Two-dimensional Signal Analysis

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Author: René Garello

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Two-Dimensional Signal Analysis



Edited by René Garello

First published in France in 2001 by Hermes Science Publications entitled “Analyse de signaux bidimensionnels” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 6 Fitzroy Square London W1T 5DX UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd, 2008 © HERMES Science Europe, 2001 The rights of René Garello to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Analyse de signaux bidimensionnels. English. Two-dimensional signal analysis / edited by René Garello. p. cm. Includes index. ISBN 978-1-84821-018-9 1. Signal theory (Telecommunication) 2. System analysis. I. Garello, René. II. Title. TK5102.5.A51813 2007 621.382'23--dc22 2007022699 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-018-9 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.

Table of Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Chapter 1. Basic Elements of 2-D Signal Processing. . . . . . . . . . . . . . . . Claude CARIOU, Olivier ALATA and Jean-Marc LE CAILLEC

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1.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . 1.2. Deterministic 2-D signals . . . . . . . . . . . . . . 1.2.1. Definition. . . . . . . . . . . . . . . . . . . . . 1.2.2. Particular 2-D signals . . . . . . . . . . . . . 1.3. Random 2-D signals. . . . . . . . . . . . . . . . . . 1.3.1. Definition. . . . . . . . . . . . . . . . . . . . . 1.3.2. Characterization up to the second order . . . 1.3.3. Stationarity . . . . . . . . . . . . . . . . . . . . 1.3.4. Characterization of orders higher than two . 1.3.5. Ergodicity . . . . . . . . . . . . . . . . . . . . 1.3.6. Specificities of random 2-D signals . . . . . 1.3.7. Particular random signals . . . . . . . . . . . 1.3.7.1. White noise . . . . . . . . . . . . . . . . 1.3.7.2. Gaussian process . . . . . . . . . . . . . 1.4. 2-D systems. . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Definition. . . . . . . . . . . . . . . . . . . . . 1.4.2. Main 2-D operators . . . . . . . . . . . . . . . 1.4.3. Main properties . . . . . . . . . . . . . . . . . 1.4.4. Linear time-invariant (LTI) system . . . . . 1.4.5. Example . . . . . . . . . . . . . . . . . . . . . 1.4.6. Separable system . . . . . . . . . . . . . . . . 1.4.7. Stability of 2-D systems . . . . . . . . . . . . 1.4.8. Support of the impulse response – causality 1.5. Characterization of 2-D signals and systems . . . 1.5.1. Frequency response of an LTI system . . . .

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1.5.2. 2-D Fourier transform . . . . . . . . . . . . . . . . . . . . . 1.5.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3. Discrete 2-D Fourier transform . . . . . . . . . . . . . . . . 1.5.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3.3. Calculation of the 2-D DFT. . . . . . . . . . . . . . . 1.5.4. 2-D z transform . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4.2. Region of convergence . . . . . . . . . . . . . . . . . 1.5.4.3. Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4.4. Transfer function of a 2-D system . . . . . . . . . . . 1.5.4.5. 2-D inverse ZT . . . . . . . . . . . . . . . . . . . . . . 1.5.4.6. Application to the study of stability of LTI systems 1.5.4.7. Minimum or non-minimum phase LTI system . . . 1.5.5. Frequency characterization of a random 2-D signal. . . . 1.5.6. Output of a 2-D system with random input . . . . . . . . . 1.6. 2-D Wold decomposition. . . . . . . . . . . . . . . . . . . . . . . 1.6.1. Innovation, determinism and regularity in the 2-D case . 1.6.2. Total decomposition of three fields . . . . . . . . . . . . . 1.6.3. Example of an outcome . . . . . . . . . . . . . . . . . . . . 1.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. 2-D Linear Stochastic Modeling . . . . . . . . . . . . . . . . . . . . . Olivier ALATA and Claude CARIOU

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2.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. 2-D ARMA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. 2-D ARMA models and prediction supports . . . . . . . . . . . . . . 2.2.2.1. Causal models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.2. Causal quarter plane model . . . . . . . . . . . . . . . . . . . . . 2.2.2.3. Causal model whose support is delimited by any two NSHPs 2.2.2.4. Semi-causal model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.5. Non-causal model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. L-Markovian fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. 2-D Markov fields and L-Markovian fields. . . . . . . . . . . . . . . 2.3.2. 2-D L-Markovian fields and Gibbs fields . . . . . . . . . . . . . . . . 2.4. “Global” estimation methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1.1. Estimation criteria by supposing the fixed order . . . . . . . .

