Wavelets in the geosciences

Lecture Notes in Earth Sciences Editors: S. Bhattacharj1, Brooklyn G. M. Friedman, Brooklyn and Troy H J Neugebauer, Bon...

Author: Roland Klees | R. H. N. Haagmans

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Lecture Notes in Earth Sciences Editors: S. Bhattacharj1, Brooklyn G. M. Friedman, Brooklyn and Troy H J Neugebauer, Bonn A Sedacher, Tueblngen and Yale

90

Roland Klees Roger Haagmans (Eds.)

Wavelets in the Geosciences With 102 Figures and 3 Tables

Springer

Editors Prof Dr. Roland Klees Roger Haagmans Delft Umverslty of Technology Delft Institute of Earth-Oriented Space Research (DEOS) Thijssewegl 1, 2629 JA Delft, The Netherlands

From January 1st, 2000 on, Mr. Haagmans wdl have the following address. Roger Haagmans Agricultural University of Norway Department of Mapping Sciences P.O Box 5034, 1432 As, Norway Cataloging-m-Pubhcation data applied for Die Deutsche Blbhothek - CIP-Emheltsaufnahme Wavelets 111the geosclences " with 3 tables / Roland Klees ; Roger Haagmans (ed) - Berlin, Heidelberg, New York, Barcelona ; Hong Kong, London, Milan ; Pans ; Singapore ; Tokyo Springer, 2000 (Lecture notes in earth sciences ; 90) ISBN 3-540-66951-5

"For all Lecture Notes in Earth Sciences published till now please see final pages of the book" ISSN 0930-0317 ISBN 3-540-66951-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the fights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or m any other way, and storage in data banks Duplication of this pubhcation or parts thereof ~s permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law © Sprxnger-Verlag Berlin Heidelberg 2000 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. m th~s publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore tree for general use. Typesetting: Camera ready by author Printed on acid-free paper SPIN: 10723309

32/3142-543210

Preface

This book is a collection of the Lecture Notes of the School of Wavelets in the Geosciences held in Delft (The Netherlands) from 4-9 October 1998. The objective of the school was to provide the necessary information for understanding the potential and limitations of the application of wavelets in the geosciences. Three lectures were given by outstanding scientists in the field of wavelet theory and applications: D r . Matthias Holschneider, Laboratoire de Geomagnetisme, Institut de Physique du Globe De Paris, France; Dr. Wire Sweldens, Mathematical Sciences Research Centre, Lucent Technologies, Bell Laboratories, Murray Hill, N J, U.S.A.; Prof. Dr. Willi Freeden, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, Germany. The lectures have been supplemented by intensive computer exercises. The school has been very successful due to the engagement and the excellent presentations of the teachers, the very illustrative and instructive computer exercises, the lively interest and participation of the participants, and the many fruitful discussions we had. Therefore I want to express my thanks to the teachers for the excellent job they did, for providing typewritten Lecture Notes, and for their excellent co-operati0n. Thanks also to the students, who were actively engaged in the lectures and exercises during the whole week. The organisation of such a School is not possible without the support of many others. First of all I want to thank Prof. Dr. Willi Freeden and his co-workers from the Geomathematics Group, Department of Mathematics, University of Kaiserslautern, who co-organised the school. They provided a considerable contribution to the success of the school. I also want to express my thanks to the support of Michael Bayer, Martin van Gelderen, Roger Haagmans and my secretary, Wil Coops-Luyten, who were heavily involved in the organisation of the school and in all practical aspects including the wonderful social program which gave the participants some flavour of Dutch culture and the beautiful city of Delft. In the work of organisation, we have been supported by all the staff of Section Physical, Geometric and Space Geodesy (FMR). Last but not least I want to thank Martin van Gelderen and Roger Haagmans for preparing the introduction. Essential to the success of the school was the support we received from various organisations: the International Association of Geodesy (IAG), the Netherlands Geodetic Commission (NCG), the Department of Geodesy at Delft University of Technology, and the Delft Institute for Earth-Oriented Space Research (DEOS). Their support is gratefully acknowledged.

Delft, October 1999

Roland Klees

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Continuous Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

XlII 1

Matthias Holschneider (Institut de Physique du Globe de Paris) Building Your Own Wavelets at Home . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Wim Sweldens (Bell Laboratories) and Peter SchrSder (California Institute of Technology) Factoring Wavelet Transforms into Lifting Steps . . . . . . . . . . . . . . . . . . . . . . .

131

Ingrid Daubechies (Princeton University) and Wire Sweldens (Bell Laboratories) Spherical Wavelets: Efficiently Representing Functions on a Sphere . . . . . . .

158

Peter SchrS"der (California Institute of Technology) and Wim Sweldens (Bell Laboratories) Least-squares Geopotential Approximation by Windowed Fourier Transform and Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Willi Freeden and Volker Michel (University of Kaiserslautern)

189

Organization

Organization Committee Program: Roland Klees (Delft University of Technology) and Willi Freeden (University of Kaiserslautern) Local organization: Michael Bayer (University of Kaiserslautern), Wil Coops, Martin van Gelderen, Roger Haagmans, Ren~ Reudink, Huiping Xu (Delft University of Technology)

Sponsors International Association of Geodesy Netherlands Geodetic Commission Department of Geodesy, Delft University of Technology Delft Institute for Earth-Oriented Space Research

Lecturers Matthias Holschneider, CPT, CNRS Luminy, Case 907, F-13288 Marseille, France, and Institut de Physique du Globe de Paris, Laboratoire de G~omagn6tisme, 4 place Jussieu, F-75252 Paris, France, hols0ipgp, jussieu, fr Wire Sweldens, Bell Laboratories, Lucent Technologies, Murray Hill NJ 07974, U.S.A., wim@bell-labs, corn Willi Freeden, Universityof Kaiserslautern,Laboratory of Technomathematics, Geomathematics Group, 67653 Kaiserslautern,P.O. Box 3049, Germany, freeden©mathematik.uni-kl.de

Other contributors Peter SchrSder, Department of Computer Science, California Institute of Technology, 1200E, California Blvd., MS 256-80, Pasadena, CA 91125, U.S.A, ps©cs, caltech, edu Ingrid Daubechies, Princeton University, Department of Mathematics, Fine Hall, Washington Rd., Princeton, NJ 08544-1000, U.S.A., b e c c a @ m a t h , princeton, edu Volker Michel, University of Kaiserslautern, Laboratory of Technomathematics, Geomathematics Group, 67653 Kaiserslautern, P.O. Box 3049, Germany, michel©mathematik.uni-kl, de

VIII

Organization

Martin van Gelderen, Roger Haagmans I and Roland Klees, Delft Institute for Earth-Oriented Space Research, Delft University of Technology, Thijsseweg 11, 2629 JA Delft, the Netherlands, gelderen@geo, t u d e l f t .nl, haagmans@geo, t u d e l f t , nl, klees@geo, t u d e l f t , nl

Participants C.M.C. Antunes, Faculty of Science, University of Lisbon, Rua da Escola Politcnica 58, 1250 Lisbon, Portugal, (351)1 392 1855, [email protected] L. Barens, Heriot-Watt University, Dept. of Petroleum Engineering, Riccarton, Edinburgh EH14 4AS, United Kingdom, (44)131 451 3691, [email protected] G. Beyer, TU Dresden, Institut ffir Planetare Geod~sie, George-B~hr-Str. 1, D-01062 Dresden, Germany, (49)35 1463 4483, beyerQipg.geo.tu-dresden.de M. Boccolari, Universita degli Studi di Modena, Dipart. Di Scienze dell'Ingegeneria, Sez. Osservatorio Geofisico, Via Campi 213/A, 1-4100 Modena, Italy, (39)59 370703, MaurobQrainbow.unimo.it F. Boschetti, CSIRO, Division of Exploration and Mining, 39 Fairway, Nedlands WA 6009, Australia, (61)8 9389 8421, [email protected] J. Bouman, DEOS, Delft University of Technology, Thijsseweg 11, NL-2629 JA Delft, The Netherlands, (31)15 278 2587, [email protected] J. Branlund, University of Minnesota, Dept. of Geology and Geophysics, 310 Pillsbury Drive SE, Minneapolis, MN 55455, U.S.A., (1)612 624 9318, [email protected] A. de Bruijne, DEOS, Delft University of Technology, Thijsseweg 11, 2629 JA Delft, The Netherlands, (31)15 278 2501, [email protected] F. Chane-Ming, Laboratoire de Physique de l'Atmosphere, Universit~ de la R@union, 15 Avenue Ren@Cassia, F-97715 St. Denis cedex 9, France, (33)262 938239, fchane'Quniv-reunion.fr I. Charpentier, Projet Idopt, Inria Rhone Alpes, LMC, Domaine Universitaire, P.O. Box 53, F-38041 Grenoble cedex 9, France, (33)4 76 63 57 14, IsabeUe.CharpentierQimng.fr G.R.T. Cooper, Geophysics Department, University of the Watersrand, Private Bag 3, Johannesburg, South Africa, (27)11 716 3159, [email protected] D. Di Manro, National Institute of Geophysics, Via di Vigna Murata 605, 1-00143 Rome, Italy, (39)6 51860328, dimauroQingrm.it A. Duchkov, Novosibirsk State University, Morskoy pr. 36-16, Novosibirsk 630090, Russia, (7)3832 34 30 35, [email protected] L. Duval, Institut Francais du Petrole, Avenue de Bois-Preau 1-4, F-92852 Rueil Malmaison Cedex, France, (33)1 47 52 61 02, lanrent.duvalQifp.fr L.I. Fern£ndez, Facultad de Ciencias Astron6micas y Geofisicas, Universidad Nacional de La Plata, Paseo del Bosue S/N, 1900 La Plata, Buenos Aires, Argentina, (54)21 83 8810, lanrafQfcaglp.unlp.edu.ar 1 Now at: Agricultural University of Norway, Dept. of Mapping Sciences, P.O. Box 5034, N-1432/~s, Norway

Organization

IX

C. Fosu, Int. of Geodesy & Navigation, University FAF Munich, D-85577 Neubiberg, Germany, (49),89 6004 3382, [email protected] G. Franssens, Belgian Institute for Space Acromomy (BISA), Ringlaan 3, B-1180 Brussels, Belgium, (32)2373 0370, [email protected] M. Giering, C.D.S., Mars Inc., 100 International Drive, Mt. Olive, NJ 07828, U.S.A., (1)973 691 3719, [email protected] A. Gilbert, University of Stuttgart, Geodetic Institute, Geschwister-Scholl-Str. 24 D, D-70174 Stuttgart, Germany, (49)7 11 121 4086, [email protected] T. Harte, University of Cambridge, The Computer Laboratory, Pembroke Street, Cambridge CB2 2GQ, United Kingdom, (44)1223 33 5089, tphl001GCL.cam.ac.uk A. Hermann, DEOS, Delft University of Technology, Thijsseweg 11, 2629 JA Delft, The Netherlands, (31)15 278 4169, [email protected] J. Krynski, Institute of Geodesy and Cartography, Jasna 2/4, P-00950 Warsaw, Poland, (48)22 827 0328, [email protected] R. Kucera, Dept. Of Mathematics, VSB-Technical University of Ostrava, Str. 17, Pistopadu 15, 70833 Ostrava Poruba, Czech Republic, (42)69 699 126, [email protected] J.N. Larsen, Geophysical Department, University of Copenhagen, Vibekegade 27, 2.th, DK-2100 Kopenhagen, Danmark, (45)3927 0105, nykjaer @math.ku.dknykjaer @math.ku.dk D.I.M. Leitner, Institute of Meteorology and Geophysics, Heinrichstrasse 55, A-8010 Graz, Austria, (43)316 382446, milQbimgs6.kfunigraz.ac.at M. Lundhavg, Nansen Environmental and Remote Sensing Center, Eduard Griegsv. 3A, N-5037 Solheimsviken, Norway, (47)55 29 72 88, [email protected] Mehmet Emin Ayhan, General Command of Mapping, Geodesy Department, TR-06100 Cebeci, Ankara, Turkey, (90)312 363 8550/2265, [email protected] R. Montelli, UMR - Geosciences Azur, 250 rue Albert Einstein Bat. 4, F-06560 Valbonne, France, (33)4 9294 2605, [email protected] G. Olsen, Dept. of Mathematical Sciences, Agricultural University of Norway, P.O. Box 5034, N-1432 As, Norway, (47)64948863, gunnar.olsenQimf.nlh.no, P. Paun, Projet Idopt, Inria Rhone Alpes, LMC, Domaine Universitaire, P.O. Box 53, F-38041 Grenoble cedex 9, France, (33)4 76 63 57 14, [email protected] G. Plank, Institute of Theoretical Geodesy, Dept. of Math. and Geoinf., TU Graz, Stein£cherstr. 36/18, A-8052 Graz, Austria, (43)316 583 662, [email protected] R. Primiceri, Dipartimento di Scienze dei Materiali, Universita'degli Studi de Lecci, Via Arnesano, C.A.P. 73100 Lecce, Italy, (39)832 320 549, [email protected]

X

Organization

I. Revhaug, The Agricultural University of Norway, Dept. of Mapping Sciences, P.O. Box 5034, N-1432 As, Norway, (47)649 48 841, [email protected], F. Sacerdote, Dipartimento di Ingegneria Civile, Universitk di Firenze, Via di S. Marta 3, 1-50139 Firenze, Italy, (39)55 479 6220, [email protected] G. Schug, Geomathematics group, University of Kaiserslautern, P.O. Box 3049, D-67653 Kaiserslautern, BRD, (49)631-205-3867, [email protected] F.J. Simons, Dept. of Earth, Atmospheric and Planetary Sciences, Massachusets Institute of Technology, Room 54-517A, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, U.S.A., (1)617 253 0741, [email protected] J. Spetzler, Universiteit Utrecht, Faculteit Aardwetenschappen, Budapestlaan 4, 3584 CD UTRECHT, The Netherlands, (31)30 253 51 35, spetzlerQgeo.uu.nl G. Strykowski, National Survey and Cadastre, Rentemestervej 8, DK-2400 Copenhagen NV, Danmark, (45)35 87 5316, [email protected] J. Vazquez, JPL/Caltech, 300/323 4800 Oak Grove Drive, Pasadena, CA 91109, U.S.A., (1)818 354 6980, JVQPacific.JPL.NASA.GOV H. Wen, TU-Graz, Steyrergasse 30, A-8010 Graz, Austria, (43)316 873 63 55, [email protected] P.M. Zeeuw, Centrum voor Wiskunde en Informatica, Postbus 94079, 1090 GB Amsterdam, The Netherlands, (31)20 592 42 09, [email protected]

Introduction

Background

The international school on Wavelets in the Geosciences fits within the tradition of the summer schools of the International Association of Geodesy that exist already since many years. The topics are generally carefully selected and cover an interesting field of current and future interest. In this respect the wavelets fulfill such expectations; they offer challenging mathematical opportunities and many fields of application. The range of applications is so wide that the focus was limited to the geosciences. In this way participants with different background were offered the opportunity to meet and exchange ideas with each other and with an excellent team of lecturers, consisting of Matthias Holschneider, Wire Sweldens and Willi Freeden. The school was organized by prof. R. Klees of the Delft University of Technology in the Netherlands and by prof. W. Freeden of the University of Kaiserslautern in Germany. The school was hosted by the Department of Geodesy of the Faculty of Civil Engineering and Geosciences of the Delft University of Technology in Delft, the Netherlands. It was held from October 4th to October 9th 1998. The number of participants was limited to 41 due to the restricted number of computers available for the exercises. The group was divers in several respects: 5 continents and 19 countries were represented. The majority of participants were Ph.D. students or members of the academic staff, and the group was completed by a few post-docs, M.Sc. students and nonacademics. Around 29% had a background in geodesy, 48% in geophysics, 19% in mathematics and the rest in other disciplines. So, from the point of view of bringing people from different background and different nationalities together in a stimulating scientific environment, which is one of thee major goals of the "summer" schools, it was a success. The basic objective of the school was to provide the necessary information to understand the potential and limitations of the application of wavelets in the geosciences. This includes: - the mathematical representation in one and more dimensions like on the sphere - the properties as compared to Fourier techniques - the signal representation and analysis ability - the use of operators in terms of wavelets - gaining experiences with wavelets using examples from geosciences in computer exercises Lectures

and Notes

The school lasted six days and contained three major subjects. Every subject was covered in two days time. All topics were supported by practical exercises o n the

0 C~

0

90

0

o.~

x

Introduction

XV

computer with examples from geodynamics, topography representation, gravity field modeling etc. The program consisted of three parts that are described in these lecture notes. The idea behind the total program is to present the current status of continuous wavelet analysis for data analysis applications in the geosciences as a first step. The second part has a more specific focus on discrete analysis with special emphasis on the second generation wavelets. This allows to loosen the link with Fourier analysis, and the extension to complex surfaces and fast and efficient data approximation. The third part presents the current status of global data analysis on the sphere with the potential field as a special field of application. This setup makes it possible for participants and readers to get a view on wavelets also beyond their own field of interest which may lead to curiosity, discussion and possibly new developments.

Part

1

The first part was lectured and written by Matthias Holschneider. The basic facts about harmonic analysis through Fourier transforms in one dimension are recalled. In particular several sampling theorems are presented linking the Fourier transform over the real line to the Fourier transform over the circle and to the so called FFT (Fourier transform over the discretize circle). This is important in applications since only the FFT is implementable on computers, however the signal one treats are at least in spirit functions over the real line. Many "errors" in data processing come from a non clear distinction between the various :transforms. To remedy with certain shortcomings of standard Fourier techniques the wavelet transform is introduced as a possible time-frequency technique. Importaut general features of wavelet transform are discussed such as energy conservation, inversion formulas, covariance properties and reproducing kernels. This is important to be able to read wavelet transforms of signals. Also wavelet transforms of functions on the circle (periodic functions) are treated. The possibility of down-sampling of wavelet coefficients is linked to the existence of frames and orthonormal bases of wavelets. Two algorithms for the implementation of continuous wavelet transform are presented. The first is based on the computation of convolutions using the FFT with a sorrow boundary treatment. The second is an algorithm based on a dyadic interpolation scheme. Several applications in the field of non stationary filtering, singularity processing, and detection of sources of potential fields are evaluated and discussed. First, upon manipulating and modifying wavelet coefficients rather than Fourier coefficients, it is shown how to construct non stationary filters. In examples this technique is applied to the decomposition of the polar motion of the earth into several components. The underlying algebra of wavelet Toeplitz operators is discussed. In particular the user is made aware of problems of non-commutativity. Wavelet transforms of noise and de-noising methods are treated based on adaptive filter design in wavelet space (thresholding). Secondly, it is shown how Io-

XVI

Introduction

cal singularities may be detected and processed through wavelet techniques. In particular how the scaling behavior of wavelet coefficients reveals the local singularity exponent. Applications of this to fractal analysis of data (extraction of generalized fractal wavelet dimension) are introduced. In particular the wavelet correlation dimension is important for the interpretation of scattering on fractal objects. In the case of isolated singularities the wavelet technique is applied to the detection analysis of geomagnetic jerks in the data of secular wariation of the magnetic field. Thirdly, a special family of wavelets based on the Poisson semigroup is used to detect hidden singularities, corresponding to remote sources of potential fields. Only the remote field is available for the analysis, but cross scale relations of the wavelet coefficients may be used to localize and characterize sources of the field inhomogeneity.

Part 2

The second part was lectured by Wim Sweldens and the lecture notes are coauthored by Peter SchrSder, and Ingrid Daubechies. Starting from historical developments they try to answer the question: Why multi-resolution? In order to illustrate this the contrast with Fourier methods is explained, and the connection with filter banks is shown. This is illustrated with a simple example: Haar wavelets. At this point the focus is on the need for spatial constructions, as opposed to Fourier based constructions, revisiting Haar. Here, the essentials of lifting are introduced: predict and update. The next step goes beyond Haar, where linear and higher order polynomial based wavelet are Considered. Finally, implementation aspects of lifting, such as speeding up, in-place memory calculations, and integer to integer transforms get special attention. The lecture proceeds with a look at the need for second generation wavelets. Aspects of importance are wavelets for boundaries, domains, irregular samples, weighted measures, and manifolds. Here the use of lifting in order to construct second generation wavelets is explained. In order to construct wavelets on the sphere one needs to build triangular grids on a sphere: geodesic sphere construction. Also the definition of multiresolution on a sphere needs to be defined. Then one can focus on constructing filter operators on a sphere. Again lifting is used now to build spherical wavelets. This is where one arrives at the fast spherical wavelet transibrm. An application is discussed from spherical image processing. The next discussion points at techniques for triangulating general 2-manifolds. For this purpose wavelets defined on 2-manifolds are required, and multi-resolution transforms "off' manifolds are introduced. This is supported by examples.

Introduction

XVII

Part 3

The third and last part of the school is lectured by Willi Freeden and the lecture notes are co-authored by Volker Michel with a focus on harmonic wavelets on the sphere. For the determination of the earth's gravity field many types of observations are nowadays available, such as terrestrial gravimetry, airborne gravimetry, satellite to satellite tracking, satellite gradiometry. The mathematicM connection between these observables on the one hand and the gravity field and the shape of the earth on the other is called the integrated concept. In this lecture windowed Fourier transforms and harmonic wavelets on the sphere are introduced tbr example for approximating the gravitational part of the gravity field progressively better and better. The classical outer harmonic models of physical geodesy, concerned with the earth's gravity field, triggered the development of the integrated concept in terms of bounded linear functionals on reproducing Hilbert (Sobolev) spaces. It is of importance to deal with completeness properties and closure theorems for dense systems of linear (observational) functionals acting on outer harmonics and reproducing kernel functions. The uncertainty principle for functions harmonic outside a sphere is treated. This leads us to the terminology of 'space-frequency localization'. It is shown that the uncertainty principle is a restriction in gravitational field determination which tells us that sharp localization in space and frequency is mutually exclusive. Finally, a space-frequency investigation is considered for the most important trial functions used in physical geodesy. Here two types of scaling functions can be distinguished, vdz. band-limited and non-band-limited. It is illustrated that in all cases the constituting elements of a multilevel approximation by convolution consist of 'dilation' and 'shifting' of a mother kernel, i.e. a potential with vanishing zero order moment. Next, the concept of multi-resolution analysis for Sobolev spaces of harmonic functions is introduced which is especially relevant for geophysicM purposes. Two possible substitutes of the Fourier transform in geopotential determination are the Windowed Fourier Transform (WFT) and the W~velet Transform (WT). The harmonic WFT and WT are introduced and it is shown how these can be used to give information about the geopotential simultaneously in the space domain and the frequency (angular momentum) domain. The counterparts of the inverse Fourier transform are derived, which allows a reconstruction of the geopotential from its WFT and WT, respectively. Moreover, a necessary and sufficient condition is derived, that an otherwise arbitrary function of space and frequency has to satisfy in order to be the WFT or WT of a potential. Finally, least-squares approximation and minimum norm (i.e. least-energy) representation, which will play a particular role in geodetic applications of both WFT and WT, are discussed in more detail.

xvIII

Introduction

Exercises

and Demonstrations

All lectures were accompanied by practical exercises using flexible wavelet programs. Academic signMs as well as true geophysical data could be treated and analysed. Next to these 'hands-on' sessions also examples from a diversity of more complex or computationally intensive applications to support the theory were demonstrated. Two videos illustrated the power of the methods in the field of computer animation and earth gravity field approximation. The enthusiasm of the lecturers and the combination of theoretical foundations and developments, and the link to the practical applications lead to a good insight into the current status and the future challenges in the field of wavelets in the geosciences. We very much enjoyed organizing and taking part in the school so we hope that you enjoy reading the book and applying the wavelets in a similar fashion.

Delft, October 1999

Roger Haagmans Martin van Gelderen

Introduction to Continuous Wavelet Analysis Matthias Holschneider 1,2 1 CPT, CNRS Luminy, Case 907, F-13288 Ma~seille, France Institut de Physique du Globe de Paris, Laboratoire de G~omagn4tisme,

4 place Jussieu, F-75252 Paris, France hols@ipgp, jussieu, fr

1

A short motivation: why time-frequency analysis?

Imagine yourself listening to a piano concerto of Beethoven. Did you ever wonder how it is possible that something like melodies and hence music exists? In more mathematical terms the miracle is the following: our ears receive a one dimensional signal p(t) over the real line I~ that ,describes how the local pressure varies with the time t. However this one-dimensional information is then somehow "unfolded" into a two-dimensional time-frequency plane: the function over the real line p(t) is mapped into a function over the time-frequency plane that tells us "when" which "frequency" occurs. Now this is in the strict sense a contradiction in its own. A pure frequency is given by the one parameter family of complex exponentials e~ = e itw. Consequently it has no time-point attached to it since it goes from - c o to + ~ . On the other hand a precise time point-represented by the delta distribution 5(t) has all frequencies in it and hence now frequency can be associated with it. Therefore neither p(t) nor it Fourier transform ~(w) give us sufficient information. The hearing process is thus based on some compromise between time-localization and frequency-localization. This is also true for wavelet transforms as we shall see. The idea is to replace the elementary frequencies e~ = e iwt on which the Fourier transform is based (see next section) by a two-parameter family of elementary functions gt,f that are localized around the time point t and the frequency f . Before we construct these functions, we first recall some basics about the central tool in signal processing, the Fourier transform.

2

The Fourier transform

We consider four types of Fourier transforms: over the real line ]~ over the circle T, and over the discretized circle Z / N Z . 2.1

Fourier transform

over

Recall that the Fourier transform of a function s(t) over the real line is obtained by "comparing" s with the one parameter family of pure oscillations e~ (t) = e iw~

2

Matthias Holschneider

by taking all possible scalar products; that is for s E L 1(]~) we have :_~-oo

s ~ Fs,

dte-i~ts(t),

(Fs)(w) = (e~ I s)R =

where we have introduced the notation

(s [ r)~t = / + - 5 dt s(t) r(t), whenever the integral converges absolutely. We usually write ~ for Fs. The Fourier transform of s then is a function over the parameter w, which may be interpreted as a frequency. We therefore call ~'(w) the frequency content or frequency representation of s, and s itself may be referred to as the time representation. It is well known, that no information is lost, since the Fourier transform preserves the energy in the sense that (s]w)=~l

(~[~),

/dt,s(t)[ ~=

/ d t [ ~ ' ( w ) [ 2.

(1)

The inverse Fourier transform is given by the adjoint operator: For r E LI(~) N L2(]~) it reads

r ~ F -lr,

(F -lr)(t)=

+ : dw ei~ r(w),

FF-I=F-1F=~. It allows us to write s as a superposition of the elementary functions e~:

~(t) = 1

/,f

l/,f

dw~(w)e~(t) = ~

We may summarize the formulas by saying that

dw~(w)ei~t.

2~r-1/2F is unitary.

