numerical differentiation: al[n] = -Ciz[n].

(c) The three coefficients and the initial conditions are passed to a differential equation solver, which solves according to i =Y,

(173)

The solver was of the Runge-Kutta type of order 4, with constant step. This compression system suffers from some drawbacks. The total bandwidth of its two transmitted coefficients is greater than the total bandwidth of the amplitude and phase representation [see Eq. (172) and Theorem 5 in Section II.F]. And besides, during solving the differential equation, there are large errors at those points where the leading coefficient is close to zero. 4. Data Compression by a Set of First-Order GDEs Using A and w Directly In order to overcome these drawbacks, we used another system (Druckmann, 1994), in which we transmit A and w = directly, and we use at the receiver a GDE in the new variables

The GDE to be solved, with initial conditions, is

e

= -w(t)v

3 = w(t)

Advances in Imaging and Electron Physics, Volume 94

Advances in Imaging and Electron Physics, Volume 148 (Advances in Imaging and Electron Physics) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 127 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 132 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 143 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 120 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 121 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 102 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 144 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 128 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 125 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics (Volume 112) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 150 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, volume 136 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 101 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 135 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 130 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 99 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 141 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 146 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 123: Advances in Electron Microscopy and Diffraction (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 134 (Advances in Imaging & Electron Physics)

Advances in Imaging & Electron Physics - Volume 100 Cumulative Index (Advances in Imaging and Electron Physics)

Advances in Imaging & Electron Physics, Volume 122 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 71 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 80 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 65 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 94

Advances in Imaging and Electron Physics, Volume 148 (Advances in Imaging and Electron Physics) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 127 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 132 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 143 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 120 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 121 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 102 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 144 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 128 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 125 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics (Volume 112) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 150 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, volume 136 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 101 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 135 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 130 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 99 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 141 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 146 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 123: Advances in Electron Microscopy and Diffraction (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 134 (Advances in Imaging & Electron Physics)

Advances in Imaging & Electron Physics - Volume 100 Cumulative Index (Advances in Imaging and Electron Physics)

Advances in Imaging & Electron Physics, Volume 122 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 71 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 80 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 65 (Advances in Imaging and Electron Physics)

Recommend Documents