65 66 66 67 67 70 71 72 72 73 73 74 76 76 76

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2.4.1.2. Probability criteria “penalized” to estimate the order of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Yule-Walker equations. . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2.1. Representation of minimum variance and formulation. . . . 2.4.2.2. Non-causal support and L-Markovian fields . . . . . . . . . . 2.4.2.3. Causal support and 2-D AR model . . . . . . . . . . . . . . . 2.4.2.4. Extension to the 2-D AR non-causal model . . . . . . . . . . 2.4.2.5. Extension to the 2-D ARMA model. . . . . . . . . . . . . . . 2.4.3. 2-D Levinson algorithm (for the parametric 2-D AR estimation) . 2.4.3.1. Recalling the 1-D case. . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2. Approach for 2-D causal and non-causal prediction models 2.4.3.3. Multichannel approach and 2-D QP AR model . . . . . . . . 2.4.3.4. Other approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. “Adaptive” or “recursive” estimation methods . . . . . . . . . . . . . . . 2.5.1. Connectivity hypotheses for adaptive or recursive algorithms . . 2.5.2. Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2.1. Methods based on a stochastic gradient. . . . . . . . . . . . . 2.5.2.2. Methods based on the recursive least squares criterion . . . 2.5.2.3. Methods based on the geometric approach of the RLS criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Application: segmentation of textured images . . . . . . . . . . . . . . . 2.6.1. Textured field and segmented field . . . . . . . . . . . . . . . . . . 2.6.2. Multiscale or hierarchical approach . . . . . . . . . . . . . . . . . . 2.6.3. Non-supervised estimation of the parameters . . . . . . . . . . . . 2.6.3.1. Multinominal distribution . . . . . . . . . . . . . . . . . . . . . 2.6.3.2. Information criterion. . . . . . . . . . . . . . . . . . . . . . . . 2.6.4. Examples of segmentation. . . . . . . . . . . . . . . . . . . . . . . . 2.6.4.1. Synthesis textures . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4.2. Brodatz textures . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4.3. SAR image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3. 2-D Spectral Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Claude CARIOU, Stéphanie ROUQUETTE and Olivier ALATA 3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 3.2. General concepts. . . . . . . . . . . . . . . . . . . . . . 3.3. Traditional 2-D spectral estimation . . . . . . . . . . . 3.3.1. Periodogram technique. . . . . . . . . . . . . . . 3.3.2. Correlogram technique . . . . . . . . . . . . . . . 3.3.3. Limits of traditional spectral analysis . . . . . . 3.4. Parametric 2-D spectral estimation . . . . . . . . . . . 3.4.1. Spectral estimation by linear stochastic models

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3.4.1.1. AR model . . . . . . . . . . . . . . . . . . . . . 3.4.1.2. ARMA model . . . . . . . . . . . . . . . . . . . 3.4.1.3. Gauss-Markov model . . . . . . . . . . . . . . 3.4.2. Maximum entropy method . . . . . . . . . . . . . . 3.4.2.1. ME principle . . . . . . . . . . . . . . . . . . . 3.4.2.2. Implementation . . . . . . . . . . . . . . . . . . 3.4.2.3. Example: alternate projection methods . . . . 3.4.2.4. Performances . . . . . . . . . . . . . . . . . . . 3.4.3. Minimum variance method . . . . . . . . . . . . . . 3.5. 2-D high resolution methods. . . . . . . . . . . . . . . . . 3.5.1. 2-D MUSIC . . . . . . . . . . . . . . . . . . . . . . . 3.5.2. Calculation of a pseudo-spectrum . . . . . . . . . . 3.5.3. Pseudo-spectrum estimation. . . . . . . . . . . . . . 3.6. Other techniques . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Comparative study of some techniques . . . . . . . . . . 3.7.1. Analysis of 2-D harmonic components . . . . . . . 3.7.1.1. Synthesis of 2-D sinusoidal signals . . . . . . 3.7.1.2. General behavior of PSD estimates . . . . . . 3.7.1.3. Statistics of frequency estimation . . . . . . . 3.7.1.4. Separability of two frequency components. . 3.7.1.5. Estimation of the power of a sinusoid. . . . . 3.7.1.6. Summary. . . . . . . . . . . . . . . . . . . . . . 3.7.2. Analysis of random fields . . . . . . . . . . . . . . . 3.7.2.1. Synthesis of parametric random fields . . . . 3.7.2.2. Spectral estimation of a GM field . . . . . . . 3.7.2.3. Summary. . . . . . . . . . . . . . . . . . . . . . 3.7.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Application: spectral analysis of remote sensing images 3.8.1. Position of the problem . . . . . . . . . . . . . . . . 3.8.2. Stochastic modeling of a radar image . . . . . . . . 3.8.3. Example of application. . . . . . . . . . . . . . . . . 3.9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