The convolution theorem. The convolution product of two signals reads F • ~(t) = f duF(t - u) ~(u). It is commutative (F * s = s * F) but the roles of F and s are distinct in applications. F is the filter and s is the signal to be filtered. The convolution theorem states

F • ~(~) = ~(~)~(~). 2.2

F o u r i e r t r a n s f o r m over Z

In practice we have only sampled signals. Mathematically these are sequences or functions over Z. The Fourier transform of a sequence in L2(~,) reads

Fzs(~) = Z ~ s ( ' ~ ) " mEZ

Continuous Wavelet Analysis

3

Again we write ~'(w) instead of FZs(w). Now, w is a priori a real number. But, since all functions e i~m are 2~ periodic, it turns out that

FZs(w + 2~r) = FZs(~o). We can identify periodic functions with functions over the circle T. Thus the Fourier transform maps functions over integers Z to functions over T F z : L2(Z) -÷ L2(T) Again it preserves the energy

E

ls(m)[2 =

['g(w)[2"

]02~d~ ~ m ~ ( ~ )

= ~e f T d ~ ( ~ ) .

mCZ The inversion formula reads

~(m) =

The convolution of two sequences reads

mEZ and again we have

F ,~(~) = 2(~)~(~) 2.3

T h e F o u r i e r t r a n s f o r m over T

For 27r-periodic functions we can define the n-th Fourier coefficient through

FTs(n) =

~

2~r

e~ns(t).

Thus the Fourier transform over the circle maps periodic functions (= functions over the 1-torus) to sequences (= functions over Z). Again we have conservation of energy

]FVs(m)l 2 = mEZ

f0

dtls(t)] 2 .

The convolutions of two functions over the 1-torus reads

F * s = ~0 27r dtF(t - u)s(u). In this formula F has to be taken as wrapped around (see Fig. 1)

4


Z

Fig. 1. The wrap-around in the convolution over the circle

2.4

The Fourier transform over Z/NE

In numerical algorithms however we still use another Fourier transform. It is defined for periodic sequences of finite length N. N

FZ/NZs(m) = ~ s(k) e2~k/N. k=l

Again this transform preserves the energy N 1 N 2 E Is(m)12= ~ k~ll F x / N x s ( k ) " m~--1

=

It can be inverted through 1

N

s(,~) = ~ ~ ( k )

e~'~k/N.

k=l

Again a convolution theorem holds. The convolution of two periodic sequences is defined in the obvious way: N

F , s(~) = Z E(m - k) s(k). k=l

Again we have to take F in a wrap-around manner. Now we have F * s(m) = F ( m ) ~ ( m ) .

2.5

Sampling and the Poisson summation formula

The various Fourier transforms we have encountered are linked together via the Poisson summation formula. It shows that there is a link between sampling and periodizing.


5

Perfect sampling. More specifically ibr a function s(t) over the real line we associate the sequence of its perfect samples at all integer points via z : (s) -+ (z),

(2s)(m) = s(,~).

In the same way we may sample a periodic function s at N evenly distributed points Z ~ : (T) -+ (Z/NE), (Zs)(m) = s(2zcm/N). A function over Z (= a sequence) may be down-sampled k-times (= throw away all points, except those situated at ..., -2k, - k , 0, k, 2k, ...)

E~: (z) -+ (z),

~ s ( m ) : s(k,~).

Finally a periodic sequence of length N may be down=sampled further by a factor k if k divides N, in which case we set

r~/Nz. (Z/NZ) -~ (Z/NZ), k z/Nz ~k s(m)=s(kmmodN), m=O, 1 , . . . N / k - 1 .

Periodizing. There is a natural map to come from functions over the real line to functions over the 1-torus, that is to periodic functions. It is the periodization operator. H : (~) -~ (T),

Us(t) = ~ s(t + 2~k). kCZ

In the anMogous way we may periodize periodic functions to obtain functions with a k-times smaller period. For this reason we write TT for the space of functions with period T. As before, T stands for T2~. k--1

HkV: (TT) --+ (TT/k),

Hs(t) = E s((t + Tm/k)modT). m=O

A sequence may be periodized to obtain a periodic sequence with period N

~:

(z) ~ (Z/NZ),

~ s ( m ) : Z s(m + kN). kEZ

Finally a periodic sequence of period N may be periodized to obtain a sequence with period N/k provided k divides N

HkZ/N~: (Z/NZ) -+ (Z/(N/k)Z), k-1

II~Z/NZs(m)=~-~s(m+l,kmodN), t-~O

m=O,l,...N/k.

6


The Poisson summation formulas. The Poisson summation formula states in its easiest form that periodizing and sampling are related through Fourier transforms. Z F = FVH,

FZ~ = HF

Or in diagram language

(a)

%

(v)

(~) -&

(z)

(~)

~

(z)

(~) -~

(T)

The analogous other relations with the above definitions are listed below

~ F z = FZ/~zrS~, Z~INZFZl Nz = Fz/(N/~)ZH2/Nz There is another useful picture for these relations. If we identify a sequence with a niodulated ~-comb,

l(m) ~ Z l(.~)~(t-.~), mEZ

and if we introduce the equally spaced delta comb as

u= Z

~(t- m)

rn6Z

then the sampling operator may be written as

Z:s~-~tl.s The periodization operator reads H : 8 ~-~ t12~r * 8,

and the Poisson formula finally reads simply

Band limited signals. A signal is band - limited if its Fourier transform is supported by some interval, say [wl, w2]. In case of a band-limited signal, we may actually compute with the samples only, provided they are sufficiently close. More precisely let Aw = w2 - wl be the bandwidth of the signal s. Suppose we sample s evenly with a frequency/2. That means we know the sequence

22./~s(k)

= s(2~k/¢2),

k c z.

Continuous W~velet Analysis

7

Suppose, $2 > Aw. Then s may be recovered from its samples via

s(t) = E ( N 2 ~ / s ? s ) ( k ) H ( t

- 21r£2k),

kCZ

where H is the function whose Fourier transform is the rectangular window of the interval [wl,w2]. Indeed, the Fourier transform of the s a m p l d sequence (Z2~/gs)(k) is the periodized Fourier transform of the original signal. Since this is band-limited and since the sampling frequency is high enough, the spectrum is just repeated without overlap. We multiply it in Fourier space with the rectangular window of the interval [Wl,aJ2] to cut out the spectrum of the original signal. Now this multiplication in Fourier space, amounts to a convolution with H of the modulated d-comb U2~/gs, and this yields the above formula. In particular, suppose, we have a band-limited filter F and a band-limited signal s, both in the same band [wl, w2]. We then may compute with the samples only. Indeed, by Poisson summation we have

The convolution product on the left-hand side is over the real line, the one on the right-hand side is the convolution of sequences.

The sampling spaces. Spaces of band-limited functions are particular instances of so called sampling spaces. These are spaces of functions over the real line, which are defined by there sample values only. For example consider the space of continuous, piecewise, affine functions with knots at the integers. A function s in this space is defined, if we know its value at the integers. The value at every intermediate point t with n < t < n + 1 is then obtained by linear interpolation s(t) = (t - n) s(n) + (n + 1 - t) s(n + 1) Consider the unique function h in this space that has h(0) = 1 and h(n) = 0 for n ~ 0. This is the triangular function h(t)=

{1-1t[ 0

for[t[ O.

Note that we write y for the complex conjugate of g. Other notations for the wavelet transform are )4;[g; s](b, a). These numbers will sometimes be called the wavelet coefficients of s with respect to the wavelet g. The wavelet transform can also be seen as a family of convolutions indexed by a parameter a. The convolution product of two function is defined as

s * r(t) = [ + ~ dt' s(t - t') r(t') = r * s(t). J-co


9

Then ~V~s(b, a) = ~a * s(b), where ~(t) = ~ ( - t ) . The two dimensional parameter space of wavelet analysis may be identified with the upper half-plane

lH={(b,a):beF~a >0}. 3.1

Wavelets and time-frequency analysis

As we explained in the introduction we want to built a kind of mathematical ear, based on atoms localized at different positions and frequencies. One way to enforce the basic wavelet to have oscillations is to require that

dtg(t)=O

~:~

~(w=O)=O.

Such a function has at least one oscillation. A bump is never a wavelet in that sense. Note however that the derivative of a bump may do very well. Now, since the 0 frequency is not contained in the spectrum of g, it is necessarily localized around a frequency different from 0, say o2. But then the scaled wavelet ga = g ( . / a ) / a is oscillating with a frequency w / a (see Fig. 2). Thus the upper half-plane of wavelet analysis is also a time-frequency half plane. The wavelet transform is a kind of mathematical ear if we identify b

time

020

- - < ~ frequency. a

3.2

Wavelets and approximation theory

By construction, the wavelet transform is a kind of mathematical microscope if we make the following identifications b < ~ position

(aA) -1 < ~ enlargement g ~

>

optics

The interpretation of a as a scale becomes even clearer if we consider the following. Suppose we want to analyse an arbitrary function s over the real line ]~ on different length scales. The first attempt to look at s at different length scales might be to look at smoothed versions of s. Therefore pick a real valued, localized, non-negative "smoothing filter" ¢(t) consider the family Ca(t) = ¢ ( t / a ) / a , a > 0 of dilated versions of ¢ and look at the smoothed versions

10


a=2 a=l

a=l/2

~a=l

t

'0 Fig. 2. The dilation operator in time and frequency. Note that small scales (a < < 1), correspond to high frequencies To~a),

of s. Since the width of the support of ¢(, is proportional to a one might say t h a t the smoothed version contains the details of s up to length-scale a, where the scale is measured in units of the size of the support of ¢a=1 = ¢. The features of s living on a smaller scale are smoothed out and not visible any more in ha- As a gets smaller, more and more detail are visible added and eventually one gets back all of s. Indeed, one can prove t h a t sa -~ s in L 2 (~). The idea of wavelet analysis is now to look at the details t h a t are added if one goes from scale a to scale a - da with da > 0 but infinitesimally small. Since aa contains all details of s up to scale a it follows that the difference cra-da -- aa are the details of s t h a t are living at length-scale a. Going to the limit da --+ 0 we are lead to look at W(b,a) = -aOaa~(b). (2) T h e factor a in front is for convenience only. Now since ¢ is smooth and compactly supported we m a y take the derivative under the integral and we obtain thanks to the identity - a O a ( ¢ ( t / a ) / a ) = ( ¢ ( t / a ) / a + (t/a) ¢ ' ( t / a ) ) / a t h a t

W(b,a)=-aOa

J-oof+~dtl¢( ~ a b

s(t)

= g, * s(b), with g(t) = (t Ot + 1) ¢(t) and g~(t) = g ( t / a ) / a . This expression looks similar to the one we have used before in the definition of the approximations a~. However one i m p o r t a n t thing has changed: ¢ which was a b u m p - as expressed by f ¢ = 1


11

- is replaced by g which rather satisfies * _ ~ dt g(t) = O,

as can be seen by partial integration: f t Ot ¢ = - f ¢. This follows also from

~(~)

=

~a~(~),

which can be verified by straight forward computation. Therefore to obtain the details of s at a length scale a we have to look at the convolution of s with the dilated version of a function g t h a t is of 0-mean; that is a wavelet. The convolutions Wa = ga * s corresponding to the details of s at length scale a is the wavelet transform of s with respect to the wavelet ~(. - t). K we sum up all the details Wa = W(., a) over all scales we might hope to recover s in the sense t h a t

/

e

P __daVi;~ =

a

in the limit e -+ O, p -+ oo.

- - ga * s --+ s

e

a

This actually holds if the wavelet g is derived from the smoothing b u m p g as before in (2). Then, since W~ = - a O ~ r ~ it follows t h a t the above integral evaluates to a~ - ap. The first t e r m goes uniformly to s in the limit e --+ 0, and the second t e r m goes to 0. Therefore the wavelet transform allows us to unfold a function over the one dimensional space I~ into a function over the two dimensional half-plane ~ of positions and details. It is this two-dimensional picture t h a t has made the success of wavelet analysis.

4

Some

elementary

algebraic

properties

We list some obvious properties of the wavelet transform for later reference. Linearity. The wavelet transform is a linear transformation, or linear operator, so t h a t the superposition principle applies; that is we have:

W A s + r ) = W ~ + W~r,

Wg(~) = ~ W ~

for every function s and r and every complex number a E C. With respect to the wavelet it is anti-linear

wig + v; s] = wig; s] + W[v; ~],

w[~g; ~] = ~wig; 4-

S y m m e t r y g ++ s. If we exchange the roles of g and s we then have the following useful formula.

12

Matthias Hotschneider Wavelet and Parity. For arbitrary functions the parity operator is defined by s ~ Ps,

(Ps)(t) = s ( - t )

A function is called even if P s = s and it is called odd if P s = - s . The analog for functions over the half-plane 7- is ~ - ~ P7-,

(P~(b,a) = T(-b,a)

By direct computation we can show that W[Pg; Ps] = 7~ )/V[g; s],

W[Pg; s] = ~Y[g; Ps].

Therefore the wavelet transform of an even (odd) function with respect to an even (odd) wavelet is itself" even P ]/Ygs = D?9 s. The wavelet transform of an even (odd) function with respect to an odd (even) wavelet is odd P W g s = -]/Ygs. Wavelet transform in Fourier space. We may write the action of dilation and translation in Fourier space

s(t) ~ s(tla)ta ~ * ~(~) ~+ ~(a~) s(t) ~ s(t - b) ¢=* ~(~) ~ ~(~) e -~b

(4)

Because of Parseval's equation for the Fourier transform (1) we may rewrite the wavelet transform in frequency space. Since by (4) we have g-~,a(w) = ~(aw) e -ib~ its follows that for g,s E L2(]~) we may compute the wavelet coefficients in Fourier space via

i /+f a~~(aw) e ~b~~(~)

(5)

The restriction )/Ygs(., a) of Wgs to one of the lines a = c is called a voice. It is given by the convolution of s with the dilated analyzing wavelet Wgs(-,a) = ga * s(.),where ~ta(t) = -~(-t/a)/a. Accordingly, since the convolution theorem states that for r E Ll(]~), s E L2(I~) we have point-wise almost everywhere

¢'~(~)

=

~(~) ~(~),

(6)

the Fourier transform of a voice reads W~s(-, a)(w) = ~(aw) ~(w).

(7)

Because a > 0, the positive (negative) frequencies of g will only interact with the positive (negative) frequencies of s; that is the wavelet transform does not mix the positive frequencies of the wavelet with the negative frequencies of the analyzed function. It therefore is natural to treat the positive and the negative part separately, and we define


13

D e f i n i t i o n 1. A function s C L 2(~) is called progressive or prograde iff its Fourier transform is supported by the positive frequencies only supp ~ _ ]~_.

It is called regressive or retrograde if the time reversed function s ( - t ) is progressive, or what is the same iff its Fourier transform is supported by the negative frequencies only. The space of progressive and regressive functions in L 2(I~) are closed subspaces and we shall denote them by H~_(1~) and H 2_(I~), respectively. The whole Hilbert space splits into an orthogonal sum L 2 (It~) = H~_(I~) @ H 2_(l~). Since both spaces are closed it follows that the orthogonal projectors on the positive (negative) frequencies are continuous. In Fourier space they read

~ ~%, 8 ~ ~-s,

~(~) ~+ o@) ~(~) ~(~) ~ o(-~)~'(~),

where the Heaviside function O is defined as {~ O(t) =

t O.

Because H + + H - = if we may use these two projectors to split the analyzing wavelet and the analyzed function into a progressive and a regressive component. For the wavelet transform we obtain

Wig; s] = W[H+g; ~ % ] + W[/I-g;/I-8]. Thus if g is a progressive wavelet only the positive frequencies of the analyzed function are "seen" by the analyzing wavelet and we have )/V[g; 8] = )/Y[g; II+s]

for g progressive.

From (7) it follows, that for progressive wavelets, each voice is a progressive function. In general the analysis with a progressive wavelet means an a priori loss of information on s for a non-p_rogressive 8 E L2(]~). However if s is real-valued, again no information is lost a priori. Indeed a real-valued function is defined by its positive frequencies alone since it satisfies the Hermitian symmetry

~(~) = ~ ( - ~ ) . Thus it may be recovered from its progressive part II+s:

7r Jo

14


where ~ z is the real part of z E C. Therefore the wavelet transform of a real function s with respect a real wavelet g may be expressed in terms of the associated progressive functions:

W[g; s] = 2 ~W[H+g; II+s] = 2 ~W[g; II+s] = 2 ~W[II+g; s]

(8)

Co-variance of wavelet transforms. We have already encountered the dilation operator and the translation operator acting on functions over the real line via Tb : s(t) -+ s(t - b),

D~ : s(t) --~ s(t/a)/a.

Correspondingly we have the dilation :Da,, a ~ > O, and translation operator 7~,, b~ E 1~ acting on functions over the half-plane.

n~+T)a,n,

n ( b , a ) ~ + a 1, T ~ ( b ~ , - ~ a ) ;

n(b,a) n(b-b',a).

\

/

\

/ ~\

7

\

/ \ \

\/

o/

\/

V T

o

b

Fig. 3. The translation operator and the dilation operator acting on a function on the half-plane

On the real axis a = 0 we get back the dilation and translation of functions over ]~. The wavelet transform satisfies now the following, co-variance property as can be verified by direct computation Wig; Das] = Z)~W[g, s]

W[g; Tbs] = 7b )A;[g; s],

or more explicitly Wig; s(t - b')](b, a) = Wig; s](b - b', a); }V[g; s(t/a')](b, a) = Fv'[g; s](b/a', ale').

(9)


15

This means that the wavelet transform of a dilated (translated) function is obtained by dilating (translating) the wavelet transform. With respect to the wavelet the following co-variance holds

)/V[g(t - b'); s](b, a) = Wig; s](b + b' a, a); )/Y[a ~g(a't); s](b, a) = Wig; s](b, a/a'). In order to gain a little more geometric intuition for these transforms let us look at the invariant subsets of the half-plane; that is those subsets of IH that are mapped into themselves when an arbitrary re-scaling (shifting) is applied. Clearly every straight line passing through the origin is mapped onto itself under all possible dilations and the same holds for any collection of such lines. Vice versa every invariant subset must contain together with the point (b, a) all its rescaled points {(b/a', a/a') : a' > 0}. But this is the straight line passing through (b, a) and the origin (0, 0). Therefore the invariant sets for the re-scaling are all cone-like structures with top at the origin. A similar argument shows that the invariant subsets for the translations are the strips parallel to the real axis. See Fig. 4 for an illustration.

,

o

b

Fig. 4. An example of a translation invariant subset (left) and a dilation invariant subset (right) of the half-plane.

4.1

T h e uncertainty relation.

As is well known, it is impossible to be simultaneously arbitrarily2 localized in time and in frequency. More precisely if we view Is(t)l 2 and I~(t)l (after suitable normalization) as probability densities for the distribution of times and

16

Matthias Hotschneider

frequencies, we may introduce the mean values < t >=

f dt t Is(t)12

< w >=

f dw w IF(co)12

f ls(t)l 2 ' and width At = ~/~ t2 >,

f ts(~)f 2

'

/~LO = X / ~ W 2 >,

where < t 2 >=

f dt t 2 Is(t)l 2

f Is(t)l 2

< w 2 >=

f dw w 2 IF(w)12

'

f t ( )12

'

then the Heisenberg uncertainty relation states that A t A w > 1. If we associate with any signal a rectangle of size A t and Aw with its center at (< t >, < w >) in the time-frequency plane, then the Heisenberg uncertainty relation gives a lower bound for the area of such "phase - space cells". How do our translation and dilation operations act in terms of these quantities? It is easy to see, that the following holds Tb : < t > ~ < t > +b, A t ~ A t , < w >~+< w >,

A w ~-~ A w

Da : < t > ~ a < t >, A t ~ aAt, < w > ~ a < w > /a, A w ~+ Aw/a. Therefore, we do not change the area in phase space of our wavelets, but we choose between time localization and frequency localization in such a way that A w / < w > remains invariant. See Fig. 5 for an illustration. 5

The

basic

functions:

the wavelets

Let us come back to the wavelets itself. As the name says, these functions are elementary oscillations. As such they should be localized in time, they should also be localized in frequency space, and they should be oscillating. Localization in time and frequency can be quantified by the following conditions

Ig(t)l 0 by

F(z) =

dt t z-1 e -t.

These wavelets are highly regular but have an at most polynomial decay at co. This is mirrored in the fast decrease of the Fourier coefficients and a lower regularity of the Fourier transform (at w = 0). ~(w)={O ~e-~

forw>0 otherwise

Here we recognize that for c~ E N these wavelets are nothing but the a t h derivative of the Cauchy kernel. Accordingly we obtain the following interpretation

24


of the wavelet transform of s C L2(I~) with respect to these wavelets: take the projection H÷s of s into the space of progressive functions H~(IR). This function can be extended to an analytic function over the upper half-plane. If we call this function T(z) we have (a E N) W~s(b,a) = a ~ Y(~)(b +

in),

Y(~)(z) = OTY(z).

Thus the wavelet transform with respect to these wavelets is closely connected to the analysis of analytic functions over the half-plane. This observation is due to Paul [20], who has used these wavelets in quantum mechanics. These wavelets are complex valued and we may distinguish between the modulus Ig(t)l and the phase argg(t). The phase is only defined up to an integer multiple of 2~r. But on every open set on which lg(t)l ~ 0 it may be chosen to be a continuous function, if g itself is continuous. At every point t where g(t) = 0 the phase is not defined. The phase speed is called the instantaneous frequency d

a(t) = ~ argg(t). This name is justified by the fact that the instantaneous frequency of the pure frequency e i ~ is equal w. By direct computation we can verify that for the Cmlchy wavelets we have [g(t)[ = (1

+ t2) -(l÷a)/2,

Therefore the phase turns (1 ÷

argg(t) = (1 + a) arctant,

a)/2 times as t goes from - o c to +c~.

+1/2

t~

-t:2

t

~

-1.5 !

I

i

+l,5

3

Fig. 10. The Cauchy wavelet for c~ = 2 in time and frequency representation. The phase turns only finitely many times. Note the singular (-=- non-smooth) behavior at (a2~ 0.


25

The Gaussian "wavelets" of J. Morlet. T h e y where first used in geophysical explorations [10] and are at the origin of the development t h a t wavelet analysis has taken since this time. These wavelets are obtained by shifting a Gaussian function in Fourier space, or what is the same, by multiplying it with an exponential g(t)

= e

=

A more convenient way to parameterize them is to fix the central oscillation at wo' = 1, say, and to change the size of the envelope

g(t) = e % -t~/2~. Strictly speaking this function is not a progressive wavelet, it is not even a wavelet, because it is not of zero mean. However the negative frequency components of g are small compared to the progressive component if w0 > 0 is large enough or (what is the same) if a is large enough. The phase p u l s e s - - u p to a c o r r e c t i o n - - a t a constant speed

d4)(t) dt - ~2(t) = Ojo + p(t)

112~

+1i2

~|

i t

5.336_.

Fig. 11. The wavelet of J. Morlet. for wo = 5.336... in time representation. Note that the real part (solid line) and the imaginary part (dotted line) are oscillating around each other with constant phase speed. The modulus and negative modulus (thick solid line) are an envelope for these oscillations. (Right) The Fourier transform of the wavelet of J. Morlet for wo = 5.336 .... Note that it does not vanish at the origin w = 0, but that it is numerically small.

26

7

Matthias Hols~_ueider

Some explicit analyzed functions and easy examples

In this section we list some examples t h a t reveal the main features of the wavelet transform. Also not all statements shall be made mathematically precise this section serves to enlarge our intuition about wavelet transforms. 7.1

The wavelet transform of pure

frequencies

These functions are related to the invariant strips in the half plane. Consider the function e~o(t ) = e i~°t, t h a t describes an oscillation with frequency ~o. The analyzing wavelet should be absolutely integrable in order to have an everywhere well-defined wavelet transform.

.4

4

a

a

1M

1t4

1116

1116

-~ -1 0 ÷1 b +~ -~ -i 0 ÷1 b ÷~ Fig. 12. The modulus of the wavelet transform of the pure oscillation with the help of the wavelet of Haar. Black corresponds to high values whereas white are small values. Note that the maximum of the amplitude is at scale a = 1/4. However there are oscillations for large scales due to the poor locMization in frequency space of the Haar wavelet. Note that the scale-parameter axis is logarithmically.

Since e~0 are progressive functions, every voice of the wavelet transform will be a progressive function too. T h e pure oscillations are up to a phase invariant under translations Tbe~o ~ e-i~o b e~o.

Some of the main features of the wavelet transform of pure oscillations follow from this translation invariance and from the co-variance of the wavelet transform. Using the co-variance property (9) we m a y write: ~42[g; e~o](b, a) = W[g; e~o (t + b)](0, a).


27

From the invariance of e~0 and the linearity of the wavelet transform it follows that W[g; e¢o](b, a) = e-i¢°ZW[g; e¢o](0, a). Therefore every voice--the scale a is fixed to a constant--is a pure oscillation with the same frequency ~o but with an amplitude and a phase, that may vary from voice to voice. The whole transform is determined by the wavelet transform along a straight line passing through the boarder line of the half-plane, e.g. b = 0. Carrying out the integration

}/~[g;e~o](O'a)=/+~dtlg( ) ~ _ ~ t

e~°t = ~( a~0 )

we obtain and thus the modulus and the phase of the wavelet transform read

M(b, a) = ~(a~o) ,

~(b, a) = arg~(a~o) + ~0 b.

(15)

Therefore the instantaneous frequency of every voice is imposed by the analyzed pure oscillation Ohm(b, a) = The modulus of the wavelet transform is constant along every voice, (a = c), and its behavior aiong the line b = 0 is given by the modulus of the Fourier transform of the wavelet. Note that the phase itself is only defined modulo 27r, but if ]~/0s is differeutiable, then the derivatives Ob~(b,a) and Oa~(b,a) are uniquely defined wherever ]M(b,a)] ~ O. Suppose that lgl has its maximal value for w = wo. Then the modulus of the wavelet transform takes its maximal value for the voice at scale

a = Wo/~o

(16)

This might indicate that Wo/ais somehow related to a frequency. The localization around this central voice depends on the localization of the spectral envelope of g around Wo. Therefore the analysis of a pure oscillation with the help of the Haar wavelet wilt give rise to a poor localization in the half-plane (see Fig. 12), whereas e.g. the Morlet wavelet gives rise to a high localization around the scale (see Fig. 13). Consider the lines of constant phase; that is the set of points in ]H, where the phase $ has a given value. If the phase of ~ is constant as e.g. if the wavelet has a real valued Fourier transform, then these lines are the straight lines b = c. Note that for wavelets with non-real Fourier transforms, these lines are curved. 7.2

T h e r e a l oscillations

In the case we analyze real valued functions it may be useful to consider however a progressive wavelet, since then we may distinguish between modulus and phase.

28


5 -4r

0

b

+4~

~r

0

b

~r

Fig. 13. (Left) The modulus of the wavelet transform of a pureoscillation with the help of-the Morlet wavelet. Note that it is well localized around the central voice a = wo/~o = 1.(Right) The phase of the wavelet transform of a pure oscillation with the help of the Bessel wavelet. The phase pulses at a constant speed ~0 -- 1, independent of the voice.

If on the contrary we use a real-valued wavelet, then the wavelet transform is oscillating itself, making the frequency analysis more difficult. Indeed, consider

s(t) = cos(~ot + ¢)

1 (e_i¢ e_~o~ + e~¢ e~Ot)"

The frequency content of s is localized at the frequencies -~o and ~o- The associated progressive function is the pure oscillation H + s ( t ) = e ~¢ e ~¢°~ / 2.

For progressive wavelets the wavelet transform of s is given by the wavelet transform of the associated progressive function H + s , which was treated in the previous example. A real-valued wavelet instead "sees" the positive and the negative frequency component. By (8) we obtain

wig; cos(~0t + ¢)](b, a) = ~ (~(a~o) e~(b~°+¢)) This implies that every voice is a real oscillation with frequency ~o and a phase and an an~lplitude that varies from one voice to the other Wig; cos(~ot + ¢)](b, a) = A(a) cos(~o b + ¢(a));

A(~) = I~(~o)1, ¢(a) = arg~(a~o) + ¢ See Fig. 14 for a sketch.


29

0

-7

0

b

+7

Fig. 14. The wavelet transform of a real oscillation with respect to the real wavelet of D. Mart. The wavelet transform is real-valued. It oscillates at the speed of the analyzed function.

7.3

The homogeneous singularities

These functions are related to the dilation invariant regions in the half plane. Consider a function t h a t is 0 for negative times t < 0 and t h a t suddenly starts more or less smoothly:

s(t) = Itl~ = o(t) t ~,

~ > -1,

(17)

where we have used the notation ]tl± = (ttl +t)/2. The bigger ~ is, the smoother is the transition at t -- 0. In order to analyze these functions, the analyzing wavelet g should be localized such t h a t tag is still in L 1 (~). For instance

Ig(t)l < c(1 + It1) -(1+~+~) with some c > 0 will be enough. The onsets a r e - - u p to a scalar--dilation invariant; t h a t is they are homogeneous functions of degree

s(~t) = ~ 8(t)

(is)

The most general dilation invariant function with exponent a > - 1 , however can be written as s(t) = c_ Itt ~- + c+ IriS. We now want to show how we recover the parameters a, c± in the wavelet coefficients. As for the pure oscillations m a n y of the general features of wavelet transforms of homogeneous functions follow from this invariance and the co-variance of the wavelet transform. From the homogeneity of s and (9) we have W[g; s](b, a) = Wig;

s(at)](b/a,

1) = a s W[g;

s](b/a,

1).

30


Therefore the wavelet transform satisfies the same type of homogeneity as the analyzed function: (A > 0) W[g; s](Ab, Aa) = A~ )/Y[g; s](b, a)

(19)

Because a is real valued, the phase of the wavelet transform remains according to (19) unchanged, along a line b/a = bc.

)A;(b, a) = aC~F(b/a),

F(b) = F¢(O, a).

Therefore the lines of constant phase converge towards the origin [9]. The modulus of a zoom instead decreases with a power law that mirrors the type of singularity of the analyzed function. We therefore may recover c~ by looking at the scaling behavior of IWl along a straight line passing through the origin. Explicit computation yields that with respect to a progressive wavelet we have

YV(O, a) = CeiCa a. Here C is a real valued constant, it encodes the size of c±. ¢ is a real valued phase, and it encodes the relative size of c±, or what is the same, the local geometry of the singularity. Finally the exponent, a is the scaling exponent of the singularity. See Fig. 15.

/fJ 0

~14

n12

3n14

.....

log(a)

r:

57t/4

3~/2

7rc/4

Fig. 15. A log-log plot reveals the scaling exponent of the singularity. For a given exponent, the phase along a straight line is constant. Its value encodes the local geometry.

Consider as first explicit example the wavelet transfbrm of a delta function localized at the origin. It is not quite of the kind (17) but it satisfies the invariance (18) with a = - 1 . By direct computation we find a


31

_5

_0

-2o 0 b +20 -2o 0 b +20 Fig. 16. (Left) The modulus of the wavelet transform of ~ with respect to a Morlet wavelet. For representational reasons we re-scale each voice to take out the trivial factor 1/a. Thus actually a IWl is shown. (Right) The associated phase picture. Note that the lines of constant phase converge towards the point, where the singularity is located. We have used a cut-off: for small valued of the modulus the phase is put to 0.

and thus a behavior of a -1 of every zoom. In Fig. 16 we have sketched the wavelet analysis of 6 with the help of a Morlet wavelet. To give an explicit example of a true onset we will use the Cauchy wavelet g~ = (2zr) -1 F(fl + 1) ( 1 - it) -(1+~) with fl > a. As we have seen in Sect. 6.2, for this wavelet we m a y identify the half-plane with the complex upper half plane by setting z = b + ia. Indeed, to give a different argument to see this, the complex conjugated, dilated and translated wavelet m a y be written as a g--~

= (27r)-1 i1+~ -P(1 -k ]~) a ~ (b + ia - t) -1-/3

=

-1 r ( 1 + Z) an

( /

Therefore the wavelet transform with respect to gz i s - - u p to the pre-factor a n an analytic function in z = b + ia. In the case of the onsets we obtain by direct computation W[g~; Itl~](b, a) = c a ~ ( - i z ) ~ - Z ; c = (2~r) -1 e -i'(1+~)/2 F(1 + a) F(fl - a). 7.4

The wavelet analysis of a hyperbolic chirp

Consider the function

8(t) = ( - i t )

(20)

32


~o

30

_1

_0A

0

b

+~5

-~5

0

b

÷~5

Fig. 17. The wavelet analysis of the onset vq with the help of some Cauchy wavelet. (Left) the re-scaled modulus x/~ lYVIis shown. Note that the a axis is logarithmically. (Right) The value of the phase of the zoom b = 0 is-37r/4 showing a specific local topologie of the singularity. The straight lines of constant phase are curved due to the logarithmic scale in a.

This is a progressive function (at least in the sense of distributions). In addition this function satisfies the discrete scaling invariance 8(~t) = s(t),

.y = e 2~/~,

which is mirrored in its wavelet transform via

The modulus of s is constant--with the exception of t = 0 where it j u m p s - - a n d a phase speed that decreases with a / t as Itl -+ cc M(t) =

cosh ~ -

+ sign(t) ~ sinh T

= 7'

which is the reason for calling s a "hyperbolic chirp". The wavelet transform of the hyperbolic chirp with respect to the Cauchy wavelet g~ is obtained using the analytic continuation in a of (20)

w i g . ; (-~t)~](b, a) = ca" (-iz) ~ - ' , c = (iTr)-1 sinh(c~Tr) F(1 + is) F ( . - ic~). For the modulus and the phase we obtain explicitly

M(b, a) -- lcl a~ (b2 + a2) -~'/~ e -'~a~¢~n'~/', ~(b, a) = arg c_,~ arctan (a/b) + ol log V/~ +

as


33

The lines of constant phase are now logarithmic spirals turning around the point (0, 0). However only the part of the spirals that lies in the upper halfplane is visible in the wavelet transform. These spirals are solution of a l o g ( N ) - # arg z = c

~

Iz]

~--- (2' e a r g z ~ / a

Thus the density of spirals is determined by the quotient #/a.

.20

.15 a

.0 -5

o

b

+15

-5

o

~h

+15

Fig. 18. (Left) The modulus of the wavelet analysis of a hyperbolic chirp with the help of a Cauchy wavelet. Note that the modulus is localized around the straight line a = b that corresponds to the instantaneous frequency of the chirp. (Right) The phase picture of the wavelet transform of a hyperbolicchirp. Note that the lines of constant phase are logarithmic spirals turning around the point (0, 0).

Let us now locate the points, where the modulus of a voice is maximal; that is we look for the points in the half-plane where &M(b, a) = 0 and O~M(b, a) < O. By direct computation we find that these points are located on the straight line a = Kb

(21)

C~

The central frequency of the analyzing wavelet gmu is given by w0 = #. If we identify as in (16) the quantity #/a with a frequency, we see from (21) that the modulus of the wavelet transform is located around the points in the upper half plane, where the time corresponds to the position b ++ t and the inverse of the scale corresponds to the instantaneous frequency iz/a ++ J2(t) of the analyzed function. The phase speed of every voice on the line (21) corresponds to the instantaneous i~equency of the analyzed function if we identify again the position parameter b with the time t:

Ohm(b, All this can be observed in Fig. 18.

ct = f2(b) = -g

34


a

.0.75

-~2 b +~ -~ 0 b ÷~2 Fig. 19. The modulus (left) and phase (right) of the wavelet analysis of the superposition of two pure frequencies with the help of the Morlet wavelet with internal frequency w = 6. The two components are clearly visible

7.5

Interactions

In the last examples we want to combine the previous elementary examples to obtain more complex functions. Clearly the superposition principle tells us that the wavelet, transform of a superposition of functions is the superposition of the respective transforms. However this does not apply to the modulus and the phase-pictures, because they are obtained by non-linear operations on the transform. Consider e.g. the superposition of two pure frequencies at frequency ~o and ~1. For the sake of simplicity we suppose that ~ is real with a single maximum in w0 and monotonic to both sides. We then have

a) = arg

e

+

Thus for voices where [g(a~o)l > > [g(a~l)l the phase is determined by ~0, whereas in the inverse case the phase is essentially determined by ~1. Thus in the phase picture we observe a transition from the phase speed ~o to ~1- Compare Fig. 19. For the modulus the same reasoning shows that it consists essentially in two strips if [g(~0/~l)[ 0.

Indeed, that mg,h is of that form follows by covariance: the combination AJ'W is an operator that comutes with both dilations and translations hence it must be a 0-degree homogeneous Fourier multiplier. The specific form can be found by apllying A/IW to the 5 distribution.

Kernel. Consider now the operation Wavelet synthesis followed by wavelet analWgAJh. It is given by a non-commutative convolution over the half-space

ysis

W g M h r = IIg,h * r, where the convolution product reads

and the kernel is given by

Hg,h(b, a) = }4;gh(b, a). All formulas can be proven by simple exchange of integrals.

38 8.3

Matthias Holschneider The reconstruction formula

A wavelet is called admissible if it satisfies I~(~)1 ~ < 0.

All wavelets we have encountered, except the Morlet wavelets, are admissible. The Morlet wavelets are numerically admissible. For admissible wavelets we can define two constants c~ = ~0 ~ dw i?(±wll2 W

For real wavelets we have c + = c~- thanks to Hermitian symmetry. For progressive wavelets we have c~ = 0, since they vanish on negative frequencies. For regressive wavelets we have c + = 0. If h is another admissible wavelet we may consider the following constants Cg,h ~

fo~ ~ (

~)~(±~,)

We then have

Mh Wgs = c~flI+s + c~,flI-s i

+

= ½(C~h + C;,DS + ~(Cg,h -- C;,h)HS, where H ± are the projectors on the positive / negative frequencies, respectively, and H denotes the Hilbert transform. Some particular cases are: • For g = h, real valued, admissible, we have

Mg Wgs = cgs. Thus s may be recovered fl'om its wavelet transform via a wavelet synthesis. • For s progressive and g, h arbitrary we have

• For s regressive and g, h arbitrary we have

A4g Wgs = c~hs. • For s arbitrary and g or h progressive we have

Thus we extract the positive frequencies of the signal. • For s arbitrary and g or h regressive we have

Mg Wgs = c~,flI-s.


39

Thus we extract the negative frequencies of the signal. • For s real valued and g or h progressive we obtain the analytic signal associated with s. In particular we have

~ M g ~ g s = cg,h + s,

-~M9 Wgs = c+h H8.

Thus wavelet analysis followed by wavelet synthesis allows us to obtain the Hilbert transform of the signal. • For s real valued and g or h regressive we obtain the conjugated of the analytic signal associated with s. In particular we have

~A49 Wgs = C~,hS, 8.4

~A49 }/Ygs = --C~hHS.

The energy conservation.

For any admissible wavelet we have t h a t the energy of the transform stays bounded if the signal has bounded energy

dtls(t) l2 . f ~ ~b~---2a a lWg~(~,a)l 2 _< (c~++ c ; ) [ JR We m a y even have equality in the following form:

/ : dbd___aaa 'Wgs(b' a ) [ 2 = C9 ]R dt Is(t)l 2 . This holds for s progressive and g arbitrary, admissible with C 9 = c+, or for s regressive and g arbitrary, admissible with C 9 = c~ or for s arbitrary and g real valued with C 9 = c + + c~" = 2c +. In all other cases we have in general only the estimate above. The proof is based in all cases on the fact that the adjoint of )4;9 is up to a factor also its inverse:

(wgs I Wgs)~ = (~ l M9w98) R = c9 (818)~. 8.5

The space of wavelet transforms and reproducing kernels

Not all functions over the half-plane are wavelet transform of some function of the real line. A function r is the wavelet transform with respect to an admissible wavelet g of some progressive function s if the coefficients r satisfy

r(b, a) = --~'D?gj~gr(b , a). cg

Indeed, if r = )/Ygs, then s = Jk49r , and thus the above equation. Now this is a non-commutative convolution with a kernel

r(b, a) = II • r(b, a),

1 ~ = -~Wgg.

c~

40


More explicitly it reads

This equation is called the equation of the reproducing kernel. Thus there are correlations with a correlation length with order of magnitude a in the wavelet coefficients.

_i

..2

-2 0 b +2 '0 b +2 Fig. 23. The wavelet transform of the Cauchy wavelet with respect to itself: the reproducing kernel.

The reproducing kernel can formally be interpreted as the minimal internal correlation of the wavelet coefficients. We only do formal computations. Consider a r a n d o m function r over the real line. We note by E (r(t)) the expectation of the r a n d o m variable r. The correlation of two time points t, u is the mean of the r a n d o m function ~(t) s(u)

¢(t, u) = E

A white noise s is a "function" t h a t is of zero mean, and for which different time points are completely un-correlated E (s(t)) = 0, E (s(t) s(u)) = 5(t - u). Consider the wavelet transform of such a random function. It is itself a random function, but this time over the half-plane. The correlation between two points in the half-plane is given by

qS(b, a; b', a') -- E (Wgs(b', a') Was(b , a))


41

We exchange formally the integration with the mean over all realizations and obtain (9b,a =

TbD~g)

qb(b,a;b',a').=E (/~+CCdtgb,a(t)s(t) f_+~dU-jb,,a,(u) s(u))

:

o

o)

In the last equation we have used the correlation function of a white noise. Integrating over the delta function we see t h a t the correlation function is given by the reproducing kernel.

IH(b-b' a) --5 a ' a'

• (b, a; b', a') = co ~

Therefore even if the analyzed function is completely un-correlated, the wavelet coefficients show a correlation because of the reproducing kernel property. The and Look at Fig. 24 where the scalecorrelation depends on dependency of the correlations is visible. Note however that we did not quite explain this phenomenon. We only showed, t h a t in the mean over m a n y realizations the correlation is given by the reproducing kernel. Nothing was said about a single realization.

(b - b')/aI

a

a/aL

a

0

-2 0 b +2 -2 0 b +2 Fig. 24. The modulus (left) and phase (right) of the wavelet transform of a white noise with respect to the Morlet wavelet. The regions of correlation change proportionally with the scale.

42 9

Matthias Holschneider Extensions

to higher

dimensions

In higher dimensions there are two distinct approaches. The first is to use the dilation and translation as before to generate a family of analyzing wavelets. T h a t is we define for functions in L 2 (]~n)

W[g,s](b,a)=~dxl-~(~ab),

bc~'n,a>O

The parameter space of wavelet analysis is now the upper half-space lI:In = {(b, a) : b E 1Rn,a > 0} Typically g has a Fourier transform that is localized in some corona. Therefore we may identify a ++

Note that a is degenerated in the sense that it contains only information about the size of the spatial frequency but no direction information. The wavelet synthesis now reads

Jt4[h'r](x)= fo~da

~ db 1----h r~ (~ab

r(b,a).

We have the following relations.

Adjoint. /~ -dbda F(b, a) )/Ygs(b,a) = fR~ dx A4gr(x) s(x). a

Inversion. J~4hWg= F-: mg,hF, with

In case that mg,h(~) = constant we say (g, h) is an analysis reconstruction pair.

Energy conservation. If (g, g) is a reconstruction pair, /

then

dbda [Wgs(b,a)lU =

=cg

a

dxls(x)l 2,

cg--

- - I~(a~) a

.


43

Kernels. Vl;g~4hr =

H * r,

//=

Wgh,

where

II .r(b,a) = / ~ db'da' t -~II \(b-b'~] Reproducing kernel. In particular, for (g, h) an analysis-reconstruction pair, we have t h a t r = Wgs for some s iff r = II~,h * r,

II~,h .........

1

W~h,

Cg,h

Cross kernel. If (g, w) is

Cg'h ----

an analysis-reconstruction pair, then

WhS = Ilg+h * ]/YgS, Hg-~h = 9.1

~o~ da a ~(a~) h(a~).

1

cg,h

)4;hr.

The rotation group

There is a second approach to higher dimensions. Let SO(n) be the group of rotations of Euclidean n-space. We denote by Ox the rotated vector x E ]~n for the rotation 0 E SO(n). We then construct a wavelet analysis based on dilation rotations, and translations.

Wgs(b,a,O) = / dx l-g (O-*(~- X) ) s(x). Now the wavelet coefficients are functions over the space ~2 = ]H ~

×

SO(n).

The rotation p a r t allows us to obtain directional information too. Now a wavelet is admissible, if

0 < ~g =

~~

1~(~){~
-

/,°

_> ,~-~

~j+l

Z

=

.

jEZ __( M

zf aEZ

Ig(~,)l

~

.,c,,

dw o7/

"

1~(~)1"

46


I

Fig. 25. The Fourier multiplier (25) of the partial reconstruction s # (24) over infinite many voices.

showing t h a t g has to be admissible. Before exploiting this relationship we do some elementary considerations in an H_~ (l~) context. First a formal considertion. Note that the sum (24) m a y be considered as a Riemann sum, namely for p regular enough: +oo

/ ~ da p(a) a

lim(~ c~---~l

1) Z

P(~)

n=--oo

Therefore if h is a reconstruction wavelet, then (a - 1)~'(w) -+ Cg,h as a -+ 1, A

and therefore (a - 1) s# -+ Cg,h~ and a complete reconstruction of s is obtained. But even for a different from 1, s # may be close to s provided the multiplier is close to a constant. The next theorem gives an estimation of the quality of the approximation of the partial reconstruction. Theorem

4. Let g, h e LI(~) N H~(IR) satisfy

jEZ

I~(~J~)i ~ -< ~ < ~,

?C I~(~J~)j ~ -< ~
O, and let s E H~ (1~). Suppose that there is a constant c, 0 < c < 1, and another constant c E C such that

ess sup ~>0 ~ .~zZh(""') ~(""') - 1 < ~. Then the following limit exists in H~ (]~) N

s#=

lim S N =

N ~oo

r+oo

~-~ ] db)/Ygs(b,a n) hb,~", N-+oo n'-~_N J_oo lim

(26)


47

and we have the following estimation IIs - ~ l h

< e Iisli:

For h = g the condition of the theorem is satisfied iff 0 < A = essinf ~>0 ~

[~(a'~w)[ 2 ,

nEZ

B = esssup a>o E

[g(anw)[2 < 0%

nEg

in which case we can set 2

C-B+~,

B-A e-B+A.

(27)

Therefore the smaller e or the closer the constants A and B are to each other, the b e t t e r the reconstruction i s - - a t least in the mean square sense. 10.3

An iteration procedure

Until now we only have obtained a partial reconstruction s # t h a t approximates s. As we shall see now, this is only the first step in an approximation scheme t h a t allows us to reconstruct s completely from the knowledge of the voices at scale a = a n. We state it in the following abstract form. T h e o r e m 5. Let K : B -~ B be a bounded operator acting in a Banach space V that satisfies IIK - ~IIB v _< e < 1.

Then I42-1 exists and is in B ( V ) , the set of bounded linear operators acting in V . Let s C V be given. If we set S n + 1 ~--- S n -~- r n + l ~

with So = s, then s~ -+ K - i s

rn+l

= 8 n -- Ks

m

as n -~ co and the difference may be estimated as fin

I I K - ~ s - s " [ [ v T(x)I ~ 0, ¢ E

[0, 2~)

__ __1 wdw 2~r n~z o

d(p 7(w, n) e -i(kw cos(¢-~)+ne)


= 2--~E

ein¢

wdw

65

d~ 7(w, n) e -i(kw cos(~)+n~)

nEZ

wdw "/(w, n) Jn(kw) e inC.

= -27r

Upon identifying the terms of both expressions that the expansion coefficients ~f and ~ are related as stated in the lemma. [] Now the following limit is known to hold uniformly for 0 E I, compact (see e.g. [22]) r----

lim J2-~FYlm(Oll , ¢) = Jm(O)e im¢. l-+c~o

v

t

We now propose to look at ~he limit a ~ 0 in oo

Note that

z~w Ty~12 2I + 1 41r

f,~t 0. One can show that R(3) > .678, R(5) > 1.272, R(7) > 1.826, R(9) _> 2.354, and asymptotically R(N) ~ .2075N [12]. 5. Refinability: ~(x) satisfies a refinement relation of the ~brm N

l=-N-bl

This follows from similar reasoning as in the interpolating ease starting from 80,k --~ (~O,k. As before subdivision can be written as: Sj+l,1 ~- E hl-2k 8j,k. k

In the case of quadratic subdivision, the filter coefficients are ht = {-1/8, 1/8, 1, 1, 1/8, - 1 / 8 ) . For quartic subdivision h~ = {3/128, -3/128, -11/64, 11/64, 1, 1, 11/64,-11/64,-3/128, 3/128} results. These constructions are part of the biorthogonal family of scaling functions described by Cohen-Daubechies-Feauveau [5], where they are named (1,3) and (1,5) respectively. An interesting connection between Deslauriers-Dubuc interpolation and average-interpolation was pointed out by Donoho [12, Lemma 2.2]: Given some sequence {s0,k} apply interpolating subdivision of order N = 2D to it. Comparing this to the sequence that results from applying average-interpolation of order N ~ = 2D - 1 to the sequence {s~,k = s0,k+~ - So,k} we find sj,k = 2j sj,t k (once again assuming equal sized intervals.) This observation follows directly from the fact that the average of the derivative of an interpolating polynomial p is simply the difference of the values that the polynomial takes on at the end and start of an interval, divided by the size of the interval. As a consequence we have

dvI(x)

=

+ 1) -

(3)

where the superscripts stand for I-nterpolating scaling function and A-verage I-nterpolating scaling function. This provides a simple recipe tbr computing the derivative of a function f(x) which results from interpolation of some sequence {s0,k}: simply take successire differences first and then apply the average-interpolating subdivision of one order less. Obviously, one could also take the differences and then apply an interpolating subdivision method, not necessarily average-interpolation. In that case the limit function would be an approximation of the derivative, while Equation 3 holds exactly. 6.5

B-Spline Subdivision

B-splines and in particular cubic B-splines are a very common primitive in computer graphics. Their popularity sterns from the fact that they are C 2, which

96

Wim Swetdens and Peter SchrSder 1.00.50.0 -0.5

I

I

I

I

I

I

I

I

-3

-2

-1

0

I

2

3

4

Fig. 16. Cubic B-sptine scaling function.

is important in many rendering applications to ensure smooth shading. They also possess the convex hull property and are variation diminishing which makes them a favorite for curve and surface modeling tasks. In this section we will show how cubic B-splines can be used as a predictor in wavelet constructions. In order to keep the exposition simple we will only consider cardinal cubic B-splines here, i.e., all control points are associated with integer parameter positions. We will atso assume a bi-infinite sequence for now and only discuss the boundary case later. Given a set of cubic B-spline control points at the integers {s0,k} subdivision tells us how to find a set of control points at the half integers which describe the same underlying B-spline curve. Repeating this process as before the control points become dense and converge to the actual curve (see Fig. 16). Typically this subdivision rule is described through a convolution with the sequence {hk} = 1/8 {1, 4, 6, 4, 1}, i.e., sj+l,2k = 1/8 (sj,k-1 + 6sj,k +

By,k+1)

sj+t,2k+1 = 1/8 (4szk + 4szk+l ). As stated this implementation of cubic B-spline subdivision does not fit into our wiring diagram framework, since it cannot be executed in place: the memory location for sj,k and Sj+l,2k are identical. Computing sj+l,2~ wilt leave us with the wrong values needed for the computation of sj+1,2~+2, for example! This is the first instance of a subdivision method which requires a sequence of elementary P function boxes, as well as a new primitive: scaling. Scaling fits into the lifting philosophy as it can be done in-place and is trivially inverted. There are a number of advantages to implementing the cubic B-spline subdivision in that way. Aside from allowing us to do the computation in place, it will be maximally parallel and, as we will see later, make it trivial to derive elementary wavelets (and through further lifting an entire class of wavelets.) We begin by averaging even coefficients into odd locations Sj+l,2k+l = 1/2 (szk + Sj,~+I).

Wavelets at Home

97

This is the Podd box in the diagram of Fig. 10. Next apply a Peven function box to get 8j+l,2k = 8j,k "]- 1/2 (Sj+l,2k--1 -1- 8 j + l , 2 k + l ) . Finally dividing all even locations to be 2 s j+ l,2k / = 2 we get our desired sequence. Substitution of the definition of the odd locations immediately verifies that the result is as intended 8j+l,2k "~ 1/2 (1/4sj,k-1 + 6/4Sj,k + 1/4Sj,k+l). We have succeeded in building cubic B-spline scaling functions with a simple inplace computation which only involves immediate neighbors. As we will see later, choosing an appropriate U function box to put before this inverse transform (see Fig. 10) can give us associated spline wavelets.

6.6

The N e x t Step

We have just seen a mlmber of ways to generate scaling functions which fit well with the overall philosophy of lifting and give the practitioner a rich set of constructions from which to choose. Before we discuss the construction of the associated wavelets, and how they "drop out of" the wiring diagrams we first consider the notion of a Multiresolution Analysis more formally in the following section. 7

Multiresolution

Analysis

In this section, we will go into some more mathematical detail about multiresolution analysis as originally conceived by Mallat and Meyer [20,19]. In the previous section we defined the notion of a scaling function T(x) and saw how all scaling functions are simply translates and dilates of one fixed function: ~j,~(x) = ~(2Jx - 1). In this section we will use these functions to build a multiresolution analysis. Assume we start a subdivision scheme of the previous section on level j from a sequence {sj,i}. Because of linearity, we can write the resulting limit function s t ( x ) as s (x) = sj, I

The scaling functions have compact support, so that for a fixed x the summation only involves a finite number of terms. We next define the space Vj of all limit functions obtained from starting the subdivision on level j. This is the linear space spanned by the scaling functions q)j,l with 1 E Z: Vj = span{~j,l(x) t l E Z}.