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122 126 127 128 129 129 130 132 132 134 135 135 137 138 138 139 140 141 150 153 157 157 159 160 160 163 163 165 165 166 167 169 170

Chapter 4. Bispectral Analysis of 2-D Signals. . . . . . . . . . . . . . . . . . . . 175 Jean-Marc LE CAILLEC and René GARELLO 4.1. Introduction. . . . . . . . . . . . . . . . . . . . 4.1.1. Higher order moments and cumulants . 4.1.2. Properties of moments and cumulants . 4.1.3. Polyspectra of stationary signals . . . . 4.1.4. Polyspectra . . . . . . . . . . . . . . . . . 4.1.5. Definition of the coherence of order p.

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4.2. Moments and spectra of order p for linear signals . . . . . . . . . . . . 4.2.1. Moments and cumulants of order p for linear signals . . . . . . . 4.2.2. Spectrum of order p for a linear signal . . . . . . . . . . . . . . . 4.2.3. General properties of the bispectra of linear signals. . . . . . . . 4.2.4. Polyspectrum of a linear signal . . . . . . . . . . . . . . . . . . . . 4.2.5. Coherence of order p for linear signals . . . . . . . . . . . . . . . 4.3. Signals in quadratic phase coupling, non-linearity and the Volterra system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Bispectrum of a signal in quadratic phase coupling . . . . . . . . 4.3.2. Volterra models and decomposition of non-linear systems . . . 4.3.2.1. General principles . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.2. Bispectrum of a Volterra model with Gaussian input data. 4.4. Bispectral estimators for 2-D signals . . . . . . . . . . . . . . . . . . . . 4.4.1. Indirect method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Autoregressive model . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4. ARMA modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5. Measure of bias and variance of estimators. . . . . . . . . . . . . 4.5. Hypothesis test for non-linearity and bicoherence tables . . . . . . . . 4.5.1. Hypothesis tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. Bicoherence tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Image restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. Artifact removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. Time-frequency Representation of 2-D Signals . . . . . . . . . . . 215 Stéphane GRASSIN and René GARELLO 5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Bilinear time-frequency representation . . . . . . . . . . . . . . . . 5.1.2. Four spaces of representation . . . . . . . . . . . . . . . . . . . . . . 5.1.3. Restriction to bilinear representation . . . . . . . . . . . . . . . . . 5.1.4. Spectral description using bilinear representations . . . . . . . . . 5.1.4.1. Bilinear representations in time-delay space. . . . . . . . . . 5.1.4.2. Bilinear representations in equivalent time-frequency space 5.2. TFR application to sampled images . . . . . . . . . . . . . . . . . . . . . 5.2.1. TFR expression of discrete images. . . . . . . . . . . . . . . . . . . 5.2.1.1. Autocorrelation function of a discrete image . . . . . . . . . 5.2.1.2. Time-frequency representation of a discrete image. . . . . . 5.2.2. Support of the sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Minimum properties and constraints on the kernel. . . . . . . . . . . . . 5.3.1. Compatibility with reversible linear transformations . . . . . . . .