98

Wim Sweldens and Peter Schrhder

For example, if ~(x) is the Haar scaling function, the ~s) is the space of functions which are piecewise constant on the intervals [k2 - j , (k + 1)2 - j ) with k C Z. If ~(x) is the piecewise linear hat (or tent) function, Vj is the space of continuous functions which are piecewise linear on the intervals [k2-j, (k + 1)2-J) with kEZ. The different Vj spaces satisfy the following properties which make them a multiresolution analysis:

1. Nestedness: k) C Vj+I. 2. T r a n s l a t i o n : if f(x) E Vj then f(x + k2 - j ) E Vj. 3. D i l a t i o n : if f(x) E Vj then f(2x) e Vj+I. 4. C o m p l e t e n e s s : every function f(x) of finite energy (E L ~) can be approximated with arbitrary precision with a function from Vj for suitably high

j. The nestedness property follows immediately from the refinement relation. If we. can write ~(x) as a linear combination of the ~ ( 2 x - k ) , then we can also write ~j,t as a linear combination of the ~j+l,k and thus Vj C ~+1. The translation and dilation properties follow from the definition of ~s~. The proof of the completeness property is beyond the scope of this tutorial. We refer to [9] for more details. A very important property of a multiresolution analysis is the order. We say that the order of a multiresolution analysis is N in case every polynomial p(x) of degree strictly less than N can be written as a linear combination of scaling functions of a given level. In other words polynomials of degree less than N belong to all spaces Vj. For example, in case of the Haar multiresolution analysis, N = 1. Constant functions belong to all Vj spaces. In case of the hat function, N = 2 because all linear functions belong to all Vj spaces. Note that the order of a multiresolution analysis is the same as the order of the predictor used to build the scaling functions. This concludes the discussion of subdivision, scaling functions, and multiresolution. Remember that the original motivation for treating subdivision was that it provides predictors which readily fit into the lifting framework. In the next section we turn to the other main component of lifting: update.

8

Update Methods

We have seen several examples of wiring diagram constructions. In the very beginning of this part we started with the Haar forward and inverse transform. In that case it was clear from inspection what the update box had to do, given that we wanted the coarser signal to be an average of the finer signal. For the linear transform, finding the update box required some algebraic manipulations. Generally update boxes are designed to ensure that the coarser signal has the same average as the higher resolution signal. in the section on subdivision methods we discussed various ways of realizing predictors in the context of an inverse transform with, effectively, zero wavelet

Wavelets at Home Interpolating subdivision even m DOFs

99

Interpolating subdivision completed even 2m-1 DOFs m-1 DOF

Q

odd

I odd

Fig. 17. On the left pure interpolating subdivision is indicated: odd samples are computed using a predictor based on even samples. Beginning with rn degrees of freedom, (2m - 1) result after one step (in the bounded interval case). Simply extending the lower wire to the left is a proper completion, i.e., it describes the extra degrees of freedom added by one subdivision step. The proof consists of observing that the completed diagram is invertible (as opposed to the pure subdivision diagram which is not). Given that it is invertible the bottom wire must represent the remaining degrees of freedom. In the language of matrices--all operations are linear--this simply states that subdivision can be completed in such a way that the resulting matrix is banded and its inverse is banded as well.

coefficients. The diagrams in Figs. 8, 9, and 10 show where the update boxes go, but we have not yet explained how they should be designed. T h a t is the subject matter of this section. We begin by making some simple observations about forward and inverse transforms and give a number of design criteria for update boxes. As in the section on subdivision we will only consider the regular setting and postpone generalizations to Part II. 8.1

Forward and Inverse Transforms

One of the nice features of the lifting formalism is the ease with which one finds the inverse transform: simply flip the diagram left to right, switch additions and subtractions, and switch multiplications and divisions. In this way forward and inverse transtbrm are equivalent from a design point of view. For some questions, it is easier to consider one rather than the other. For example, in the case of borrowing predictors from subdivision methods it is most natural to first consider the inverse transform, and later derive the forward transform. These relationships between forward and inverse "wiring diagrams" lead directly to important consequences. Let us consider the case of subdivision more closely. It is a linear transformation which takes in m degrees of freedom on the top wire, say coefficients associated with the integers, and outputs (2m - 1) degrees of freedom, i.e., coefficients associated with the half integers due to interpolating subdivision. We are a bit sloppy here when ignoring issues such as the boundary, but for purposes of our argument we may do so for now (later on when we discuss the generalization to boundaries we will see that the present argument is solid.) Fig. 17 indicates this idea on the left.

100

Wire Sweldens and Peter SchrSder

...0,0,1,0,0...

Limit functions

l

I ---0,0,-5,1,.5,O,O---

/1

/ \

\\

//

/ \

\%

t Limit functions

Merge

o,o,

f

: f

,

,

, ~ /

,,!,

,

/

v,

Fig. 18. The result of putting a delta sequence on the top (smooth) or bottom (detail) wire of an interpolating subdivision (inverse transform) in the case of the linear predictor. On the right the resulting functions, after subdivision ad infinitum, for a sequence of consecutive delta inputs. Note how the detail functions sit "inbetween" the scaling functions. The picture is essentially the sarne for higher order interpolating predictors: the detail functions are inbetween the scaling functions and dilated (by a factor of 2) versions of the scaling functions.

A question t h a t immediately arises is whether we can characterize the remaining m - 1 degrees of freedom in a simple and useful fornl. This is an important aspect of multiresolution analysis where we have the spaces Vj C Vj+I and we m a y ask how to characterize the difference between two such consecutive spaces: what is the difference in "resolution" between the two spaces? W h a t are we loosing when we go from a finer resolution to a coarser resolution? Since subdivision is a linear transformation we m a y think of it as a (2m 1) x m matrix. In t h a t language the question of characterizing the extra degrees of freedom, is to ask for a set of columns to add to this matrix so t h a t it becomes square, is banded, and its inverse is banded as well. At first sight these requirements appear to be rather hard to satisfy. However, we already know the answer to this question! Suppose we have the inverse transform diagram with the detail wire added (on the right side of Fig; 17) then we can immediately build a diagram with the forward transform ("running everything backwards" ). Concatenating the two we get the identity by construction. Since the whole operation is linear we have just convinced ourselves t h a t the linear operator represented by the inverse transform is invertible, if we also consider the lower detail wire. In other words, writing subdivision in our somewhat nonstandard way, we can immediately read off a

W'avelets at Home ...0,0,1,0,0... k'Y

~

- I'~"

.

1 ...0,1,4,6, 4 ,1,0... ~ .

101

Limitfunctions ~.

1,0,0... //I functionjlx Limit / iI~__ .o,o._~ .1.4. ~I \ ~ // ~ \ ~, ' \

...0,0,1,0,0..+ Fig. 19. The result of putting a delta sequence on the top (smooth) or bottom (detail) wire of a cubic B-spline subdivision (inverse transform). On the right the resulting functions, after subdivision ad infinitum, for a sequence of consecutive delta inputs.

representation of the additional degrees of freedom introduced as one goes from Vj to Vj+I: simply put the delta sequence (~o,t on the detail wire. In the case of interpolating subdivision (Fig. 18 shows the example of linear interpolating subdivision) the result is a delta sequence 61,2t+t. If we continue the subdivision process ad infinitum the functions shown result. Note how the functions due to (successive) smooth coefficients overlap. The functions due to detail coefficients, the actual wavelets, are of the same shape in this case, but smaller and sit inbetween. A more complicated (and interesting) example is provided by the subdivision diagram for the cubic B-splines. Fig. 19 illustrates the behavior of this subdivision diagram as a function of putting successive delta sequences on the smooth and detail wire respectively. For the smooth wire 50,k results in the well known sequence 1/8{1,4, 6, 4, 1} of refinement coefficients. Putting 60,1 on the detail wire results in the sequence 1/4 {1,4, 1}. From our considerations above we know that this sequence must be a completion of the space spanned by the 1/8 {1, 4, 6, 4, 1} and in this way captures the degrees of freedom inbetween Vj and ~+1. In principle we could stop here and would not have to worry about update boxes. The problem is that there are many possible completions and the ones we get "for free" from the subdivision diagram are not necessarily the only ones or the best ones. Note that the detail functions which we got in the linear (Fig. 18) and cubic B-spline case (Fig. 19) do not have a zero integral, a condition we need to ensure that the coarser version of the signal has the same integral as the finer version (see the next subsection). Designing update boxes is all about manipulating the representation of these differences, of "inbetween" degrees of freedom. That this should be so is easy

102

Wim Sweldens and Peter SchrSder

to see. The update box on the inverse transform side makes a contribution to the smooth wire based on a signal on the odd wire (see Figs. 8, 9, and 10). Considering the sequence 50,1 on the detail wire we can see how having a U box will alter the sequence generated at the end (see Fig. 20). From the examples of the Haar and linear transform we remember that the main purpose of the update box is to ensure that the average of the coarser level signals is maintained, i.e, 2j - 1

2- j ~

s~,t

(4)

l=0

is independent of j. From this we see that there is no need for an extra update box in the case of wavelet transforms built from average-interpolation. Given that we can write the forward transform as a Haar transform followed by one average-interpolation prediction operation, the update box which is part of the Haar transform already assures (4). We next discuss how to design update boxes for interpolating subdivision and cubic B-spline subdivision.

8.2

Update for Interpolation

We already encountered one update step for the linear example in the very beginning. It consisted of computing the coarser level coefficients as sj-l,z = sj,2t + 1/4

(dj-l,l-1 -1- dj-l,1).

We show here how such an update can be derived in general. The main purpose of the update step is to ensure that the average is maintained. Saying that the average of the signal sj is equal to the average of the signal sj-1 is equivelent to saying that the average of the detail or difference signal dj-1 is zero. Given that the detail signal is nothing else but a linear combination of complementary sequences as derived in the previous section, it is sufficient to construct complementary sequences with zero average. We consider Fig. 18 again. The complementary sequence, i.e., the result from putting a 1 on the odd wire is {0, 0, 1, 0, 0}. Obviously this does not have a zero average. Therefore we use the update box from the odd wire to the even wire (see Fig. 20). The result of putting a 1 on the even wire is {1/2,1,1/2, 0, 0} or {0, 0, 1/2, 1, 1/2} depending on the position of the original 1. The update box now combines these sequences to build a complementary sequence with average zero. We propose an update which adds a fraction A of the odd element into the two neighboring evens. This leads to a complementary sequence of the form

{O,O,l,O,O}-A{1/2,1,1/2,0,O}-A{O,O, 1/2,1,1/2}. Choosing A = 1/4 leads to the complementary sequence {-1/8,-1/4,3/4,-1/4,-1/8}, which has average zero as desired.

Wavelets at Home

103

...0,0,..25,-.25,0...

----"@

I

OOlOO

...0,-.125,-.25,.75,-.25,-.125,0...

Limit functions •..0,-•125,75,-. 125,0... "~ "

/ ~ \L,

/\ ',,/

/ ~,.

'

Fig. 20. Starting with the delta sequence on the bottom wire, the update box make a contribution to the even wire, which in turn causes further changes in the odd wire after prediction• The result is the sequence at the end which corresponds to the indicated limit functions. The average of the coefficients is zero: as expected.

Other types of update can be used as well. They ensure t h a t not only the average but also the first ?~ moments of the sequences are preserved, i.e., 2 j --1

m p = 2 -j E

lp sj,l

(s)

l=o is independent of j for 0 < p < 2¢. The update weights can be found easily as well. In [26] it is shown t h a t as long as fit < N one can simply take the same weights for update as one used for prediction, divided by two. For example, in case the predictor is of order N _> 4, one can get an update w i t h / Y = 4 by letting

s j - l j = sj,2l - 1/32 d j - l , t - 2 + 9/32 dj-l,l-i + 9/32 dj-l,I -- 1/32 dj-l,~+l. 8.3

Update

f o r c u b i c B-splines

T h e u p d a t e box for the B-splines can be found using the same reasoning as in the previous section. The complementary sequence now is

{O,O, 1 / 4 , 1 , 1 / 4 , 0 , O } - A/8{1,4,6,4,1, O,O}- A/8{O,O, 1,4,6,4,1}~ which leads to A = 3/8.

9

Wavelet Basis Functions

In this section we formally establish the relationship between detail coefficients and the wavelet functions. In the earlier section on multiresolution analysis (Section 7) we described spaces Vj which are spanned by the fundamental solutions

104

Wim Sweldens and Peter Schrhder

of the subdivision process, the scaling functions ~j,z. Here we will examine the differences Vj÷I \Vj and the wavelets Cj,t (x) which span these differences. Consider an initial signal Sn = {Sn,z I 1 E Z}. We can associate a function s•(x) in Vn with this signal:

8o(x) = Z t

Calculate one level of the wavelet transform as described in an earlier section. This yields coarser coefficients sn-l,1 and coefficients d~-l,l. The coarser coefficients s n - l j corresponds to a new function in K~-I:

8n--l (X) -~ E

8n-l'l~On-l'l(X)" l

Recall that ~n-z,l(x) is the function that results if we run the inverse transform on the sequence 5n-l,l. With this the above equation expresses the fact that the function sn-1 (x) is the result of "assembling" the functions associated with a 1 in each of the positions (n - 1,/), each weighted by sn-l,~. At first sight it is unclear which function the signal du-1 corresponds to. Somehow we feel that it corresponds to the "difference" of the signals Sn(X) and sn-1 (x). To find the solution we need to define the wavelet function. Consider a detail coefficient d0,0 = 1 and set all other detail coefficients d0,k with k ¢ 0 to zero. Now compute one level of an inverse wavelet transform. This corresponds to what we did in Figs. 18 and 19 when putting a single 1 on the lower wire. After a single step of an inverse transform this yields coefficients gl of a function in V1, e.g., gl = {0,1, 0} in case of linears without lifting in Fig. 18 or gl = { - 1 / 8 , - 1 / 4 , 3 / 4 , - 1 / 4 , - 1 / 8 } in the case of linears with lifting. We define this function to be the wavelet function ¢(x) = ¢0,0(x). Thus:

¢0,0(x) = Z

= Z

l

l

- z)

This wavelet function is often called the mother wavelet. All other wavelets ¢j,k are obtained by setting dj,k = 1 and dj,l with I ¢ k to zero, calculating one level of the inverse wavelet transforms and then subdividing ad infinitum to get the corresponding function in I/)+1. Using an argument similar t-o the case of scaling functions, one can show that all wavelets are translates and dilates of the mother wavelet:

% , ( x ) = ¢(2J

- k).

Fig. 21 shows several mother wavelets coming from interpolation, Fig. 22 shows wavelets resulting from average-interpolation, and Fig. 23 shows a wavelet assod a t e d with cubic B-spline scaling functions. Now we can answer the question from the start of this section. We first define the detail function dn-l(x) to be

dn-1 (x) = E dn-lj ~3n--1,l. l

Wavelets at Home

105

0.50.5 t 0.0

02

-0.5

i

-4

i

i

t

1

I

-0.5

- 3 - 2 - ~ 0 1 2 3 4

!

-4-3

0.5-

0.5-

0.0

0.0

-0.5 -4

I

-3

J

-2

I

-1

J

0

~

1

J

2

I

3

I

-0.5

4

-4

I

-3

I

I

I

I

I

I

I

-2

-1

0

1

2

3

4

I

-2

I

-1

I

0

I

1

I

2

I

3

I

4

Fig. 21. Wavelets from interpolating subdivision. Going from left to right, top to bottom are the wavelets of order N = 2, 4, 6, and 8. Each time/~r = 2.

Because of linear superposition of the wavelet transform, it follows that

sn(x) = Sn-l (X) 4- dn-1(x) = E 8n-l'l ~)n-l'l(X) q- E dn-l,l Cn-l,l. l

l

In words, the function defined by the sequence sn,t is equM to the result of performing one forward transform step, computing Sn-l,t (upper wire) and dn-l,t (lower wire), and then using these coefficients in a linear superposition of the elementary functions which result when running an inverse transform on a single 1 on the top or b o t t o m wire, in the respective position. So what we expected is true: the detail function dn-l(X) is nothing but the difference between the original function and the coarser version. As we will see later, there are many different ways to define a detail function dn-1. They typically depend on two things: first, the detail coefficients dn-l,l (which are computed in the forward transform) and secondly the wavelet functions Cn-l,t (which are built during the inverse transform). The n-level wavelet decomposition of a function Sn(X) is defined as n--1

8 (x) = 8o(x) + Z dj(x). j=0

The space Wj is defined to be the space that contains the difference functions:

wj = span{¢j, (z) i l e z}.

106

Wim Sweldens and Peter SchrSder

?

0.5~

0.5-

0.0

0.0

-0,5 -

-0,5 -

-4

t

I

I

!

I

t

t

I

-3

-2

-1

0

1

2

3

4

0.5

0,5

0.0

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-0.5

-0.5 -4

1

I

I

I

1

l

1

t

-3

-2

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0

1

2

3

4

I

I

t

t

I

I

1

-4

-3

-2

-1

0

2

3

4

I

I

I

I

I

I

I

I

-4

-3

-2

-I

0

l

2

3

4

Fig. 22. Wavelets from average-interpolation. Going from left to right, top to bottom are the wavelets which correspond to the orders N = 1, 3, 5, and 7. Each time/~r _- 1.

It then follows that Wj is a space complementing ~ in

~+1

5+1 = yj • wj. In an earlier section we defined the notion of order of a muttiresolution analysis. If the order is N, then the wavelet transform started from any polynomial p(x) = sn(x) of degree less than N will only yield zero wavelet coefficients dj,z. Consequently all detail functions dj (x) are zero and all sj (x) with j < n are equal to p(x). In this section we introduce the dual order N of a multiresotution analysis. We say that the dual order is N in case the wavelets have 2¢ vanishing moments:

Z Z x p Cj, (x) dx = 0 for 0 _ 3. An interesting question is how the new sample points should be placed. A disadvantage of always adding midpoints is that imbalances between the lengths of the intervals are maintained. A way to avoid this is to place new sample points only in intervals whose length is larger than the average interval length. Doing so repeatedly will bring the ratio of largest to smallest interval length ever closer to 1. Another possible approach would add new points so that the length of the intervals varies in a smooth manner, i.e., no large intervals neighbor small intervals. This can be done by applying an interpolating subdivision scheme, with integers as sample locations, to the xj,k themselves to find the xj+l,2k+l. This would result in a smooth mapping from the integers to the xj,k. After performing this step the usual interpolating subdivision would follow. Depending on the application one of these schemes may be preferable. Next we took some random data over a random set of 16 sample locations and applied linear (N = 2) and cubic (N = 4) interpolating subdivision to them. The resulting interpolating functions are compared on the right side of Fig. 33. These functions can be thought of as a linear superposition of the kinds of scaling functions we constructed above for the example j = 3. Note how sample points which are very close to each other can introduce sharp features in the resulting function. We also note that the interpolation of order 4 exhibits some of the overshoot behavior one would expect when encountering long and steep sections of the curve followed by a reversal of direction. This behavior gets worse for higher order interpolation schemes. These experiments suggest that it might be desirable to enforce some condition on the ratio of the largest to the smallest interval in a random sample construction.

124

Wire Sweldens a n d Peter SchrSder

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ib(z)h then there always exists a Laurent polynomial q(z) (the quotient) with Iq(z)i = t a ( z ) l - Ib(z)l, and a Laurent polynomial r(z) (the remainder) with ir(z)l < Ib(z)l so that

a(z) = b(z) q(z) + r(~). We denote this as (C-language notation):

q(z) = ~(~) / b(~) and r(z) = a(z) ~ ~(~). If Ib(z)l = 0 which means b(z) is a monomial, then r(z) = 0 and the division is exact. A Laurent polynomial is invertible if and only if it is a monomial. This is the main difference with the ring of (regular) polynomials where constants are the only polynomials that can be inverted. Another difference is that the long division of Laurent polynomials is not necessarily unique. The following example illustrates this.

Example 1. Suppose we want to divide a(z) = z -1 + 6 + z by b(z) = 4 + 4z. This means we have to find a Laurent polynomial q(z) of degree 1 so that r(z) given by

r(z) = a(z) - b(z) q(z) is of degree zero. This implies that b(z)q(z) has to match a(z) in two terms. If we let those terms be the term in z -1 and the constant then the answer is q(z) = 1/4 (z -1 + 5). Indeed,

~(z) =

(~-1 + 6 + z) - (4 + 4 ~ ) ( 1 / 4 z -1 + 5/4) = - 4 ~ .

The remainder thus is of degree zero and we have completed the division. However if we choose the two matching terms to be the ones in z and z -1 , the answer is q(z) = 1/4 (z -1 + 1). Indeed,

r(z) = (z -1 + 6 + z) - (4 + 4 z ) ( 1 / 4 z -1 + 1/4) = 4. Finally, if we choose to match the constant and the term in z, the solution is

q(z) = 1/4 (5 z -1 + 1) and the remainder is r(z) = - 4 z -1. The fact that division is not unique will turn out to be particularly useful later. In general b(z)q(z) has to match a(z) in at least la(z)t - Ib(z)l + 1 terms, but we are free to choose these terms in the beginning, the end, or divided between the beginning and the end of a(z). For each choice of terms a corresponding long division algorithm exists. In this paper, we also work with 2 × 2 matrices of Laurent polynomials, e.g.,

M(z) : [~(z) b(~)]

[c(z) d(z) j "

Factoring Wavelet Transforms into Lifting Steps

137

These matrices also form a ring, which is denoted by M(2;l~[z,z-1]). If the d e t e r n ~ a a t of such a matrix is a monomial, then the matrix is invertible. The set of invertible matrices is denoted GL(2; ]~[z,z-l]). A matrix from this set is unitary (sometimes also referred to as para-unitary) in case -1

3

Wavelet

transforms

Fig. 3. Discrete wavelet transform (or subband transform): The forward transform consists of two analysis filters h (low-pass) and ~ (high-pass) followed by subsampling, while the inverse transform first upsamples and then uses two synthesis filters h (lowpass) and g (high-pass).

Fig. 3 shows the general block scheme of a wavelet or subband transform. The forward transform uses two analysis filters h (low-pass) and ~ (band pass) followed by subsampling, while the inverse transform first upsamples and then uses two synthesis filters h (low-pass) and g (high-pass). For details on wavelet and subband transforms we refer to [43] and [57]. In this paper we consider only the case where the four filters h, g, h, and ~, of the wavelet transform are FIR filters. The conditions for perfect reconstruction are given by

h(z) h(z -1) + g(z) "~(z-1) = 2 h(z)'h(-z -1) + g(z) g ( - z -1) : 0. We define the

modulation matrix M(z) as M(z) = [h(z) h ( - z ) ] [g(z) g ( - z ) J "

We similarly define the dual modulation matrix condition can now be written as

M(z). The perfect reconstruction

M(z-1) t M(z) = 2I,

(1)

138

Ingrid Daubechies and Wire Sweldens

• LP

P(z) BP

Fig. 4. Polyphase representation of wavelet transform: first subsample into even and odd, then apply the dual polyphase matrix. For the inverse transform: first apply the polyphase matrix and then join even and odd.

where I is the 2 × 2 identity matrix. If all filters are FIR, then the matrices M(z) and M(z) belong to GL(2; ]R[z,z-l]). A special case are orthogonaI wavelet transforms in which case h = h and g = ~. The modulation matrix M(z) = M(z) is then v ~ times a unitary matrix. The polyphase representation is a particularly convenient tool to express the special structure of the modulation matrix [3]. The polyphase representation of a filter h is given by h(z) = h~(z 2) + z-lho(z2), where he contains the even coefficients, and ho contains the odd coefficients:

ho(z) =

h kz

and ho(Z) =

k

Z-k, k

or

he(z~) _ h(z) + h(-z) and ho(z 2) - h(z) - h(-z) 2 2z -1 We assemble the polyphase matrix as P(z) =

[he(z)

ho(z) go(Z)J'

so that

We define/5(z) similarly. The wavelet transform now is represented schematically in Fig. 4. The perfect reconstruction property is given by

P(z)/5(z-1)t = I.

(2)

Again we want P(z) and P(z) to contain only Laurent polynomials. Equation (2) then implies that det P(z) and its inverse are both Laurent polynomials; this is possible only in case d e t P ( z ) is a monomial in z: d e t P ( z ) = Czl; P(z) and/~(z) belong then to GL(2; t~[z, z-l]). Without loss of generality we assume that d e t P ( z ) = 1, i.e, P(z) is in SL(2;~[z,z-1]). Indeed, if the determinant is


~BP

139

-

Fig. 5. The hfting scheme: First a classical subband filter scheme and then lifting the tow-pass subband with the help of the high-pass subband.

•

LP ~

~

'

~

Fig. 6. The dual lifting scheme: First a classical subband filter scheme and later lifting the high-pass subband with the help of the low-pass subband. not one, we can always divide go(z) and go(z) by the determinant. This means t h a t for a given filter h, we can always scale and shift the filter g so that the determinant of the polyphase matrix is one. The problem of finding an FIR wavelet transform thus amounts to finding a matrix P(z) with determinant one. Once we have such a matrix, P ( z ) and the four filters for the wavelet transform follow immediately. From (2) and Cramer's rule it follows that

he(z)=go(Z-1),

[to(Z)~--ge(z-1),

[le(z)=-ho(z-1),

~o(z) mhe(z-1).

This implies

~(z) = z -1 h ( - z -1) and h(z) = - z -1 g(-z-1). The most trivial example of a polyphase matrix is P(z) = I. This results in h(z) = h(z) = 1 and g(z) = ~(z) = z -1. The wavelet transform then does nothing else but subsampling even and odd samples. This transform is called the polyphase transform, but in the context of lifting it is often referred to as the Lazy wavelet transform [44]. (The reason is that the notion of the Lazy wavelet can also be used in the second generation setting.) 4

The

Lifting

Scheme

The lifting scheme [44, 45] is an easy relationship between perfect reconstruction filter pairs (h, g) that have the same low-pass or high-pass filter. One can then start from the Lazy wavelet and use lifting to gradually build one's way up to a multiresolution analysis with particular properties.

140

Ingrid Daubechies and Wim Sweldens

Definition 1. A filter pair (h, g) is complementary in case the corresponding polyphase matrix P(z) has determinant 1. If (h, g) is complementary, so is (h, ~ . This allows us to state the lifting scheme. T h e o r e m 1 (Lifting). Let (h, g) be complementary. Then any other finite filter gnaw complementary to h is of the form:

g°°w(z) = g(z) + h(z)

s(z2),

where s(z) is a Laurent polynomial. Conversely any filter of this form is complementary to h. Proof. The polyphase components of h(z) s(z 2) are for even he(z) s(z) and for odd ho(z) s(z). After lifting, the new polyphase matrix is thus given by

This operation does not change the determinant of the polyphase matrix. Fig. 5 shows the schematic representation of lifting. Theorem 1 can also be written relating the low-pass filters h and h. In this formulation, it is exactly the Vetterli-Herley lemma [56, Proposition 4.7]. The dual polyphase matrix is given by:

We see that lifting creates a new h filter given by

T h e o r e m 2 (Dual lifting). Let (h, g) be complementary. Then any other finite filter h ~°w complementary to g is of the form:

h~°~(z) = h(z) + g(z) t(z2), where t(z) is a Laurent polynomial. Conversely any filter of this form is complementary to g. After dual lifting, the new polyphase matrix is given by

DuN lifting creates a new ~ given by =

-

Fig. 6 shows the schematic representation of dua~ lifting. In [44] lifting and dual lifting are used to build wavelet transforms starting from the Lazy wavelet. There


141

a whole family of wavelets is constructed from the Lazy followed by one dual lifting and one primal lifting step. All the filters h constructed this way are half band and the corresponding scaling functions are interpolating. Because of the many advantages of lifting, it is natural to try to build other wavelets as well, perhaps using multiple lifting steps. In the next section we will show that any wavelet transform with finite filters can be obtained starting from the Lazy followed by a finite number of alternating lifting and dual lifting steps. In order to prove this, we first need to study the Euclidean algorithm in closer detail. 5

The Euclidean

Algorithm

The Euclidean algorithm was originally developed to find the greatest common divisor of two natural numbers, but it can be extended to find the greatest common divisor of"two polynomials, see, e.g, [4]. Here we need it to find common factors of Laurent polynomials. The main difference with the polynomial case is again that the solution is not unique. Indeed the gcd of two Laurent polynomials is defined only up to a factor z p. (This is similar to saying that the gcd of two polynomials is defined only up to a constant.) Two Laurent polynomials are relatively prime in case their gcd has degree zero. Note that they can share roots at zero and infinity. T h e o r e m 3 (Euclidean A l g o r i t h m for L a u r e n t P o l y n o m i a l s ) . Take two

Laurent polynomials a(z) and b(z) ~ 0 with la(z)l > Ib(z)l. Let ao(z) = a(z) and bo(z) = b(z) and iterate the following steps starting from i = 0

a +l(Z)

= b (z)

bi+l (z) = ai(z) % bi(z).

(3) (4)

Then an(z) = gcd(a(z), b(z) ) where n is the smallest number/or which b~(z) = O. Given that Ib~+l(z)l < Ibi(z)l, there is an m so that lb,~(z)l = 0. The algorithm then finishes for n = m + l . The number of steps thus is bounded by n < tb(z)l+l. If we let qi+l(z) = ai(z) / bi(z), we have that

=

i=71.

1-qi(z)

[b(z)]"

Conseqnently a(z) ] = r I [q~z)

[an(OZ)

and thus a~(z) divides both a(z) and b(z). If a~(z) is a monomial, then a(z) and b(z) are relatively, prime.

Example 2. Let a(z) = ao(z) = z -1 + 6 + z and b(z) = bo(z) = 4 + 4 z . Then the first division gives us (see the example in Section 2): al(z) = 4 + 4 z


142

bt(z) = 4 ql(z) = 1 / 4 z -1 + 1/4.

The next step yields a2(z) = 4 b2(z) = 0 q2(z) = 1 + z.

Thus,

a(z)

and

b(z)

are relatively prime and

The number of steps here is n = 2 = 6

The

Factoring

Ib(z)l + 1.

Algorithm

In this section, we explain how any pair of complementary filters (h, g) can be factored into lifting steps. First, note that he (z) and ho (z) have to be relatively prime because any common factor would also divide det P(z) and we already know that det P(z) is 1. We can thus run the Euclidean algorithm starting from he(z) and ho(z) and the gcd will be a monomial. Given the non-uniqueness of the division we can Mways choose the quotients so that the gcd is a constant. Let this constant be K . We thus have that

Note that in case Iho(z)l > Ih~(z)l, the first quotient ql(z) is zero. We can always assume that n is even. Indeed if n is odd, we can nmltiply the h(z) filter with z and g(z) with - z -1. This does not change the determinant of the polyphase matrix. It flips (up to a monomial) the polyphase components of h and thus makes n even again. Given a filter h we can always find a complementary filter g 0 by letting

10][0

•

Here the final diagonal matrix follows from the fact that the determinant of a polyphase matrix is one and n is even. Let us slightly rewrite the last equation. First observe that


143

Using the first equation of (5) in case i is odd and the second in case i is even yields: P°(z) = ~

[~

[q2)(z) 01ILK I ~ K ] "

(6)

Finally, the original filter g can be recovered by applying Theorem 1. Now we know that the filter g can always be obtained from gO with one lifting Or:

Combining all these observations we now have shown the following theorem: T h e o r e m 4. Given a complementary filter pair (h,g), then there always exist Laurent polynomials si(z) and ti(z) for 1 < i < m and a non-zero constant K so that m

P(z)= ~

[10 s~z) ] [t~(lz)01] [ K I ~ K ] .

The proof follows from combining (6) and (7), setting m = h i 2 + 1, tin(z) = O, and sin(z) = K e s(z). In other words every finite filter wavelet transform can be obtained by starting with the Lazy wavelet followed by m lifting and dual lifting steps followed with a scaling. The dual polyphase matrix is given by

From this we see that in the orthogonal case ( P ( z ) = P(z)) we immediately have two different factorizati0ns. Figs. 7 and 8 represent the different steps of the forward and inverse transform schematically.

7

Examples

We start with a few easy examples. We denote filters either by their canonical names (e.g. Haar), by (N, N) where N (resp. /V) is the number of vanishing moments of ~ (resp. g), or by (la - ls) where la is the length of analysis filter and Is is the length of the synthesis filter h. We start with a sequence x = {xt I 1 E Z} and denote the result of applying the low-pass filter h (resp. highpass filter g) and downsampling as a sequence s = {sl I l C Z} (resp. d). The intermediate values computed during lifting we denote with sequences s (i) and d (i). All transforms are instances of Fig. 7.

144


BP

Fig. 7. The forward wavelet transform using lifting: First the Lazy wavelet~ then alternating lifting and dual lifting steps, and finally a scaling.

Fig. 8. The inverse wavelet transform using lifting: First a scaling, then alternating dual lifting and lifting steps, and finally the inverse Lazy transform. The inverse transform can immediately be derived from the forward by running the scheme backwards.

7.1

Haar wavelets

In the case of (unnormalized) Haar wavelets we have that h(z) = 1 + z -1, g(z) = - 1 / 2 + 1/2z - t , h(z) = 1/2 + 1/2z -1, and ~(z) = - 1 + lz -1. Using the Euclidean algorithm we carl thus write the polyphase matrix as:

[1:/:]= [110] Thus on the analysis size we have:

This corresponds to the following implementation of the forward transform:

8} 0) ---~X2l d}0) ~---x21+l d, =d} ° ) - s}°) sl = s}°) + 1/2 dl, while the inverse transform is given by: 8} °) = 8~ - 1 / 2 dz


145

d~°) = dt + s~°) x2t+l = d~°)

X21 = 8~0). 7.2

Givens rotations

Consider the case where the polyphase matrix is a Givens rotation (a ~ 7r/2). We then get cosa-sins]= sin a cos a j

[sinai

cos a

01] [ l o - s i n a cosa] [co~a 1/cOosa]. 1

We can also do it without scaling with three lifting steps as (here assuming [cosa-sina] = (cosa - 1 ) / s i n a ] (cos a - l l ) / s i n a ] [sina cosa j [~ [sinlaO1] [~ " This corresponds to the well known fact in geometry that a rotation can always be written as three shears. The lattice factorization of [51] allows the decomposition of any orthonormal filter pair into shifts and Givens rotations. It follows that any orthonormal filter can be written as lifting steps, by first writing the lattice factorization and then using the example above. This provides a different proof of Theorem 4 in the orthonormal case. 7.3

Scaling

These two examples show that the scaling from Theorem 4 can be replaced with four lifting steps:

[

-- [10

[

°1]

or

Given that one can always merge one of the four lifting steps with the last lifting step from the factorization, only three extra steps are needed to avoid scaling. This is particularly important when building integer to integer wavelet transforms in which case scaling is not invertible [6].

146

Ingrid Danbechies and Wim Sweldens

7.4

I n t e r p o l a t i n g filters

In case the low-pass filter is half band, or h(z) + h ( - z ) = 2, the corresponding scaling function is interpolating. Since he(z) = 1, the factorization can be done in two steps: P ( z ) = [hol(z) l + ho(z) g~(z) j =

The filters constructed in [44] are of this type. This gives rise to a family of (N, N) (N and N even) symmetric biorthogonal wavelets built from the DeslauriersDubuc scaling functions mentioned in the introduction. The degrees of the filters are Ihol = N - 1 and lgel = N - 1. In case N < N, these are particularly easy as .q!N)(z) = --1/2 h (R) (z-l). (Beware: the normalization used here is different from the one in [44].) Next we look at some examples that had not been decomposed into lifting steps before. 7.5

4-tap o r t h o n o r m a l filter w i t h two vanishing m o m e n t s (D4)

Here the h a n d g filters are given by [16]: h(z) = ho + hi z -1 + h2 z -2 + h3 z -3 g(z) = - h 3 z 2 + h2 z 1 - hi + ho z - t ,

with ho-

4v/~ ,

hi-

4vr~ ,

h2=

4 - v ~ ' and h 3 =

4---~

The polyphase matrix is P(z) = P(z) =

h0 + h2 z -1 -h3 z 1 - hi ] hl + h3 z -1 h2z l + ho J '

(8)

and the factorization is given by: P(z) =

@

.

(9)

As we pointed out in Section 6 we have two options. Because the poljphase matrix is unitary, we can use (9) as a factorization for either P ( z ) or P ( z ) . In the latter case the analysis polyphase matrix is factored as: [ ~

0 1 1

+

1


147

This corresponds to the following implementation for the forward transform:

d}1) "~ X21÷l -- vfax21 s}1) = x2, + 4 ~ / 4 ~} ~) + ( ~ -

2)/4 ~}$)1

O) d}2) = d}1) + s 1--1 81 = (v/3 + 1)/x/2s} 1)

dl

The inverse transform follows from reversing the operations and flipping the signs: d} 2) = ( v ~ + 1)/v/2dt s} 1) = ( v ~ = dT)

1)/V~st _

x~, = ~}') - ~ / 4 ~

1) - ( ~ -

2)/4 ~}~)~

x21+l = d}1) + x/3x2t. The other option is to use (9) as a factorization for P ( z ) . The analysis polyphase matrix then is factored as:

[7 0]

1

and leads to the following implementation of the forward transform:

s} 1) = x21 + v ~ z : l + l

~}t) = x2z+I - ¢ 5 / 4 ~}~) - ( ¢ ~ - 2)/4 s}g~

dt = ( V ~ + 1)/v~d} 2). Given that the inverse transform always follows immediately from the forward transform, from now on we only give the forward transform. One can also obtain an entirely different lifting factorization of D4 by shifting the filter pair corresponding to: h(z) = ho z + hl + h2 z - l + ha z -2 g(z) = h3 z - h2 + hi z -1 - ho z -2,

with P ( z ) .= P ( z ) -~- [ hi -}- h3 z-1 - h 2 - hO z-1 t [ hoz+h2 h3z+hl J

148


as polyphase matrix. This leads to a different factorization: ~(z) =

[10 ][+,

~

z + ~

3¢~

0

01

3-~ ' 3v~ J

and corresponds to the following implementation: d~ 1) = x21+l - 1 / v ~ x 2 1 + 2

s~1) = x~, + ( 6 - 3¢~)/4@) + ¢~/4 ~1)~ d l) _

d, = (3 - v/-3)/(3vf2) 42). This second factorization can also be obtained as the result of seeking a factorization of the original polyphase matrix (8) where the final diagonal matrix has (non-constant) monomial entries. 7.6

6-tap orthonormal filter with three vanishing moments (D6)

Here we have

3

h(~) = ~

h~ z -~,

k-=- 2

with [16] h_2 = ~/2 (1 + V~-0+ ~/5 + 2 vrio)/32 h_l : v/2 (5 + v ~ + 3 ~/5 + 2 v/l-0)/32 ho = v~ (10 - 2 ~T0 + 2 V/5 + 2 v~-0)/32 hi : v~ (10 - 2 v / ~ - 2 ~5 + 2 v/~)/32 h2 : v~ (5 + v~-0- 3 ~5 + 2 ~/~)/32

The polyphase components are h e ( z ) = h - 2 z "4- ho + h2 z - 1

ge(z) = - h a z - h i - h - 1 z - 1 h o ( z ) = h - 1 z + hi + h3 z - 1 go(Z) = h2 z + ho + h - 2 z - 1 .


149

In the factorization algorithm the coefficients of the remainders are calculated as:

ro = h - 1 - h3 * h - 2 / h 2 rl = h i - h2 * h o / h 2 sl = ho - h - 2 * r l / r o - h2 * to~r1 t = -h3/h-2

* s~.

If we now let -0.4122865950 h 2 / r l ~ -1.5651362796 h - 2 / r o ~ 0.3523876576 r l / s l ~ 0.0284590896 to~s1 ~ 0.4921518449 5 = - h 3 / h - 2 * s~ ~ -0.3896203900 4= sl ~ 1.9182029462, h3/ht ~

then the factorization is given by: P ( z ) = [1a 01] [10/3z-t1+ fl'] [ 7 1 7 ' z ~1 [ ; ~1] [~ 1 ~ ] " We leave the implementation of this filter as an exercise for the reader. 7.7

(9-7) filter

Here we consider the popular (9-7) filter pair. The analysis filter h has 9 coefficients, while the synthesis filter h has 7 coefficients. Both high-pass filters g and have 4 vanishing moments. We choose the filter with 7 coefficients to be the synthesis filter because it gives rises to a smoother scaling function than the 9 coefficient one (see [17, p. 279, Table 8.3], note that the coefficients need to be multiplied with v~). For this example we run the factoring algorithm starting from the analysis filter: he(z)

= h4 (z 2 + z -2) + h2 (z + z -1) + ho and

ho(z)

The coefficients of the remainders are computed as: ro = ho - 2h4

hi~h3

r l = h2 - h4 - h4 h i ~ h 3 so -~ hi - h3 - h3 r o / r l to = ro - 2 r l .

= h3 (z 2 + z -1) + hi (z + 1).

150


Then define a = h4/h3 ~ = h3/rl

~

-1.586134342 -0.05298011854

0.8829110762 = so~to ~ 0.4435068522 = to = ro - 2rl ~ 1.149604398.

7 = rl/so

~

Now

[Z(I~_z) °1] LO [1~(1+1 z-~)]J [5(11+ z ) ° l ]

[~1~]

Note that here too many other factorizations exist; the one we chose is symmetric: every quotient is a multiple of (z + 1). This shows how we can take advantage of the non-uniqueness to maintain symmetry. The factorization leads to the following implementation: s~ O) = x21

d~O) =

X2/+l

6(0) S~1) = 8~O) -{-

(d ~- I(~) + ,~(1) ~I--1]

d~ 2) = d/(1) + ff (S~ 1) .aQ(1)~ . o/+ 1]

=

+

+ d"),,_.

st = ~ s~2)

~ = ~)/~. 7.8

C u b i c B-splines

We finish with an example that is used frequently in computer graphics: the (4,2) biorthogonal filter from [12]. The scaling function here is a cubic B-spline. This example can be obtained again by using the factoring algorithm. However, there is also a much more intuitive construction in the spatial domain [46]. The filters are given by

h(z)

= 3/4 + 1/2 (z + z -1) + 1/8 (z 2 + z -2)

a(z) = 5/4z -1 - 5/32 (1 + z-2) _ 3/8 (z + z -3) - 3/32 (z ~ + ~-4),

and the factorization reads:


8

151

Computational complexity

In this section we take a closer look at the computational complexity of the wavelet transform computed using lifting. As a comparison base we use the standard algorithm, which corresponds to applying the polyphase matrix. This already takes advantage of the fact that the filters will be subsampled and thus avoids computing samples that will be subsampled immediately. The unit we use is the cost, measured in number of multiplications and additions, of computing one sample pair (sz, dl). The cost of applying a filter h is Ihl + 1 multiplications and lhl additions. The cost of the standard algorithm thus is 2(th I + Igl) + 2. If the filter is symmetric and Ihl is even, the cost is 3 Ihl/2 + 1. Let us consider a general case not involving symmetry. Take Ihl = 2N, Igl = 2M, and assume M _> N. The cost of the standard algorithm now is 4(N + M) + 2. Without loss of generality we can assume that lhel = N, lhol = N - 1, tgel = M, and Igol = M - 1. In general the Euclidean algorithm started from the (h~, ho) pair now needs N steps with the degree of each quotient equal to one (iqil = I for 1 < i < N). To get the (g~,go) pair, one extra lifting step (7) is needed with Isl = M - N. The total cost of the lifting algorithm is: scaling: 2 N lifting steps: 4N final lifting step: 2(M - N + 1) total

2(N+M+2)

We have shown the following: T h e o r e m 5. Asymptotically, for long filters, the cost o/the lifting algorithm for

computing the wavelet transform is one hal/of the cost of the standard algorithm. In the above reasoning we assumed that the Euclidean algorithm needs exactly N steps with each quotient of degree one. In a particular situation the Euclidean algorithm might need fewer than N steps but with larger quotients. The interpolating filters form an extreme case; with two steps one can build arbitrarily long filters. However, in this case Theorem 5 holds as well; the cost for the standard algorithm is 3(N + N) - 2 while the cost of the lifting algorithm is 3/2(N +/~). Of course, in any particular case the numbers can differ slightly. Table I gives the cost S of the standard algorithm, the cost L of the lifting algorithm, and the relative speedup ( S / L - 1) for the examples in the previous section. One has to be careful with this comparison. Even though it is widely used, the standard algorithm is not necessarily the best way to implement the wavelet transform. Lifting is only one idea in a whole tool bag of methods to improve the speed of a fast wavelet transibrm. Rioul and Duhamel [39] discuss several other schemes to improve the standard algorithm. In the case of long filters, they suggest an FFT based scheme known as the Vetterli-algorithm [56]. In the case of short filters, they suggest a "fast running FIR" algorithm [54]. How these ideas combine with the idea of using lifting and which combination will be optimal

152


Table 1. Computational cost of lifting versus the standard algorithm. Asymptotically the lifting algorithm is twice as fast as the standard algorithm. Wavelet

Standard

Lifting Speedup

Haar 3 3 0% D4 14 9 56% D6 22 14 57% (9-7) 23 14 64% (4,2) B-spline 17 10 70% (N, N) Interpolating 3(N + N) - 2 3/2(N + N) ~ 100% Ihl=2N, Igl=2M 4 ( N + M ) + 2 2 ( N + M + 2 ) ~100%

for a certain wavelet goes beyond the scope of this paper and remains a topic of future research. 9

Conclusion

and Comments

In this tutorial presentation, we have shown how every wavelet filter pair can be decomposed into lifting steps. The decomposition amounts to writing arbitrary elements of the ring SL(2; ~[z, z-l]) as products of elementary matrices, something that has been known to be possible for a long time [2]. The ibllowing are a few comments on the decomposition and its usefulness. First of all, the decomposition of arbitrary wavelet transforms into lifting steps implies that we can gain, for all wavelet transforms, the traditional advantages of lifting implementations, i.e. 1. Lifting leads to a speed-up when compared to the standard implementation. 2. Lifting allows for an in-place implementation of the fast wavelet transform, a feature similar to the Fast Fourier Transform. This means the wavelet transform can be calculated without allocating auxiliary memory. 3. All operations within one lifting step dan be done entirely parallel while the only sequential part is the order of the lifting operations. 4. Using lifting it is particularly easy to build non linear wavelet transforms. A typical example are wavelet transforms that map integers to integers [6]. Such transforms are important for hardware implementation and for lossless image coding. 5. Using lifting and integer-to-integer transforms, it is possible to combine biorthogonal wavelets with scalar quantization and still keep cubic quantization cells which are optimal like in the orthogonal case. In a multiple description setting, it has been shown that this generalization to biorthogonality allows for substantial improvements [58]. 6. Lifting allows for adaptive wavelet transforms. This means one can start the analysis of a function from the coarsest levels and then build the finer levels by refining only in the areas of interest, see [40] for a practical example.


153

The decomposition in this paper also suggests the following comments and raises a few open questions: 1. Factoring into lifting steps is a highly non-unique process. We do not know exactly how many essentially different factorizations are possible, how they differ, and what is a good strategy for picking the "best one"; this is an interesting topic for future research. 2. The main result of this paper also holds in case the filter coefficients are not necessarily real, but belong to any field such as the rationals, the complex numbers, or even a finite field. However, the Euclidean algorithm does not work when the filter coefficients themselves belong to a ring such as the integers or the dyadic numbers. It is thus not guaranteed that filters with binary coefficients can be factored into lifting steps with binary filter coefficients. 3. In this paper we never concerned ourselves with whether filters were causal, i.e., only had filter coefficients for k > 0. Given that all subband filters here are finite, causality can always be obtained by shifting the filters. Obviously, if both analysis and synthesis filters have to be causal, perfect reconstruction is only possible up to a shift. By executing the Euclidean algorithm over the ring of polynomials, as opposed to the ring of Laurent polynomials, it can be assured that then all lifting steps are causal as well. 4. The long division used in the Euclidean algorithm guarantees that, except for at most one quotient of degree 0, all the quotients will be at least of degree 1 and the lifting filters thus contain at least 2 coefficients. In some cases, e.g., hardware implementations, it might be useful to use only lifting filters with at most 2 coefficients. Then, in each lifting step, an even location will only get information from its two immediate odd neighbors or vice versa. Such lifting steps can be obtained by not using a full long division, but rather stopping the division as soon as the quotient has degree one. The algorithm still is guaranteed to terminate as the degree of the polyphase components will decrease by exactly 1 in each step. We are now guaranteed to be in the setting used to sketch the proof of Theorem 5. 5. In the beginning of this paper, we pointed out how lifting is related to the multiscate transforms and the associated stability analysis developed by Wolfgang Dahmen and co-workers. Although their setting looks more general than firing since it allows for a non-identity operator K on the diagonal of the polyphase matrix, while lifting requires identities on the diagonal, this paper shows that, in the first generation or time invariant setting, no generality is lost by restricting oneself to lifting. Indeed, any invertible polyphase matrix with a non-identity polynomial K(z) on the diagonal can be obtained using lifting. Note that some of the advantages of lifting mentioned above rely fundamentally on the K = I and disappear when allowing a general K. 6. This faetorization generalizes to the M-band setting. It is known that a M x M polyphase matrix with elements in a Euclidean domain and with determinant one can be reduced to an identity matrix using elementary row and column operations, see [24, Theorem 7.10]. This reduction, also known as the Smith normal form, allows for lifting factorizations in the M-band case.

154


In [48] the discussion of the decomposition into ladder steps (which is the analog, in different notation, of what we have called here the factorization into lifting steps) is carried out for the general M-band case; please check this paper for details and applications. 7. Finally, under certain conditions it is possible to construct ladder like structures in higher dimensions using factoring of multivariate polynomials. For details, we refer to [37]. Acknowledgments The authors would like to thank Peter SchrSder and Boon-Lock Yeo for many stimulating discussions and for their help in computing the factorizations in the example section, Jelena Kovaucevi~ and Martin Vetterli for drawing their attention to reference [28], Paul Van Dooren for pointing out the connection between the M-band case and the Smith normM form, and Geert Uytterhoeven, Avraham Melkman, Mark Masten, and Paul Abbott for pointing out typos and oversights in an earlier version. Ingrid Daubechies would like to thank NSF (grant DMS-9401785), AFOSR (grant F49620-95-1-0290), ONR (grant N00014-96-1-0367) as well as Lucent Technologies, Bell Laboratories for partial support while conducting the research for this paper. Wim Sweldens is on leave as Senior Research Assistant of the Belgian Fund of Scientific Research (NFWO).

References 1. A. Aldroubi and M. Unser. Families of multiresolution and wavelet spaces with optimal properties. Numer. Funet. Anal. Optim., 14:417-446, 1993. 2. H. Bass. Algebraic K-theory. W. A. Benjamin, Inc., New York, 1968. 3. M. G. Bellanger and J. L. Daguet. TDM-FDM transmultiptexer: Digital polyphase and FFT. IEEE Trans. Commun., 22(9):1199-1204, 1974. 4. R. E. Blahut. Fast Algorithms for Digital Signal Processing. Addison-Wesley, Reading, MA, 1984. 5. A. A. M. L. Bruekens and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. Selected Areas Commute., 10(1), 1992. 6. R. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo. Wavelet transforms that map integers to integers. Appl. Comput. Harmon. Anal., 5(3):332-369, 1998. 7. J. M. Carnicer, W. Dahmen, and J. M. Pefia. Local decompositions of refinable spaces. AppL Comput. Harmon. Anal., 3:127-153, 1996. 8. C. K. Chui. An Introduction to Wavelets. Academic Press, San Diego, CA, 1992. 9. C. K. Chui, L. Montefusco, and L. Puccio, editors. Conference on Wavelets: Theory, Algorithms, and Applications. Academic Press, San Diego, CA, 1994. 10. C. K. Chui and J. Z. ~¢Vang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113:785-793, 1991. 11. C. K. Chui and J. Z. Wang. A general framework of compactly supported splines and wavelets. J. Approx. Theory, 71(3):263-304, 1992.


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Spherical Wavelets: Efficiently Representing Functions on a Sphere* Peter SchrSder 1 and Wim Sweldens2 1 Department of Computer Science, California Institute of TecImology, Pasadena, CA 911257 U.S.A. ps~cs, caltech, edu 2 Bell Laboratories, Lucent Technologies, Murray Hill NJ 07974, U.S.A. wim~bell-labs, corn

1 1.1

Introduction Wavelets

Over the last decade wavelets have become an exceedingly powerful and flexible tool for computations and data reduction. They offer both theoretical characterization of smoothness, insights into the structure of functions and operators~ and practical numerical tools which lead to faster computational algorithms. Examples of their use in computer graphics include surface and volume illumination computations [17, 31], curve and surface modeling [18], and animation [20] among others. Given the high computational demands and the quest for speed in computer graphics, the increasing exploitation of wavelets comes as no surprise. While computer graphics applications can benefit greatly from wavelets~ these applications also provide new challenges to the underlying wavelet technology. One such challenge is the construction of wavelets on general domains as they appear in graphics applications and in geosciences. Classically, wavelet constructions have been employed on infinite domains (such as the real line ~ and plane ~ ) . Since most practical computations are confined to finite domains a mlmber of boundary constructions have also been developed [6]. However, wavelet type constructions for more general manifolds have only recently been attempted and are still in their infancy. Our work is inspired by the ground breaking work of Lounsbery et a/.[22, 21] (hereafter referred to as LDW). While their primary goal was to efficiently represent surfaces themselves we examine the case of efficiently representing functions defined on a surface, and in particular the case of the sphere. Although the sphere appears to be a simple manifold, techniques from ]~2 do not easily extend to the sphere. Wavelets are no exception. The first construction of wavelets on the sphere was introduced by Dahlke et a/.[7] using a tensor product basis where one factor is an exponential spline. To our knowledge a computer implementation of this basis does not exist at this moment. A * P. SchrSder and W. Sweldens. from Computer Graphics Proceedings, 1995~ 161 - 172, ACM Siggraph. Reprinted by permission of ACM.

Spherical Wavelets: Efficiently Representing Functions on a Sphere

159

continuous wavelet transform and its semi-discretization were proposed in [14]. Both these approaches make use of a (~, t~) parametrization of the sphere. This is the main difference with our method, which is parametrization independent. Aside from being of theoretical interest, a wavelet construction for the sphere leading to efficient algorithms, has practical applications since many computational problems are naturally stated on the sphere. Examples from computer graphics include: manipulation and display of earth and planetary data such as topography and remote sensing imagery, simulation and modeling of bidirectional reflection distribution functions, illumination algorithms, and the modeling and processing of directional information such as environment maps and view spheres. In this paper we describe a simple technique for constructing biorthogona] wavelets on the sphere with customized properties. The construction is an incidence of a fairly general scheme referred to as the lifting scheme [29, 30]. The outline of the paper is as follows. We first give a brief review of applications and previous work in computer graphics involving functions on the sphere. This is followed by a discussion of wavelets on the sphere. In Section 3 we explain the basic machinery of lifting and the fast wavelet transform. After a section on implementation, we report on simulations and conclude with a discussion and suggestions for further research.

Fig. 1. The geodesic sphere construction starting with the icosahedron on the left (subdivision level 0) and the next 2 subdivision levels.

1.2

R e p r e s e n t i n g Functions on t h e Sphere

Geographical information systems have long had a need to represent sampled data on the sphere. A number of basic data structures originated here. Dutton [11] proposed the use of a geodesic sphere construction to model planetary relief, see Fig. 1 for a picture of the underlying subdivision. More recently, Fekete [13] described the use of such a structure for rendering and managing spherical geographic data. By using hierarchical subdivision data structures these workers naturally built sparse adaptive representations. There also exist many non-hierarchical interpolation methods on the sphere (for an overview see [24]).

160

Peter Schr6der and Wire Sweldens

An important example from computer graphics concerns the representation of functions defined over a set of directions. Perhaps the most notable in this category are bi-directional reflectance distribution functions (BRDFs) and radiance. The BRDF, fr(wi, X, Wo), describes the relationship at a point x on a surface between incoming radiance from direction wi and outgoing radiance in direction Wo. It can be described using spherical harmonics, the natural extension of Fourier basis functions to the sphere, see e.g. [32]. These basis functions are globally supported and suffer from some of the same difficulties as Fourier representations on the line such as ringing. To our knowledge, no fast (FFT like) algorithm is available for spherical harmonics. Westin et at. [32] used spherical harmonics to model BRDFs derived from Monte Carlo simulations of micro geometry. Noting some of the disadvantages of spherical harmonics, Gondek et a/.[16] used a geodesic sphere subdivision construction [11, 13] in a similar context. The result of illumination computations, the radiance L(x, w), is a function which is defined over all surfaces and all directions. For example, Sillion et al. [28] used spherical harmonics to model the directional distribution of radiance. As in the case of BRDF representations, the disadvantages of using spherical harmonics to represent radiance are due to the global support and high cost of evaluation. Similarly no locally controlled level of detail can be used. In finite element based illuminations computations wavelets have proven to be powerful bases, see e.g. [26, 4]. By either reparameterizing directions over the set of visible surfaces [26], or mapping them to the unit square [4], wavelets defined on standard domains (rectangular patches) were used. Mapping classical wavelets on some parameter domain onto the sphere by use of a parameterization provides one avenue to construct wavelets on the sphere. However, this approach suffers from distortions and difficulties due to the fact that. no globally smooth parameterization of the sphere exists. The resulting wavelets are in some sense "contaminated" by the parameterization. We will examine the difficulties due to an underlying parameterization, as opposed to an intrinsic construction, when we discuss our construction. We first give a simple example relating the compression of surfaces to the compression of functions defined on surfaces.

Fig. 2. Recursive subdivision of the octahedral base shape as used by LD~Vfor spherelike surfaces. Level 0 is shown on the left followed by levels 2 and 4.


1.3

161

An Example

LDW constructs wavelets for surfaces of arbitrary topological type which are parameterized over a polyhedral base complex. For the case of the sphere they employed an octahedral subdivision domain (see Fig. 2). In this framework a given goal surface such as the earth is parameterized over an octahedron whose triangular faces are successively subdivided into four smaller triangles. Each vertex can now be displaced radially to the limit surface. The resulting sequence of surfaces then represents the multiple levels of detail representation of the final surface. As pointed out by LDW compressing surfaces is closely related to compressing functions on surfaces. Consider the case of the unit sphere and the function f(s) = f(~, ~) = cos2 ~ with s E S 2. We can think of the graph of this function as a surface over the sphere whose height (displaced along the normal) is the value of the function f . Hence an algorithm which can compress surfaces can also compress the graph of a scalar function defined over some surface. At this point the domain over which the compression is defined becomes crucial. Suppose we want to use the octahedron O. Define the projection T : 0 --+ S 2, s = T(p) = P/ItP[I- We then have ](p) = f(T(p)) with p e O. Compressing f(s) with wavelets on the sphere is now equivalent to compressing / ( p ) with wavelets defined on the octahedron. While f is simply a quadratic function over the sphere, ] is considerably more complicated. For example a basis over the sphere which can represent quadratics exactly (see Section 3.4) will trivially represent f . The same basis over tile octahedron will only be able to approximate ]. This example shows the importance of incorporating the underlying surface correctly for any construction which attempts to efficiently represent functions defined on that surface. In case of compression of surfaces themselves one has to assume some canonical domain. In LDW this domain was taken to be a polyhedron. By limiting our program to functions defined on a fixed surface (sphere) we can custom tailor the wavelets to it and get more efficiency. This is one of the main points in which we depart from the construction in LDW.

0.5 1 0.5

|

-0.5 1-0.5

Fig. 3. A simple example of refinement on the line. The basis functions at the top can be expressed as linear combinations of the refined functions at the bottom.

162 2

2.1

Peter Schr5der and Wire Sweldens Wavelets

on the

Sphere

Second Generation Wavelets

Wavelets are basis functions which represent a given function at multiple levels of detail. Due to their local support in both space and frequency, they are suited for sparse approximations of functions. Locality in space follows from their compact support, while locality in frequency follows from their smoothness (decay towards high frequencies) and vanishing moments (decay towards low frequencies). Fast O (n) algorithms exist to calculate wavelet coefficients, making the use of wavelets efficient for many computational problems. In the classic wavelet setting, i.e., on the real line, wavelets are defined as the dyadic translates and dilates of one particular, fixed function. They are typically built with the aid of a scaling ]unction. Scaling functions and wavelets both satisfy refinement relations (or two scale relations). This means that a scaling function or wavelet at a certain level of resolution (j) can be written as a linear combination of scaling basis functions of the same shape but scaled at one level finer (level j + 1), see Fig. 3 for an example. The basic philosophy behind second generation wavelets is to build wavelets with all desirable properties (localization, fast transform) adapted to much more general settings than the real line, e.g., wavelets on manifolds. In order to consider wavelets on a surface, we need a construction of wavelets which are adapted to a measure on the surface. In the case of the real line (and classical constructions) the measure is dx, the usual translation invariant (Haar) Lebesgue measure. For a sphere we will denote the usual area measure by dw. Adaptive constructions rely on the realization that translation and dilation are not fundamental to obtain the wavelets with the desired properties. The notion that a basis function can be written as a finite linear combination of basis functions at a finer, more subdivided level, is maintained and forms the key behind the fast transform. The main difference with the classical wavelets is that the filter coefficients of second generation wavelets are not the same throughout, but can change locally to reflect the changing (non translation invariant) nature of the surface and its measure. Classical wavelets and the corresponding filters are constructed with the aid of the Fourier transform. The underlying reason is that translation and dilation become algebraic operations after Fourier transform. In the setting of second generation wavelets, translation and dilation can no longer be used, and the Fourier transform thus becomes worthless as a construction tool. An alternative construction is provided by the lifting scheme. 2.2

M u l t i r e s o l u t i o n Analysis

We first introduce multiresolution analysis and wavelets and set some notation. For more mathematical detail the reader is referred to [10]. All relationships are summarized in Table 1.


163

T a b l e 1. Quick reference to the notation and some basic rdationships for the case of second generation biorthogonal wavelets. Y~nctions primal scaling functions dual scaling functions primal wavelets dual wavelets Biorthogonality relationships (~o~,k, ~ , k ' ) = ak,k,

(¢j,~, Cy,~,)

=a,~,~,a~,~,

Tj,k and ~_j,k, are biorthogonal Cj,m and Cj,,m, are biorthogonal

(¢~ .... ~ , ~ ) = 0 Vanishing moment relations ~a,-~ has N vanishing moments Cj,m h a s / ~ vanishing moments

~oj,kreprod, polyn, degree < N ~j,k reprod, polyn, degree
0, so that I Vj C ~ + 1 , (finer spaces have higher index) II Uj_>0 1/3. is dense in L 2, III for each j, scaling functions ~j,~ with k e ](:(j) exist so that {~j,k I k E/~(j)} is a Riesz basis I of ~ . Think of/(:(j) as a general index set where we assume that K:(j) C / C ( j + 1). In the case of the real line we can take E ( j ) = 2 - J Z , while for an interval we might have E(j) = {0, 2 - J , . . . , 1 - 2 - J } . Note that, unlike the case of a classical multiresolution analysis, the scaling functions need not be translates or dilates of one particular function. Property (I) implies that for every scaling function ~j,~, coefficients {hj,kj} exist so that ~j,k = ~ l hj,k,t ~j+l,1.

(1)

The hj,k,t are defined for j >_ o, k c /~(j), and l E E ( j + 1). Each scaling function satisfies a different refinement relation. In the classical case we have hj,k,l = hl-2k, i.e., the sequences hi,k4 are independent of scale and position. Each muttiresolution analysis is accompanied by a dual multiresolution analysis consisting of nested spaces ~ with bases given by dual scaling functions ~Sj,~, which are biorthogonal to the scaling functions:

(~j,k,~,k') = 5k,k' for k,k' ~ lC(j), where (f, g/ = f f g dw is the inner product on the sphere. The dual scaling functions satisfy refinement relations with coefficients {hj,k,l}In case scaling functions and dual scaling functions coincide, (~j,k = ~Sj,k for all j and k) the scaling functions form an orthogonal basis. In case the mnltiresolution analysis and the dual multiresolution analysis coincide (V./= ~ for all j but not necessarily ~j,k = qSj,k) the scaling functions are semi-orthogonal. Orthogonality or semi-orthogonality sometimes imply globally supported basis functions, which has obvious practical disadvantages. We will assume neither and always work in the most general biorthogonal setting (neither the multiresolution analyses nor the scaling functions coincide), introduced for classical wavelets

in [5] One of the crucial steps when building a multiresolution analysis is the construction of the wavelets. They encode the difference between two successive levels of representation, i.e., they form a basis for the spaces Wj where Vj • Wj = Vj+I. Consider the set of functions {¢j,m l J >- 0, m E )el(j)}, where 2,4(j) C ~ ( j + 1)is again an index set. If 1 A Riesz basis of some Hilbert space is a countable subset {]~} so that every element f of the space can be written uniquely as f ---~e ck .fk~ and positive constants A and B exist with AHfl] 2 < ~ k tCkt2 J0 set to zero then results in Aj,~ coefficients which converge to function values of ~jo,ko as j -+ co. In case the cascade algorithm converges in L 2 for both primal and dual scaling functions, biorthogonal filters (as given by the lifting scheme) imply biorthogonal basis functions.


177

One of the fundamental questions is how properties, such as convergence of the cascade algorithm, Riesz bounds, and smoothness, can be related back to properties of the filter sequences. This is a very hard question and at this moment no general answer is available to our knowledge. We thus have no mathematical proof that the wavelets constructed form an unconditional basis except in the case of the Haar wavelets. A recent result addressing these questions was obtained by Dahmen [8]. In particular, it is shown there which properties in addition to biorthogonality are needed to assure stable bases. Whether this result can be applied to the bases constructed here needs to be studied in the future. Regarding smoothness, we have some partial results. It is easy to see that the Haar wavelets are not continuous and that the Linear wavelets are. The original Butterfly subdivision scheme is guaranteed to yield a C i limit function provided the connectivity of the vertices is at least 4. The modified Butterfly scheme that we use on the sphere, will also give C i limit functions, provided a locally smooth (C i) map from the spherical triangulation to a planar triangulation exists. Unfortunately, the geodesic subdivision we use here does not have this property. However, the resulting functions appear visually smooth (see Fig. 6). We are currently working on new spherical triangulations which have the property that the Butterfly scheme yields a globally C i function. In principle, one can choose either the tetrahedron, octahedron, or icosahedron to start the geodesic sphere construction. Each of them has a particular number of triangles on each level, and therefore one of them might be more suited for a particular' application or platform. The octahedron is the best choise in case of functions defined on the hemisphere (cfr. BRDF). The icosahedron will lead to the least area inbalance of triangles on each level and thus to (visually) smoother basis functions.

Q 13

1E-01 0

a

13 0

a 0

13 D

0

1E-02.

Q

0

.e

O

rl o

Q o

i= o

1E-03,

Linear * Linearlifted

13 o

.

D o

D o

a o

1E-04.

1E+01

"

" ' "'"il

' '

1E+02

.......

I

IE+03

........

i

o

1E+04

numberof coefficients Fig. 10. Relative li error as a function of the number of coefficients for the example function f(s) = ~ and (lifted) Linear wavelets. With the same number of coefficients the error is smaller by a factor of 3 or conversely a given error can be achieved with about 1/3 the number of coefficients if the lifted basis is used.

178 3.8

Peter SchrSder and Wim Sweldens An Example

We argued at the beginning of this section that a given wavelet basis can be made more performant by lifting. In the section on interpolating bases we pointed out that a wavelet basis with Diracs for duals and a primal wavelet, which does not have 1 vanishing moment, unconditional convergence of the resulting series expansions cannot be insured anymore. We now give an example on the sphere which illustrates the numerical consequences of lifting. Consider the function f ( s ) = X/'lsxi for s = (sx, sy, sz) e S2. This function is everywhere smooth except on the great circle sx = 0, where its derivative has a discontinuity. Since it is largely smooth but for a singularity at 0, it is ideally suited to exhibit problems in bases whose primal wavelet does not have a vanishing moment. Fig. 10 shows the relative ll error as a function of the number of coefficients used in the synthesis stage. In order to satisfy the same error threshold the lifted basis requires only approximately 1/3 the number of coefficients compared to the unlifted basis.

4

Implementation

We have implemented all the described bases in an interactive application running on an SGI Irix workstation. The basic data structure is a forest of triangle quadtrees [11]. The root level starts with 4 (tetrahedron), 8 (octahedron), or 20 (icosahedron) spherical triangles. These are recursively subdivided into 4 child triangles each. Naming edges after their opposite vertex, and children after the vertex they retain (the central child becomes To) leads to a consistent naming scheme throughout the entire hierarchy. Neighbor finding is a simple 0(1) (expected cost) function using bit operations on edge and triangle names to guide pointer traversal [11]. A vertex is allocated once and any level which contains it carries pointers to it. Each vertex carries a single A and 7 slot for vertex bases, while face bases carry a single A and 7 slot per spherical triangle. Our actual implementation carries other data such as surface normals and colors used for display, function values for error computations, and copies of all 7 and )~ values to facilitate experimentation. These are not necessary in a production system however. Using a recursive data structure is more memory intensive (due to the pointer overhead) than a flat, array based representation of all coefficients as was used by LDW. However, using a recursive data structure enables the use of adaptive subdivision and results in simple recursive procedures for analysis and synthesis and a subdivision oracle. For interactive applications it is straightforward to select a level for display appropriate to the available graphics performance (polygons per second). In the following subsections we address particular issues in the implementation.

Spherical Wavelets: Efficiently Representing Functions on a Sphere 4.1

179

Restricted Quadtrees

In order to Support lifted bases and those which require stencils that encompass some neighborhood the quadtrees produced need to satisfy a restriction criterion. For the Linear vertex bases (lifted and unlifted) and the Bio-Haar basis no restriction is required. For Quadratic and lifted Bio-Haar bases no neighbor of a given face may be off by more than 1 subdivision level (every child needs a proper set of "aunts"). For the Butterfly basis a two-neighborhood must not be off by more than 1 subdivision level. These requirements are easily enforced during the recursive subdivision. The fact that we only need "aunts" (as opposed to "sisters") for the lifting scheme allows us to have wavelets on adaptively subdivided hierarchies. This is a crucial departure from previous constructions, e.g., tree wavelets employed by Gortler et al.[17] who also needed to support adaptive subdivision. 4.2

Boundaries

In the case of a hemi-sphere (top 4 spherical triangles of an octahedral subdivision), which is important for BRDF functions, the issues associated with the boundary need to be addressed. Lifting of vertex bases is unchanged, but the Quadratic and Butterfly schemes (as welt as the lifted Bio-Haar bases) need neighbors, which may not exist at the boundary. This can be addressed by simply using another, further neighbor instead of the missing neighbor (across the boundary edge) to solve the associated matrix problem. It implicitly corresponds to adapting filter coefficients close to the boundary as done in interval constructions, see e.g. [6]. This construction automatically preserves the vanishing moment property even at the boundary. In the implementation of the Butterfly basis, we took a different approach and chose in our implementation to simply reflect any missing faces along the boundary. 4.3

Oracle

One of the main components in any wavelet based approximation is the oracle. The function of the oracle is to determine which coefficients are important and need to be retained for a reconstruction which is to meet some error criterion. Our system can be driven in two modes. The first selects a deepest level to which to expand all quadtrees. The storage requirements for this approach grow exponentially in the depth Of the tree. For example our implementation cannot go deeper than 7 levels (starting from the tetrahedron) on a 32MB Indy class machine without paging. Creating full trees, however, allows for the examination of all coefficients throughout the hierarchies to in effect implement a perfect oracle. The second mode builds sparse trees based on a deep refinement oracle. In this oracle quadtrees are built depth first exploring the expansion to some (possibly very deep) finest level. On the way out of the recursion a local A n a l y s i s I is performed arid any subtrees whose wavelet coefficients are all below a user

180

Peter SchrSder and Wire Swetdens

Table 2. Representative timings for wavelet transforms beginning with 4 spherical triangles and expanding to level 9 (22° faces and 219 + 2 vertices). All timings are given in seconds and measured on an SGI R4400 running at 150MHz. The initial setup time (allocating and initializing all data structures) took 100 seconds. Basis

Analysis SynthesislLifted Basis Analysis Synthesis

Linear Quadratic Butterfly Bio-Haar

3.59 21.79 8:43 4.31

3.55 Linear 21.001Quadratic 8.42 Butterfly 6.09 Bio-Haar

5.85 24.62 10.64 42.43

5.83 24.68 10.62 36.08

supplied threshold are deallocated. Once the sparse tree is built the restriction criterion is enforced and the (possibly lifted) analysis is run level wise. The time complexity of this oracle is still exponential in the depth of the tree, but the storage requirements are proportional to the output size. With extra knowledge about the underlying function more powerful oracles can be built whose time complexity is proportional to the output size as well.

4.4

T r a n s f o r m Cost

The cost of a wavelet transform is proportional to the total number of coefficients, which grows by a factor of 4 for every level. For example, 9 levels of subdivision sta~,ing from 4 spherical triangles result in 220 coefficients (each of £ and 9/) for face bases and 219 + 2 (each of/~ and V) for vertex bases. The cost of analysis and synthesis is proportional to the nulnber of basis functions, while the constant of proportionality is a function of the stencil size. Table 2 summarizes timings of wavelet transforms for all the new bases. The initial setup took 100 seconds and includes allocation and initialization of all data structures and evaluation of the £9,k. Since the latter is highly dependent on the evaluation cost of the function to be expanded we used the constant function 1 for these timings. None of the matrices which arise in the Quadratic, and Bio-Haar bases (lifted and unlifted) was cached, thus the cost of solving the associated 4 × 4 matrices with a column pivoted QR (for Quadratic and lifted Bio-Haar) was incurred both during analysis and synthesis. If one is willing to cache the results of the matrix solutions this cost could be amortized over multiple transforms. We make three main observations about the timings: (A) Lifting of vertex bases adds only a small extra cost, which is almost entirely due to the extra recursions; (B) the cost of the Butterfly basis is only approximately twice the cost of the Linear basis even though the stencil is much larger; (C) solving the 4 × 4 systems implied by Quadratic and lifted Bio-Haar bases increases the cost by a factor of approximately 5 over the linear case (note that there are twice as many coefficients for face bases as for vertex bases). While the total cost of an entire transform is proportional to the number of basis functions, evaluating the resulting expansion at a point is proportional to


181

the depth (tog of the number of basis functions) of the tree times a constant dependent on the stencil size. The latter provides a great advantage over such bases as spherical harmonics whose evaluation cost at a single point is proportional to the total number of bases used.

5

Results

In this section we report on experiments with the compression of a planetary topographic data set, a B R D F function, and illumination of an aaisotropic glossy sphere. Most of these experiments involved some form of coefficient thresholding (in the oracle). In all cases this was performed as follows. Since all our bases are normalized with respect to the Loo norm, L 2 thresholding against some user supplied threshold c becomes if 17j,ml~/supp(~'j]m)< c, 7j,m : : O. Furthermore c is scaled by (max(f) - min(f)) for the given function f to make thresholding independent of the scale of f. IE+00

IE~0

cA o

a~ ieg~

IE-Ol

a Linear * Quadratic + * * x

lB-O2 -

~(~ l~ ~ ~ t

Butterfly

Linear lifted Quadratic lifted Butterfly lifted Bio-Haar G Bio-Haar lifted .......

~ ~

~ ~,~g ~ ~,,~g ~ ~A g ~

t . . . . . . . . I . . . . . . . . I . . . . . . al ~ " IE+02 1E+03 IE+04 IE+05

number of coefficients

IE-OI

'=

IE-02

Q Linear g *, Quadratic $ * Butterfly * Linear lifted * Quadratic lifted x Butterfly lifted g Bio-Haar O Bio-Haar lifted IE+02

IE+03

g 1~ g

IE+04


IE+05

IE+01

IE-~02

IE+03

tE404

IE+05


Fig. 11. Relative 11 error as a function of the number of coefficients used during the reconstruction of the earth topographic data set (left and middle) and BRDF function (right). The six vertex bases and two face bases perform essentially the same for the earth. On the left with full expansion of the quadtrees to level 9 and thresholding. In the middle the results of the deep refinement oracle to level 10 with only a sparse tree construction. The curves are identical validating the refinement strategy. On the right the results of deep refinement to level 9 for the BRDF. Here the individual bases are clearly distinguished.

5.1

Compression

of Topographic Data

In this series of experiments we computed wavelet expansions of topographic data over the entire earth. This function can be thought of as both a surface,

182

Peter SchrSder and Wim Sweldens

and as a scalar valued function giving height (depth) for each point on a sphere. The original data, ETOPO5 from the National Oceanographic and Atmospheric Administration gives the elevation (depth) of the earth from sea level in meters at a resolution of 5 arc minutes at the equator. Due to the large size of this data set we first resampled it to 10 arc minutes resolution. All expansions were performed starting from the tetrahedron followed by subdivision to level 9. Fig. 11 shows the results of these experiments (left and middle). After computing the coefficients of the respective expansions at the finest level of the subdivision an analysis was performed. After this step all wavelet coefficients below a given threshold were zeroed and the function was reconstructed. The thresholds were successively set to 2 -~ for i = 0 , . . . , 17 resulting in the number of coefficients and relative 11 error plotted (left graph). The error was computed with a numerical quadrature one level below the finest subdivision to insure an accurate error estimation. The results are plotted for all vertex and face bases (Linear, Quadratic, Butterfly, Bio-Haar, lifted and unlifted). We also computed 12 and loo error norms and the resulting graphs (not shown) are essentially identical (although the l~ error stays initially high before falling off due to deep canyon features). The plot reaches to about one quarter of all coefficients. The observed regime is linear as one would expect from the bases used. The most striking observation about these error graphs is the fact that all bases perform similar. This is due to the fact that the underlying function is non-smooth. Consequently smoother bases do not perform any better than less performant ones. However, when drawing pictures of highly compressed versions of the data set the smoother bases produce visually better pictures (see Fig. 12). Depending on the allowed error the compression can be quite dramatic. For example, 7200 coefficients are sufficient to reach 7% error, while 119000 are required to reach 2% error. In a second set of experiments we used the deep refinement oracle (see Section 4.3) to explore the wavelet expansion to 10 levels (potentially quadrupling the number of coefficients) with successively smaller thresholds, once again plotting the resulting error in the middle graph of Fig. 11. The error as a function of coefficients used is the same as the relationship found by the perfect oracle. This validates our deep refinement oracle strategy. Memory requirements of this approach are drastically reduced. For example, using a threshold of 2 -9 during oracle driven refinement to level 10 resulted in 4 616 coefficients and consumed a total of 27MB (including 10MB for the original data set). Lowering the threshold to 2 - l ° yielded 10 287 coefficients and required 43MB (using the lifted Butterfly basis in both cases). Finally Fig. 12 shows some of the resulting adaptive data sets rendered with RenderMan using the Butterfly basis and a pseudo coloring, which maps elevation onto a piecewise linear color scale. Total runtime for oracle driven analysis and synthesis was 10 minutes on an SGI R4400 at 150MHz. C o m p a r i s o n w i t h L D W The earth data set allows for a limited comparison of our results with those of LDW. They also compressed the ETOPO5 data set


183

Fig. 12. Two views of the earth data set with 15 000 and 190 000 coefficients respectively using the Butterfly basis with 1t errors of 0.05 and 0.01 respectively. In the image on the left coastal regions are rather smoothed since they contain little height variation (England and the Netherlands are merged and the Baltic sea is desertificated). However, such spiky features as the Cape Verde Islands off the coast of Africa are clearly preserved.

using pseudo orthogonalized (over a 2 neighborhood) Linear wavelets defined over the octahedron. They subdivide to 9 levels (on a 128MB machine) which corresponds to twice as many coefficients as we used (on a 180MB machine), suggesting a storage overhead of about 3 in our implementation. It is hard to compare the quality of the bases without knowing the exact basis used or the errors in the compressed reconstruction. However, LDW report the number of coefficients selected for a given threshold (741 for 0.02, 15 101 for 0.002, and 138 321 for 0.0005). Depending on the basis used we generally select fewer coefficients (6 000 - 15 000 for 0.002 and 28 000 - 65 000 for 0.0005). As timings they give 588 seconds (on a 100 MHz R4000) for analysis which is significantly longer than our smoothest basis (lifted Butterfly). Their reconstruction time ranges from 75 (741 coefficients) to 1 230 (138 058 coefficients) seconds which is also significantly longer than our times (see Table 2). We hypothesize that the timing and storage differences are lar'gely due to their use of flat array based data structures. These do not require as much memory, but they are more compute intensive in the sparse polygonal reconstruction phase. 5.2

BRDF Compression

In this series of experiments we explore the potential for efficiently representing B R D F functions with spherical wavelets. BRDF functions can arise from measurements, simulation, or theoretical models. Depending on the intended application different models may be preferable. Expanding BRDF functions in terms

184

Peter SchrSder and Wire Sweldens

of locally supported hierarchical functions is particularly useful for wavelet based finite element illumination algorithms. It also has obvious applications for simulation derived BRDF functions such as those of Westin et at. [32] and Gondek et al. [16]. The domain of a complete BRDF is a hemisphere times a hemisphere. In our experiments we consider only a fixed incoming direction and expand the resulting function over all outgoing directions (single hemisphere). To facilitate the computation of errors we used the BRDF model proposed by Schlick [25]. It is a simple Pad@ approximant to a micro facet model with geometric shadowing, a microfacet distribution function, but no Fresnel term. It has roughness (r [0,1], where 0 is Dirac mirror reflection and 1 perfectly diffuse) and anisotropy (p e [0, 1], where 0 is Dirac style anisotropy, and I perfect isotropy) parameters. To improve the numerical properties of the BRDF we followed the suggestion of Westin et al. [32] and expanded cOsOofr(wi,O, .).

Fig. 13. Graphs of adaptive oracle driven approximations of the Schlick BRDF with 19, 73, and 203 coefficients respectively (left to right), using the lifted Butterfly basis. The associated thresholds were 2-3, 2-6, and 2-9 respectively, resulting in relative 11 errors of"0.35, 0.065, and 0.015 respectively.

In the experiments we used all 8 bases but specialized to the hemisphere. The parameters were 0~ = ~r/3, r = 0.05, and p = 1. The results are summarized in Fig. 11 (rightmost graph). It shows the relative 11 error as a function of the number of coefficients used. This time we can clearly see how the various bases differentiate themselves in terms of their ability to represent the function within some error bound with a given budget of coefficients. We make several observations all lifted bases perform better than their unlifted versions, confirming our assertion that lifted bases are more performant; increasing smoothness in the bases (Butterfly) is more important than increasing the number of vanishing moments (Quadratic); - dual lifting to increase dual vanishing moments increases compression ability dramatically (Bio-Haar and lifted Bio-Haar); - overall the face based schemes do not perform as well as the vertex based schemes. -

-


185

Fig. 13 shows images of the graphs of some of the expansions. These used the lifted Butterfly basis with an adaptive refinement oracle which explored the expansion to level 9 (i.e., it examined 219 coefficients). The final number of coefficients and associated relative 11 errors were (left to right) 19 coefficients (ll = 0.35), 73 coefficients (11 = 0.065), and 203 coefficients (11 = 0.015). Total runtime was 170 seconds on an SGI R4400 at 150MHz.

Fig. 14. Results of an illumination simulation using the lifted Butterfly basis. A red, glossy, anisotropic sphere is illuminated by 2 area light sources. Left: solution with 2 000 coefficients (11 = 0.017). Right: solution with 5 000 coefficients (11 = 0.0035)

5.3

Illumination

To explore the potential of these bases for global illumination algorithms we performed a simple simulation computing the radiance over a glossy, anisotropic sphere due to two area light sources. We emphasize that this is not a solution involving any multiple reflections, but it serves as a simple example to show the potential of these bases for hierarchical illumination algorithms. It also serves as an example of applying a finite element approach to a curved object (sphere) without polygonalizing it. Fig. 14 shows the results of this simulation. We used the lifted Butterfly basis and the B R D F model of Schlick with r = 0.05, p = 0.05, and an additive diffuse component of 0.005. Two area light sources illuminate the red sphere. Note the fine detail in the pinched off region in the center of the "hot" spot and also at the north pole where all "grooves" converge.

186 6

Peter Schr5der and Wire Sweldens Conclusions

and Future

Directions

In this paper we have introduced two new families of wavelets on the sphere. One family is based on interpolating scaling functions and one on the generalized Haar wavelets. They employ a generalization of multiresolution analysis to arbitrary surfaces and can be derived in a straightforward manner from the trivial multiresolution analysis with the lifting scheme. The resulting algorithms are simple and efficient. We reported on the application of these bases to the compression of earth data sets, BRDF functions and illumination computations and showed their potential for these applications. We found that - for smooth functions the lifted bases perform significantly better than the unlifted bases; - increasing the dual vanishing moments leads to better compression; smoother bases, even with only one vanishing moment, tend to perform better for smooth functions; - our constructions allow non-equally subdivided triangulations of the sphere. -

We believe that many applications can benefit from these wavelet bases. For example, using their localization properties a number of spherical image processing algorithms, such as local smoothing and enhancement, can be realized in a straightforward and efficient way [27]. While we limited our examination to the sphere, the construction presented here can be applied to other surfaces. In the case of the sphere enforcing vanishing polynomial moments was natural because of their connection with spherical harmonics. In the case of a general, potentially non-smooth (Lipschitz) surface, polynomial moments do not necessarily make much sense. Therefore, one might want to work with local maps from the surface to the tangent plane and enforce vanishing moment conditions in this plane. It is possible to generalize this construction to wavelets on arbitrary surfaces, see, e.g. [19]. An interesting question for future research is the construction of compactly supported surface wavelets with smooth C 2 scaling functions such as Loop. These functions are not necessarily interpolating, so two lifting steps may not suffice. Applications of these bases lie in the solution of differential and integral equations on the sphere as needed in, e.g., illumination, climate modeling or geodesy. Acknowledgments The first author was supported by DEPSCoR Grant (DoD-ONR) N00014-94-11163. The second author was supported by NSF EPSCoR Grant EHR 9108772 and DARPA Grant AFOSR F49620-93-1-0083. He is also Senior Research Assistant of the National Fund of Scientific Research Belgium (NFWO). Other support came from Pixar Inc. We would also like to thank Princeton University and the GMD, Germany, for generous access to computing resources. Help with geometric data structures was provided by David Dobkin. Finally, the comments of the referees were very helpful in revising the paper.

Spherical Wavelets: Efficiently Representing Ftmctions on a Sphere

187

References 1. Alfeld, P., Neamtu, M., and Schumaker, L. L. Bernstein-B4zier polynomials on circles, sphere, and sphere-like surfaces. Preprint. 2. Carnicer, J. M., Dahmen, W., and Pefia, J. M. Local decompositions of refinable spaces. Tech. rep., Insitut ffir Geometrie und angewandete Mathematik, RWTH Aachen, 1994. 3. Certain, A , J. Popovid, T. DeRose, T. Duchamp D. SMesin and W~erner Stuetzle Interactive Muttiresolution Surface Viewing Computer Graphics (SIGGRAPH '96 Proceedings) (1996), 91-98 4. Christensen, P. H., Stollnitz, E. J., Salesin, D. H , and DeRose, T. D. Wavelet Radiance. In Proceedings of the 5th Eurographics Workshop on Rendering, 287302, June 1994. 5. Cohen, A., Daubechies, I., and Feauveau, J. Bi-orthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45 (1992), 485-560. 6. Cohen, A., Daubechies, I., Jawerth, B., and Vial, P. Multiresolution analysis, wavelets and fast algorithms on an interval. C. R. Acad. Sci. Paris S~r. I Math. I, 316 (1993), 417-421. 7. Dahlke, S., Dahmen, W., Schmitt, E., and Weinreich, I. Multiresolution analysis and wavelets on S2 and S% Tech. Rep. 104, Institut f~r Geometrie und angewandete Mathematik, RWTH Aachen, 1994. 8. Dahmen, W. Stability of multiscale transformations. Tech. rep., Institut ffir Geometrie und angewandete Mathematik, RWTH Aachen, 1994. 9. Dahmen, W., PrSssdorf, S., and Schneider, R. MultiscMe methods for pseudodifferential equations on smooth manifolds. In Conference on Wavelets: Theory~ Algorithms, and Applications, C. K. C. et al., Ed. Academic Press, San Diego, CA, 1994, pp. 385-424. 10. Daubechies, I. Ten Lectures on Wavelets. CBMS-NSF Regional Conf. Series in Appl. Math., Vol. 61. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992. 11. Dutton, G. Locational Properties of Quaternary Triangular Meshes. In Proceedings of the Fourth International Symposium on Spatial Data Handling, 901-910, July 1990. 12. Dyn, N., Levin, D., and Gregory, J. A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control. Transactions on Graphics 9, 2 (April1990), 160-169. 13. Fekete, G. Rendering and Managing Spherical Data with Sphere Quadtrees. In Proceedings of Visualization 90, 1990. 14. Freeden, W., and Windheuser, U. Spherical Wavelet Transform and its Discretization. Tech. Rep. 125, Universit~it KMserslautern, Fachbereich Mathematik, 1994. 15. Girardi, M., and Sweldens, W. A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general measure spaces. Tech. Rep. 1995:2, IndustriM Mathematics Initiative, Department of Mathematics, University of South

Carolina, 1995. (ftp://lip.math. scarolina, edn/pub/imi_95/imi95_2 .ps). 16. Gondek, J. S., Meyer, G. W., and Newman, J. G. Wavelength Dependent Reflectance Functions. In Computer Graphics Proceedings, Annual Conference Series, 213-220, 1994. 17. Gortler, S., SchrSder, P., Cohen, M., and Hanrahan, P. Wavelet Radiosity. In Computer Graphics Proceedings, Annum Conference Series, 221-230, August 1993.

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18. Gortler, S. J., and Cohen, M. F. Hierarchical and Variational Geometric Modeling with Wavelets. In Proceedings Symposium on Interactive 319 Graphics, 35-42, April 1995. 19. Lee A., W. Sweldens, P. Schr5der, L. Cowsar and D. Dobkin MAPS: Multiresolution Adaptive Parameterization of Surfaces Computer Graphics (SIGGRAPH '98 Proceedings), 95-104, 1998 20. Liu, Z., Gortler, S. J., and Cohen, M. F. Hierarchical Spacetime Control. Computer Graphics Proceedings, Annual Conference Series, 35-42, July 1994. 21. Lounsbery, M. Multiresolution Analysis ]or Surfaces of Arbitrary Topological Type. PhD thesis, University of Washington, 1994. 22. Lounsbery, M., DeRose, T. D., and Warren, J. Multiresolution Surfaces of Arbitrary Topological Type. Department of Computer Science and Engineering 93-1005, University of Washington, October 1993. Updated version available as 93-1005b, January, 1994. 23. Mitrea, M. Singular integrals, Hardy spaces and Clifford wavelets. No. 1575 in Lecture Notes in Math. 1994. 24. Nietson, G. M. Scattered Data Modeling. IEEE Computer Graphics and Applications 13, 1 (January 1993), 60-70. 25. Schlick, C. A customizable reflectance model for everyday rendering. In Fourth Eurographies Workshop on Rendering, 73-83, June 1993. 26. Schr5der, P., and Hanrahan, P. Wavelet Methods for Radiance Computations. In Proceedings 5th Eurographies Workshop on Rendering, June 1994. 27. Schr5der, P., and Sweldens, W. Spherical wavelets: Texture processing. Tech. Rep. 1995:4, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, 1995.

(ftp ://ftp. math. scarolina, edu/pub/imi_95/imi95_4, ps). 28. Sillion, F. X., Arvo, J. R., Westin, S. H., and Greenberg, D. P. A global illumination solution for general reflectance distributions. Computer Graphics (SIGGRAPH '91 Proceedings), Vol. 25, No. 4, pp. 187-196, July 1991. 29. Sweldens, W. The lifting scheme: A construction of second generation wavelets. Department of Mathematics, University of South Carolina. 30. Sweldens, W. The lifting scheme: A custom-design construction of biorthogonal wavelets. Tech. Rep. 1994:7, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, 1994.

(ftp ://ftp. math. scarolina, edu/pub/imi_94/imi94_7, ps). 31. Westerman, R. A Multiresolution Framework for Volume Rendering. In Proceedings ACM Workshop on Volume Visualization, 51-58, October 1994. 32. Westin, S. H., Arvo, J: R., and Torrance, K. E. Predicting reflectance functions from complex surfaces. Computer Graphics (SIGGRAPH '92 Proceedings), Vol. 26, No. 2, pp. 255-264, July 1992.

Least-squares Geopotential Approximation by Windowed Fourier Transform and Wavelet Transform Willi Freeden and Volker Michel University of Kaiserslautern, Laboratory of Technomathematics, Geomathematics Group, 67653Kaiserslautern, P.O. Box 3049, Germany [email protected] [email protected] http://w~w.mathematlk.uni-kl.de/~wwwgeo

1

Introduction

The spectral representation of the potential U of a continuous mass distribution such as the earth's external gravitational potential by means of outer harmonics is essential to solving many problems in today's physical geodesy and geophysics. In future research, however, Fourier expansions (x) 2 n + 1

U = E E fU(y)Hn,j(c~;y)dw(y)H~,j(c~,.) n----0 j = l A

(&0 denotes the suface element on the sphere A around the origin with radius a) in terms of outer harmonics

o=o,1 .....

will not be the most natural or useful way of representing (a harmonic function such as) the earth's gravitational potential. In Order to explain this in more detail we think of the earth's gravitational potential as a signal in which the spectrum evolves over space in significant way. We imagine that at each point on the sphere A the potential refers to a certain combination of frequencies, and that in dependence of the mass distribution inside the earth, the contributions to the frequencies and therefore the frequencies themselves are spatially changing. This space-evolution of the frequencies is not reflected in the Fourier transform in terms of non-space localizing outer harmonics, at least not directly. In theory, a member U of the Sobolev space 740 (of harmonic functions in the outer space A~xt of the sphere A with square-integrable restrictions on A) can be reconstructed from its Fourier transform, i.e. the 'amplitude spectrum'

190

Willi Freeden and Volker Michel

with

/,

(U, H,,,j (a, "))no = ] U(x)Hn,j (a; x) dw(x), A

but the Fourier transform contains information about the frequencies of the potential over all positions instead of showing how the frequencies vary in space. This paper will present two methods of achieving a space-dependent frequency analysis in geopotential determination, which we refer to as the windowed Fourier transform and the wavelet transform. Essential tool is the concept of a harmonic scaling function { ~(2)}, P E (0, ec). Roughly speaking, a scaling func-

Ae,t U A, of the

tion is a kernel ~(2) : Aext x Aext ~ I~, Ae~t = oo

form

2n+l

~(2)(x'Y) = E

(q°"(n))2 E

n-----0

Hnj(a;x)Hnj(a;y)

j=l

converging to the 'Dirac-kernel' 5 as p tends to 0. The Dirac kernel is given by oo

2nq-1

=Z

n=0 j=l

Consequently {~pp(n)}n=0,i .... is a (suitable) sequence satisfying

lira qpp(n) = 1 p ---:-0 p>0

for each n = O, 1,.... According to this construction principle, ¢ ~ *E)/[",~f l (0, c~), constitutes an approximate convolution identity, i.e. the convolution integral

A

formally converges to

U(x) = (5 • V)(x)

= (5(x, .), U)n ° =

] 5(x,y)U(y) dw(y) A

for all x E Aext as p tends to 0. Therefore, if U is a potential of class 7-/0, then lim I U - q5(2) * U 1

=0.

(1)

p>o

The windowed Fourier transform and the wavelet transform are two-parameter representations of a one-parameter (spatial) potential in n0. This indicates the existence of some redundancy in both transforms which in turn gives rise to

LS Geopotential Approximation

191

establish the least-squares approximation properties (promised in the title of the paper). The windowed Fourier transform (WFT) can be formulated as a technique known as 'short-space Fourier transform'. This transform works by first dividing a potential (signal) into short consecutive (space) segments and then computing the Fourier coefficients of each segment. The windowed Fourier transform is a space-frequency localization technique in that it determines the frequencies associated with small space portions of the potential. The windowed Fourier segments are constructed by shifting in space and modulating in frequency the 'window kernel' ~p given by 2n+l

~p(X,y) = ~ p ( n ) n=0

~H~,j(a;x)H,~,j(~;y),

(x, y) C Aext x Aext -

j=l

Note that ~2) (x, y) = (~p * ~p) (x, y) = / ~p (x, z)~p (z, y) dw(z) A

for all (x, y) C Aext x Aext- Once again, the way of describing the windowed Fourier transform (i.e. the 'short-space Fourier transform') is as follows: Let U be a potential of class 7-/o. The Fourier transform FT is given by

(FT)(U)(n,j) = f U(y)Hn,j(a; y) dw(y) A

((n,j) CAf, A / = {(k, l) I k = 0, 1,... ;l = 1,..., 2k + 1}), i.e., FT maps no into the space 7/0(N') of all sequences {H(n,j)}(nj)ear with (H(n,j)) 2 < c~ . (n,j)eN

Now ~p, with p arbitrary but fixed, is a space window ('cutoff kernel'). Chopping up the potential amounts to multiplying U by the kernel ~p, i.e. U(y)~p(x, y) with x E Aext, y ~ A, where the fixed value p measures the length of the window (i.a., the 'cutoff cap') on the sphere A. The Fourier coefficients of the product in terms of outer harmonics {Hn,j(a; -)} ~=~,.o.:~.$1 are then given by

(FT)(~p(x, .)V)(n, j) = I U(y)~Sp(x,y)Hn,j (a; y) dw(y), A

(n,j) E N', x E Aext. In other words, we have defined the 7-/o-inner product of U with a discrete set of 'shifts' and 'modulations' of U. The windowed Fourier transform is the operator (WFT)¢p, which is defined for potentials U E ~/0 by

A

192


for in, j) E /¢" and x E Aext (note that ~(2)(1 ) is a normalization constant determined by the choice of ~@(p~'!~). If ~p is concentrated in space at a point ) k x E A, then (WFT)¢,(U)(n,j; x) gives intbrmation of U at position x E A and frequency (n,j) EAf. The potential U E 7/0 is completely characterized by the values of

{ (W FT) (U)(n, j; x) } (.a)es , ~EAext

U can be recovered via the reconstruction formula --1/2 co 2 n 9 - 1 /

E E

U = (~(2)(1))

(WFT)~. iU)(n'j;x)q~oi''x) dw(x)Hn,jia;')

n----0 j = l A

in the ll" Ilno-sense. Obviously, (WFT)~, converts a potential U of one spatial variable into a function of two variables x E Aext and (n, j) E Af without changing its 'total energy', i.e.: co 2n-l-i

II II 0 = n=O

j=l

S( (WFT)+. (U))in,S; ")'

k

But, as we shall see in this paper, the space ~ = (WFT)@. (7/o) of all windowed Fourier transforms is a proper subspace of the space 7/olAF x A----~t)of all (twoparameter) functions G:(n,j;x) ~-~ G(n,j;x), in,j) E AF, x E Aext, such that Gin, j; .) E 7/o for all in,j) E N and liGllno(ArxHXT:0< co (this simply means that G is a subspace of 7/0(H x Aext) but not equal to the latter).Thus being a member of 7/o(X x A---~xt)is a necessary but not sufficientcondition for G E (note that the extra condition that is both necessary and sufficient is called consistency condition). The essential meaning of ~ = (WFT)~p (7/0) in the framework of the space No (Af x A--~t) can be described by the following least-squares property: Suppose we want a potential with certain properties in both space and frequency. We look for a potential U E No such that (WFT)~p(U)(n,j;x) is closest to H(n,j;x), (n,j) EAf, x E Aext in the '?/o(Af x Aext)-metric', where H E 7/0(Af × Aext) is given. Then the solution to the least-squaresproblemis provided by the function --1/2

eo 2n-l-I f

which indeed is the unique potential in 7/0 that minimizes the '7/o(Af x Aex~)error':

Ilg - (WFT)v, (UH)l]no(nrxA~) = inf ] I H - (WFT)op(V)ll~to(Arx~). UET/o

(3)

LS Geopotential Appr0ximation

193

Moreover, if H E 6, then Eq. (2) reduces to the reconstruction formula. In the context of oversampling a signal Eq. (3) means that the tendency for minimizing errors is expressed in the least-squares property of the windowed Fourier transform. Although the oversampting of a potential (signal) might seem inefficient, such redundancy has certain advantages: it can detect and correct errors, which is impossible when only minimal information is given. Although the shape of the window may vary depending on (the space width) p, the uncertainty principle (see the considerations of M. Holschneider in this volume for the Euclidean case and [6,7,10] for the spherical and harmonic case) gives a restriction in space and frequency. This relation is optimal (cf. [7]), in the sense of localization in both domains, when ~p is a Gaussian, in which case the windowed Fourier transform is called the Gabor transform (cf. [14]). An essential problem of the windowed Fourier transform is that it poorly resolves phenomena of spatial extension shorter than the a priori chosen (fixed) window. Moreover, shortening the window to increase spatial resolution can result in unacceptable increases in computational effort. In practice, therefore, the use of the windowed Fourier transform is limited. This serious calamity, however, will be avoided by the wavelet transform. The wavelet transform acts as a space and frequency localization operator in the following way. Roughly speaking, if {~(2)}, P C (0, oc), is a harmonic scaling function and p is a small positive value, then ~2)(y, .), Y E A, is highly concentrated about the point y. Moreover, as p tends to infinity, ~(2)(y, .) becomes more and more localized in frequency. Correspondingly, the uncertainty principle (cf. [10]) states that the space localization of ~2)(y, .) becomes more and more decreasing. In conclusion, the products y ~ qS(~)(x,y)U(y), y E A, x E Aext, for each fixed value p, display information in U ~ n0 at various levels of spatial resolution or frequency bands. Consequently, as p approaches infinity, the convolution integrals ~5(p2) * U = ~p * 4ip, U

= Af =f

A

y)u(y) d (y)

A

display coarser, lower-frequency features. As p approaches zero, the integrals give sharper and sharper spatial resolution. In other words, like a windowed Fourier transform, the convolution integrals can measure the space-frequency variations of spectral components, but they have a different space-frequency resolution. Each scale-space approximation ~(2) • U = ~p * qhp. U of a potential U E 7/o must be made directly by computing the relevant convolution integrals. In doing so, however, it is inefficient to use no information from the approximation ~2) *U within the computation of ~(2,) • U provided that pt < p. In fact, the efficient p

194


construction of wavelets begins by a multiresolution analysis, i.e. a completely recursive method which is therefore ideal for computation. In this context we observe that oo

R A

-~ R

A

UET/o,

R--+O,

A

i.e.

lis

nm u a>0

ep(-,z)

R A

s

eAz, y)U(y) d~(y) d~(~)

A

?

= 0, 7/o

provided that c~

!l~2)(x'y) = E

2 n-{-1

(Co(n))2 E

n=0

Hnd(a;x)Hnd(a;Y)'

(x,y) E Aext × Aext ,

j=l

is given such that d (¢p(n)) 2 = -p-~p (~Op(n))2

(4)

for n = 0, 1,... and all p C (0, oo). Conventionally, the family {~p}, p E (0, oo), is called a (scale continuous) wavelet. The (scale continuous) wavelet transform W T : 7-/o -+ 7-/o((0, oo) x Aext) is defined by (WT)(U)(p; x) = ( ~ , U)(x) =

(eAx,-), U)~0

In other words, the wavelet transfbrm is defined as the 7-/0-inner product of U E 7/o with the set of 'shifts' and 'dilations' of U. The (scale continuous) wavelet transform W T is invertible on 7/o, i.e. o0

A

o

in the sense of II" I]~to. From Parseval's identity in terms of outer harmonics it follows that oo

iS (vp(.,y), U)~o ~

A

d~(y) = (u, u)~0,

0

i.e., W T converts a potential U of one variable into a function of two variables x E Aext and p E (0, oo) without changing its total energy.


195

low-pass filter and band-pass filter, respectively. Correspondingly, the convolution operators are given by

Pp(U)=~p*~p*U,

U ET{o

R,(u)

u e T{o .

=

,

, u,

The Fourier transforms read as follows:

(FT)(Pp(U))(n,j) = (FT)(V)(n,j)(~p(n)) 2, (FT)(Rp(U))(n,j) = (FT)(U)(n,j)(¢p(n)) 2,

(n,j) E 2¢" (n,j) EAf .

These formulae provide the transition from the wavelet transform to the Fourier transform. The scale spaces 12n = Pp(7{o) form a (continuous) multiresolution analysis

~2pc Vp, C ?-to,

0 [l] + 1. Then sup xEA~x~

N 2n+l E (U'HmJ(a;'))L2(A)

(VtU)(x)- E

(V1Hn'J) (a; x)

I

n = O j----I

0

But this is the desired result. In connection with (6) we obtain the following result. C o r o l l a r y 1. Under the assumptions of Theorem 6 lira sup

R-..~ O

R>o x C ~

t s; U(x) -

Wm)(U)(p;y)g%y(X

dw(y) = 0 .

A R

In other words, a constructive approximation by wavelets defined on Aext is found to approximate the solution of the Dirichlet boundary-value problem for the Laplace equation on Zest.

222

7.2


Least-squares Property

Denote by 7/0 ((0, c¢) x A--~xt) the space of all functions U:(0, c¢) x Aext --+ IS such that U(p; .) E 7to for every p E (0, oc) and oo

oo

f llu(p;")112o-dp7 = f 0

f lU(P;y)l~ dw(y)

< c¢ .

(29)

0 A

On 7/0((0, (XD)X Aext) we are able to impose an inner product (-, ")7/o((O,~)xA-~0 by letting

(f(',),W(','))7/o((O,c~)xA~xt)-/f U(p~Y)V(p~y)d(..d(Y)d~Pp o A = f (u(p; .), V(p, "))~o aPp 0 for U, V E 7/0((0, co) x A----~xt).Correspondingly,

=

(oi,,

g(p, .)ll 7to 2

From Theorem 6 we obtain the following result telling us that the wavelet transform does not change the total energy. L e m m a 9. Let {~Pp}, p e (0, co), be a wavelet. Suppose that U, V are of class

7-[0 with (u, H0,1 (~; ))7/0 = (V, go,1 (~; "))no = 0. Then

oo

0

A

As we have seen, W T is a transform from the one-parameter space 7/o into the two-parameter space 7/o((0, oo) x A----~t). However, the transform W T is not surjective on 7-/o((0, oo) x A---~xt)(note that 7/o((0, oo) x Aext) contains unbounded elements, whereas it is not hard to see that (WT)(U) is bounded for all U £ 7/o). This means that W = (WT)(no) (3i) is a proper subspace of 7/0((0, oc) x A--~t):

w ~ ~o((0, oo) × Ae~0-


223

Therefore, one may ask the question of how to characterize W within the framework of ?-/0((0, oc) x Aext). For that purpose we consider the operator P : n 0 ((0, oo) × A--~t) + W

(32)

defined by oo 7

y')= / / , /

, J

O

A

Ip; )v (p;y)

(0, oo),

c A~xt,

t-

(33) where we have introduced the kernel

K(p';v'lp;v)

= / ~p,;y,(x)~;~(.) d~(x) = (~¢;~'(),~;y())~o

m

A First our purpose is to verify the following lemma.

The operator P : ~ o ( ( 0 , o c ) x Aext) -+ W defined by (32), (33) is a projection operator.

L e m m a 10.

Proof. Assume that H = ~" = for x E Aext

(WT)(U) E W. Then it is not difficult to see that

(x)

P(H)(p; x) = / / K (p; x [ a; y)(WT)(U)(a; y) d~(y)a-~ .J

,I

o

A

= O(p; x)

= (WT)(U)(p; ~). Consequently, P ( H ) ( - , - ) = H(., .) for all H(-,-) E "W. Next we want to show that for all H ± ( -, .) E W ± we have P(H±( ., .)) = O. For that purpose, consider an element H i ( -, -) of W ±. Then, for all U C 7"/o we have (H i (., .), (WT)(U)(., .)) go((O,oo)×Aox~) -- 0 . (34) Now, tbr p E (0, co) and x e Aext, we obtain under the special choice U = ~mx(')

o = (H ± (., .1, (WT) oo

o

A

cx~

o

A

= P(H±)(p;x) .

(ep;.~) ())~o((0,~) ×~-:7)

224


In other words, P(H±( ., -)) = 0 for all H i ( -, -) E W ±. Therefore we find P (7/o((0,oo) ×

= w,

P W ± = 0, []

p2 = p , as desired.

FV = (WT)(7/o) is characterized as follows: L e m m a 11. H E W if and only if the 'consistency condition' O0

P o

.4

is satisfied. Obviously,

K(p', y'[.;-) c W, K(.;.Ip, y) e W ,

p' c (0, oo), y' C Aext, pC(O, oo),yCAext,

i.e.:

(p'; y' l p; y) ~+ K(p', Y' I P;Y) is the (uniquely determined) reproducing kernel in ]/Y. Sunnnarizing our results we therefore obtain the following theorem. T h e o r e m 7. Let H be an arbitrary element of 7-/0((0, oo) x Aext). Then the

unique function UIt E 7io satisfying the property

(with 8 = WT(U)) is gi.en by O0

o

A

Theorem 7 means that UH defined above comes closest in the sense that the '7/((0, c~) × A---~t)-distance' of its wavelet transform ~rH to H assumes a minimum. In analogy to the windowed Fourier theory we call UH the least-squares approximation to the desired potential U E 7/0. Of course,, for H E )4;, Eq. (35) reduces to the reconstruction formula. All aspects of least-squares approximation discussed earlier for the windowed Fourier transform remain valid in the same way. The coefficients in 7/0((0, oo) x A-~t) for reconstructing a potential U E 7/0 are not unique. This can be readily developed from the following identity OO

o

A

LS GeopotentiaI Approximation

225

where 0 = (WT)(U) is a member of ]42 and U ± is an arbitrary member of W ±. Our considerations enable us to formulate the following minimum norm representation: T h e o r e m 8. For arbitrary U 6 74o the function 0 = (WT)(U) e ½' is the unqiue element in 74((0, co) × A~t) satisfying inf

HET/O ((0,oo)X Aex%)

tlHttn((o,~)×~)

(WT)--I(H)=U

8

Scale Discrete

Wavelet

Transform

Until now emphasis has been put on the whole scale interval. In what follows, however, scale discrete wavelets will be discussed. We start from a strictly decreasing sequence {pj}, j E Z, such that lim)_~ pj = 0 and limj_~_~ pj = co. For reasons of simplicity, we choose pj = 2 -3 , j E Z, throughout this paper. Definition 9. Let ~oD = ~po be the generator of a scaling function (as defined above). Then the piecewise continuous function eD : [0, c~) -+ ~ is said to be the 740-generator of the mother wavelet kernel ~D (of a scale discrete harmonic wavelet) if it satisfies the admissibility condition and satisfies, in addition, the difference equation D

2

(¢2(t)) 2 = (~2 ( t / 2 ) ) ~ - (Vo (t)) ,

t c ]0,co)

For ~D and ¢0D, respectively, we introduce functions ~D and ~bD , respectively, in the canonical way

~D(t)=DY~D(t)=(pDo (2-Jr),

tE[O, co),

eD(t ) = DDeD(t) = eD ( 2 - i t ) ,

t E [0, c~) .

Then, each function ~D and ~bD, respectively, j E Z, satisfies the admissibility condition. This enables us to write ¢ ~ = D1¢~_~, j ~ Z, whenever ~o~ satisfies the admissibility condition. Correspondingly, for the 74o-kernel ~ D j E Z, generated by cD via (@D)A(n) : ¢ f ( n ) , n e No, we let

~D = D I ~ D I , j E Z . Definition 10. The subfamily {q~D}, j E Z, of the space 74o generated by ~D via OPD) A (n) = eD (n), n = O, 1,..., is called a scale discrete harmonic wavelet. The generator eD : [0, co) -+ ~ and its dilates eD = DD¢0D satisfy the following properties: eD(0) = 0, j E Z ,

226


(¢~(t)) 2 = (~y÷l(t)) ~ - ( ~ ( t ) ) ~,

j e z, t e [0,~),

J

(TD(t))2+E(¢D(t))2=(~D+l(t))2,

J e N o , tE[O, cc) •

(36)

j=0

It is natural to apply the operator D D directly to the mother wavelet. In connection with the "shifting operator" Sy, this will lead us to the definition of the kernel ~j;y. More explicitly, we have ~ D = D D ~ 0 D,

jEZ,

(37)

and (Sy~D) (x) =~PDy(x) = ~ D ( x , y ) =Opi(x,y),

(x,y) e A~xt × gext •

(38)

Putting together (37) and (38) we therefore obtain for (x, y) e Aext × Aext, D ~;;~(x) = (S~D?~0~) (x) =

~-~2n+l

o:o 4 ~

(xy)

A

(~°~) (2-J~) ~ ]

~.~

Po

Definition 11. Let the 7~o-kernel ~oD be a mother wavelet kernel corresponding to a scaling function ~oD = q~po. Then, the scale discrete wavelet transform at scale j C Z and position y E Aext is defined by

(WT)D(u)(j;y) = (~.,y,U)no,

U eT/o •

It should be mentioned that each scale continuous wavelet {~p}, p E (0, c~), implies a scale discrete wavelet {~D}, j C Z, by letting

((~?)~(~))~: ((~?+~)~(~))~- ((~)A(~))~, where

((~)~(~))~: /

c~

n ~ No,

(,:(n))~ dp P

PJ

i.e.,

i/2

Note that this construction leads to a partition of unity in the following sense

0

3:-c~

j:0

for n E N. Our investigations now enable us to reconstruct a potential U C 7to from its discrete wavelet transform as follows.


227

T h e o r e m 9. Any potential U E 7/0 can be approximated by its J-level scale discrete wavelet approximation J

qSO;y)n° ~2y(.)dw(y) +

= A

(WT)D(u)(j; y)~Dy(.)dw(y)

(39)

j=o A

in the sense that lira IIV - VJiluo = 0

J--+~

Proof. Let U be a member of class 7/0. From (36) it follows that

/

o

J

(U'~o;y)n0 ~OD;y(')dw(y) + E

A

/ (WT)D(u)(j;Y)~£Dy (')dw(y)

j=0 A

= f(V,

o o ' ~j+~;~)~o ~j+1;y(-)d~(y)

•

A

Letting J tend to infinity the result follows easily from Theorem 2.

[]

As an immediate consequence we obtain C o r o l l a r y 2. Let Z be a regular surface. Under the assumptions of Theorem 9, lira

sup [U(x) - Uj(x)[ = O .

J--+co x E~Text

8.1

Multiresolution

Next we come to the concept of multiresolution by means of scale discrete harmonic wavelets. For U • 7i0 denote by R D (band-pass filters), pD (low-pass filters), the convolution operators given by

,u,

UET/o U E~o

respectively. The scale spaces 1)o and the detail spaces D?)D are defined by v? = Pf(Uo),

w? = R?(Uo), respectively. The collection {)?D} of all spaces 1;D, j E Z, is called multiresolution analysis of 7/o. Loosely spoken, t}o contains all j-scale smooth functions of 7-/0. The notion "detail space" means that 14)D contains the "detail" information needed to go from an approximation at resolution j to an approximation at resolution j + 1.

228


To be more concrete, WD denotes the space complementary to Vff in ]2j+l D in the sense that D v;+l = v? + w~

Note that

J ~;-}-1 •

j=O

Any potential U • 7/0 can be decomposed in the following way: Starting from POD(U) we have J

P?+I(U) = P°D(u) + E RD(u)

"

j=O

The partial reconstruction RD(u) is nothing else than the difference of two "smoothings" PD+I(U) and PjD(U) at consecutive scales

R:(U) p?+l(u)- pl'(u). =

Moreover, in spectral language we have

(~?(u),~o,,(.;.)).o = (~,~o,,(.;.)).0 ((~j.)A (.))., (n, l) • N (R~(u), Ho,,(.;-)).o = (u,~o,,(-;-))~o ((~?)" (n))',

(40)

(n, 0 • ~" •

The formulae (40) give (scale discrete) wavelet decomposition an interpretation in terms of Fourier analysis by means of outer harmonics by explaining how the frequency spectrum of a potential U C VD is divided up between the space ];D_1 and W j -DI " The multiresotution can be illustrated by the following scheme:

POD(U) tT~

ft~

Vo~ VoD klJ

+

c

v~

w~

+

kl)

Po~(U) + R2(u) 8.2

Pg(U)

P?(U)...

P~+I(U)...-~ U

#T~

... ...+

rr~

c

V

c

v j+l '~

... C 740

w j-1 ~

+

WD

+

....

7/o

+

R~(U)

+

....

U

klJ

+

... + R~_l(u)

Least-squares Approximation

The reconstruction formula (Theorem 9) may be rewritten as follows:

lira flU - uJl[~o = o,

J---~oo

u c 74o,


229

where the J-level scale discrete wavelet approximation now reads in shorthand notation as follows: J

E

uz =

/(WT)D(u)(j;y)~Dy (') d~(y) .

j=--c~ A

As in the continuous case we can make use of the projection property in the scale discrete case. We know already that (WT)D:74o --+ 740(Z x A~xt)~ where 740(Z × ~4ext) is the space of all functions U:(j;x) ~-~ U(j;x) with U(j; .) 6 74o for every j 6 Z and (:~

(yD

llu(j;)ll~0 = ~ j=-~

f lu(j;~)l~ d~(~)

< ~

.

j=-c~ A

It is not hard to see that

W D = (WT)D(74o) ~ 740(Z × Aext). Hence, we are able to define the projection operator pD ~74O(Z × ~e~t) "+ W D by oo

pD(u)(j';Y') = E

f KD(j';y' lj;y)U(j;y) dw(y),

j=--oo A

where

KD(j ',y'Ij;y) = / ~j,;y,(x)~j;y(X) D D ~D dw(x) = (~j,i~,(.), D j;y(.))~o

.

(41)

A

In similarity to results of the scale continuous case it can be deduced that pD is a projection operator. Therefore we have the following characterization of )/yD: L e m m a 12. U(., .) 6 fleD i f and only if the 'consistency condition' co

V(j';y') = ~

P

/ g D (J';Y' IJ;Y) V(j;y) d~(y) J

J=-~ A

= E

(KD(j';y' l J;')V(J;'))no

j~--oo

is satisfied. Summarizing our results we obtain the following theorem. T h e o r e m 10. Le{ H be an arbitrary element of 74o(Z x Aext). Then the unique function UD 6 740 sati]ying the property [g(',-)

~D(.,.) -

~o(ZxAcx~----) = Ve~toinfI H ( . , - ) - ~D(.,.)l~to(ZxAo~0

230


(with (]D = ( W T ) D ( U ) ) is given by co

U (x) = Z f v(j; J=-co A

Moreover, we have T h e o r e m 11. For arbitrary U E ~o the function ~T = ( W T ) D ( u ) e kV D is the unqiue element in 7-/o(Z x Aext) satisfying [

9

~(Z xA--'~0

n eno(~xA.~t-----)

IlH]ln(z ×A-~,)

((wT)D)--I(H)=U

Examples

Now we are prepared to introduce some important examples of scaling functions and corresponding wavelets (cf. [6]). We distinguish two types of wavelets, namely non-band-limited and band-limited wavelets. Since there are only a few conditions for a function to be a generator of a scaling function, a large number of wavelet examples may be listed. For the sake of brevity, however, we have to concentrate on a few examples. 9.1

Non-band-limited Wavelets

All wavelets discussed in this subsection share the fact that their generators have a global support. Rational Wavelets: Rational wavelets are realized by the function qOl : [0, ec) -+ given by (t) = (z + t) t e [0, Indeed, ~1(0) -- 1, ~1 is monotonically decreasing, it is continuously differentiable on the interval [0, co), and we have ~l(t) = O(ltl-l-~), t --~ oc, for s = 1 + e, e > 0. The (scale continuous) scaling function {~p},p E (O, oc), is given by 47ra 2 ( l + p n ) s \ ~ [ ]

~p(x,y)=

p~

x . y

, (x,y) e Aextxdext •

n~0

It is easy to see that

¢1 (t) = v ~ ( 1

+ t) -~-~/~

so that the scale continuous harmonic wavelets {~Pp},p e (0, ec), are obtained from CAn) = 2 x / ~ ( 1 + p n ) - ~ - l / ~ , s > 1, n e No, whereas the scale discrete wavelets {~pD}, j E Z, are generated by CD(n) = ((1 + 2 - J - i n ) - ~ -- (1 + 2-Jn)-2~) 1/~ j e Z , n e No


Exponential Wavelets. We choose ~l(t) = e-Rt,R > O, t

231

• [0, oc). Then it

follows that

~p(t) = e -Rpt, p

•

(0, ~ )

and

¢~(t) = 2 v ~

e -R'~,

p • (0, ~ )

.

Moreover,

¢~(n)

9.2

~" ~/~

e -2-j-IR~ ~

Band-limited Wavelets

All wavelets discussed in this subsection are chosen in such a way that the support of their generators is compact. As a consequence the resulting wavelets are band-limited. A particular result is that the Shannon wavelets provide us with an orthogonal multiresolution. Shannon Wavelet. The generator of the Shannon scaling function is defined by

~l(t) = {~ f°r t • [0'1) for t • [1, co) . The scale continuous harmonic scaling function {~p}, p • (0, c~), is given via 1 for n • [0, p - l ) 0forn•[p-1 ~)

~p(n)=

A scale continuous wavelet does not make sense. However, a scale discrete wavelet {~D}, j • Z, is available. More precisely, 1 for n • [2J,2 j+]) 0 elsewhere.

CD(n) =

But this means that the scale discrete multiresolution is orthogonal (i.e., ~D j + l ~--yD ® w D is an orthogonal sum for all j). Modified Shannon Wavelet. The generator of the modified Shannon scaling function reads as follows

1 ~l(t) = 0

for t • [0, }) for t • [}, 1) for t • [t, c~)

The scale continuous harmonic wavelets {~p}, p E (0, c~), are given by 0

¢~(~) =

fornE[

,;p

)

{OVr~ for n E "0 1 -F,p-~) [}p-~, for n e [p-*, ce).

232


The scale discrete harmonic wavelets {~D}, j E Z are obtainable via 0 for n e (1 + ln(2-Jn)) 1/2 forn E CD(n) = ~ (-- l n ( 2 - J - i n ) + ln(2-Jn)) t/2 for n e [ ~--ln(2-J-ln)) 1/2 for n E for n ~

[0, 2J ~) [2J~,2J+l~) [2j+l ~,2J)

[2J,2j+l) [25+1, co).

C(ubic) P(olynomial) Wavelet (CP-Wavelet). In order to have a higher intensity of the smoothing effect than in the case of modified Shannon wavelets we introduce a function ~1 : [0, co) ~ ]~ in such a way that ~1[[o,1] coincides with the uniquely determined cubic polynomial p : [0, 1] ~ [0, 1] with the properties: ; ( o ) = 1 , p(1) = o ; ' ( 0 ) = 0 , p'(1) = o .

It is easy to see that these properties are fulfilled by p(t)=(1-t)2(l+2t),

re[0,1]

.

This leads us to a function ~1 : [0, co] ~ ~ given by ~1(t)

f ( 1 - t ) 2 ( 1 + 2t) fort E [0,1) for t E [1, co). lo

It is clear that ~1 is a monotonically decreasing function. The (scale continuous) scaling function {~p}, p E (0, co), is given by

¢flp(n)

~l(pn)

]" (1 - pn)2(1 + 0

/

2pn) for n

e [0,p -1) for n E [p-l, co) .

Scale continuous and discrete wavelets are obtainable by obvious manipulations.

10

Fully S c a l e D i s c r e t e W a v e l e t s

For j = 0,1,... let bNj, i = 1 , . . . , N j , be the generating coefficients of the approximate integration rules (cf. [1]) Nj

2(YiN~" ) +sj(F), /F(y) ~w(y)= Z oi "N~-" A

F E'Nj,

i=1

}

corresponding lim ] s j ( F ) [ = 0

5-+00

forall

F E I ; 5,


233

(i.e. the coefficients bh~ , i = 1 , . . . , Nj, are supposed to be determined by an a priori calculation using approximate integration, interpolation, etc). Note that the coefficients bNj and the nodes Yigj , i = 1,... ,Nj, are independent of the choice of F E "~j. Assume that U is a potential of class Pot(°)(Aext) C 7/0. Then the J-level scale discrete wavelet approximation can be represented in fully discrete form as follows:

with

No

z

°)

()

i~1

J ~ \

]j;y~ J

j=0 i=1

This leads us to the following result. T h e o r e m 12. Any potential U E Pot(Aext) can be approximated in the form

lim

J-+oo

Iu(y)-Uy(y)l 2

=o.

Moreover, lira

sup IU(x)-UjD(x)[=O

J-+co xEZ~xt

for all (geodetically relevant) regular surfaces Z satisfying (5). Fast evaluation methods (i.e., fast summation techniques, tree algorithms, and pyramid schemata) for numerical computation have been presented in [6]. The details will be omitted here. In what follows we illustrate the multiresolution analysis of the bandlimited NASA, GSFC, and NIMA Earth Gravitational Model (EGM96) using exact outer harmonic approximate integration rules (cf. [6,8]). Further graphical illustrations are available for download at ftp://www.mathematik.nni-kl.de/pub/geodata . It includes software for Windows 95/98/NT. The data themselves are also available on internet by http: //www.mathematik.uni-kl.de/~wwwgeo/research/geodata . The numerical calculations of the representations of EGM96 presented in this work have been done by Dipl.-Math. Gunnar Schug, Geomathematics Group, University of Kaiserslautern.

234


References 1. Driscolt J.R., Healy D.M.: Computing Fourier Transforms and Conv01utions on the 2-Sphere. Adv. Appl. Math., 15, (1996), 202-250. 2. Freeden, W.: 0ber eine Klasse yon Integralformeln der Mathematischen Geod~sie. VerSff. Geod. Inst. RWTH Aachen, Heft 27, 1979. 3. Preeden, W.: On the Approximation of External Gravitational Potential With Closed Systems of (Trial) Functions., Bull. Geod., 54, (1980), 1-20. 4. Freeden, W.: Least Squares Approximation by Linear Combinations of (Multi-) Poles. Dept. Geod. Sci., 344, The Ohio State University, Columbus, 1983 5. Freeden, W.: A Spline Interpolation Method for Solving Boundary-value Problems of Potential Theory from Discretely Given Data. Numer. Meth. Part. Diff. Eqs. , 3, (1987), 375-398. 6. Freeden, W.: MultiscaIe Modelling of Spaceborne Geodata, Teubner-Verlag, Stuttgart Leipzig, 1999. 7. F~eeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere (with Applications to Geomathematics), Oxford Science Publications, Clarendon Press, Oxford, 1998. 8. Freeden, W , Glockner, O, Schreiner, M.: Spherical Panel Clustering and Its Numerical Aspects. J. of Geod., 72, (1998), 586-599. 9. Freeden, W., Kersten, H.: A Constructive Approximation Theorem for the Oblique Derivative Problem in Potential Theory. Math. Meth. in the Appl. Sci., 3, (1981), 104-114. 10. Freeden, W., Michel, V.: Constructive Approximation and Numerical Methods in Geodetic Research Today - An Attempt at a Categorization Based on an Uncertainty Principle., J. of Geod. (accepted for publication). 11. Freeden, W., Schneider, F.: An Integrated Wavelet Concept of Physical Geodesy. J. of Geod., 72, (1998), 259-281. 12. Freeden, W., Schneider, F.: Wavelet Approximation on Closed Surfaces and Their Application to Bounday-value Problems of Potential Theory. Math. Meth. Appl. Sci., 21, (t998), 129-165. 13. Freeden, W., Windheuser, U.: Spherical Wavelet Transform and its Discretization. Adv. Comp. Math., 5, (1996), 51-94. 14. Gabor, D.: Theory of Communications. J. Inst. Elec. Eng. (London), 93, (1946), 429-457. 15. Heil, C.E., Walnut~ D.F.: Continuous and Discrete Wavelet Transforms. SIAM Review, 31, (1989), 628-666. 16. Kaiser, G.: A Friendly Guide to Wavelets. Birkh~user-Verlag, Boston, 1994. 17. Kellogg, O.D.: Foundations of Potential Theory. Frederick Ungar Publishing Company, 1929. 18. Lemoine, F.G., Smith, D.E., Kunz, L., Smith, R., Pavlis, E.C., Pavlis, N.K., Klosko, S.M., Chinn, D.S., Torrence, M.H., Wilhamson, R.G., Cox, C.M., Rachlin, K.E., Wang, Y.M., Kenyon, S.C., Salman, R.., Trimmer, R., Rapp, R.H., Nerem, I~.S.: The Development of the NASA, GSFC, and NIMA Joint Geopotential Model. International Symposium on Gravity, Geoid, and Marine Geodesy, The University of Tokyo, Japan, Springer, IAG Symposia, 117, 1996, 461-469. 19. Michel, V.: A Multiscale Method for the Gravimetry Problem - Theoretical and Numerical Aspects of Harmonic and Anharmonic Modelling, PhD University of Kaiserslautern, Geomathematics Croup, Shaker-Verlag, Aachen, 1999.


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20. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman and Company, New York, 1970. 21. Miiller, C.: Spherical Harmonics, Lecture Notes in Mathematics, no. 17, SpringerVerlag, Heidelberg, 1966. 22. l~pp, R.H., Wang, M., Pavhs, N.: The Ohio State 1991 Geopotential and Sea Surface Topography Harmonic Coefficient Model. Dept. Geod. Sci., 410, The Ohio State University, Columbus, 1991. 23. Schwintzer, P., Reigber, C., Bode, A., Kang, Z., Zhu, S.Y., Massmann, F.H., Raimondo, J.C., Biancale, R., Ballnino, G, Lemoine, J.M., Moynot, B., Marty, J.C., Barber, F., Boudon, Y.: Long Wavelength Global Gravity Field Models: GRIM4$4, CRIM4-C4. J. of Geod., 71, (1997), 189-208.

236

Wilti F r e e d e n a n d Volker Michel

-100.0

0.0

100,0

200.0

-200.0

0.0

[100 Ovalm]

-200.0

0.0

200.0

[100 Gal m]

400.0

-100,0

[100 Gal m]

-500.0

0.0

200.0

0,0

100.0

[100 Gal m]

500,0

0.0

[100 Gal m] F i g . 2. R e p r e s e n t a t i o n of E G M 9 6 in CP-scale spaces a t h e i g h t 0 kin; scales 3 (top) t o 5 (bottom)

100,0

[100 Gal m]

(left)

a n d C P - d e t a i l spaces

(right)

LS G e o p o t e n t i a l A p p r o x i m a t i o n

-500.0

0.0

500.0

-50.0

0.0

50.0

[100 Gat m]

[100 Gal m]

-500.0

0.0

237

500.0

-50.0

0.0

50.0

[100 Gal m]

[100 Gal m]

i

-500.0

0.0

500.0

[100 Gal m] F i g . 3. R e p r e s e n t a t i o n of E G M 9 6 in C P - s c a l e spaces at h e i g h t 0 km; scales 6 (top) to 8 (bottom)

(left)


(right)

238

Willi F r e e d e n a n d Volker Michel

-100.0

0.0

100.0

-100.0

[100 Gat m]

-200,0

0.0

0,0

100.0

[100 Gal m]

-100.0

200.0

0.0

[100 Gat m]

-200.0

0.0

100.0

[lOOUal m]

200.0

400.0

-40.0

[100Galm] F i g . 4. R e p r e s e n t a t i o n of E G M 9 6 in C P - s c a l e spaces at h e i g h t 200 km; scales 3 (top) t o 5 (bottom)

-20.0

0,0

20.0

40.0

[100Galm]

(left) a n d

C P - d e t a i l spaces

(right)


~400,0

-200.0

0,0

200.0

400.0

-10.0

[i00 Gal m]

-400,0

-200.0

0.0

-200.0

0.0

10.0

[100 Gal m]

200.0

400.0

[100 Gal m]

-400.0

0.0

239

200.0

0.0 [100 Gal m]

400.0

[100 Gal m]

Fig. 5. Representation of EGM96 in CP-scale spaces at height 200 kin; scales 6 (top) to 8 (bottom)

(left) and CP-detail spaces (right)

240

Willi Freeden and Votker Michel

-100.0

0.0

100.0

-100.0

[100 Gal m]

-200.0

0.0

0.0

100.0

[100 Gal m]

200.0

-50.0

[100 Gal m]

-200.0

0.0

0,0

50.0

[100 C~alm]

200.0

-20.0

[100 Gal ml

Fig. 6. Representation of EGM96 in CP-scale spaces at height 400 kin; scales 3 (top) to 5 (bottom)

0.0

20.0

[100 Gal m]

(left)

and CP-detail spaces

(right)

LS G e o p o t e n t i a l A p p r o x i m a t i o n

-200.0

0.0

200.0

-5.0

[100 Gal m]

-200.0

0.0

0.0

5.0

10.0

[100 Gal nil

200.0

-2.0

0.0

2.0

[100 Gal m]

[100 Gal m]

-200.0

0.0

241

200.0

[100 Gal m] F i g . 7. R e p r e s e n t a t i o n of E G M 9 6 in C P - s c a l e spaces at height 400 kin; scales 6 (top) to 8 (bottom)

(left)


(right)

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