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215 215 216 217 218 218 219 219 219 220 222 223 223 224

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5.3.2. Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. TFR with real values . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5. Spectral estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5.1. Case of second order stationary images. . . . . . . . . . . . 5.3.5.2. Case of signals with bilinear frequency modulation . . . . 5.3.6. Evolution of properties of a modified kernel . . . . . . . . . . . . 5.3.6.1. Pseudo-smoothed version . . . . . . . . . . . . . . . . . . . . 5.3.6.2. Simply smoothed version . . . . . . . . . . . . . . . . . . . . 5.3.6.3. Simply masked version . . . . . . . . . . . . . . . . . . . . . 5.4. Notion of analytic images . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Formulation of the problem for the images . . . . . . . . . . . . . 5.4.2. Traditional solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3. Symmetric solution with reference to a hyperplane . . . . . . . . 5.4.4. Solution with a non-symmetric half-plane . . . . . . . . . . . . . 5.4.5. Choice of spectral division . . . . . . . . . . . . . . . . . . . . . . 5.5. Spectral analysis application of SAR images . . . . . . . . . . . . . . . 5.5.1. Analysis of an internal waveform . . . . . . . . . . . . . . . . . . 5.5.1.1. Characteristics of the image and preprocessing . . . . . . . 5.5.1.2. Analysis method . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1.3. Presentation of the results and interpretation . . . . . . . . 5.5.2. Analysis of an internal wave field with superimposition . . . . . 5.5.3. Analysis of a small area internal wave field . . . . . . . . . . . . 5.5.4. Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Approximation of an internal wave train . . . . . . . . . . . . . . . . . 5.6.1. Benefit of approximation of the frequency law . . . . . . . . . . 5.6.2. Problem resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3. Adequacy of bilinear modulation with instantaneous frequency estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3.1. Overview of the benefit of this adequacy. . . . . . . . . . . 5.6.3.2. Verification of the adequacy hypothesis . . . . . . . . . . . 5.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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225 225 225 226 227 227 228 228 229 229 230 230 231 233 233 237 241 243 243 244 246 249 249 250 252 252 252

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255 255 256 257

Chapter 6. 2-D Wavelet Representation . . . . . . . . . . . . . . . . . . . . . . . 259 Philippe CARRÉ, Noël RICHARD and Christine FERNANDEZ 6.1. Introduction. . . . . . . . . . . . . . . . . . . . 6.2. Dyadic wavelet transform: from 1-D to 2-D 6.2.1. Multiresolution analysis . . . . . . . . . 6.2.2. Wavelets and filter banks . . . . . . . . 6.2.3. Wavelet packets . . . . . . . . . . . . . . 6.2.4. 2-D extension by the simple product. . 6.2.5. Non-separable 2-D wavelets . . . . . .

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259 260 260 262 264 266 272

Table of Contents

6.2.6. Non-decimated decomposition . . . . . . . . . . . . . . . . . . . 6.3. Trigonometric transform to adaptive windows . . . . . . . . . . . . . 6.3.1. Malvar wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.1. Window function . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.2. Local trigonometric bases. . . . . . . . . . . . . . . . . . . 6.3.2. Folding operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3. Windowed orthonormal base . . . . . . . . . . . . . . . . . . . . 6.3.4. Extension of Malvar wavelets to 2-D . . . . . . . . . . . . . . . 6.4. Transform by frequency slicing . . . . . . . . . . . . . . . . . . . . . . 6.4.1. Continuous theory of 1-D Meyer wavelets . . . . . . . . . . . . 6.4.2. Definition of Meyer wavelet packets . . . . . . . . . . . . . . . 6.4.3. Numerical outcome of decomposition in 1-D Meyer wavelet packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3.1. Restriction on positive frequencies . . . . . . . . . . . . . 6.4.3.2. Initialization: calculation of the first scale . . . . . . . . . 6.4.3.3. Definition of window wI s1 for i ! 0 . . . . . . . . . . . . . 2i

11

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278 282 282 282 283 284 287 288 292 293 295

. . . .

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295 296 297 297

6.4.3.4. Definition of window wI s1 for i ! 0 . . . . . . . . . . . . . . . . 299 2 i 1

6.4.3.5. Calculation of the coefficients of Meyer wavelet packets for i > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 6.4.3.6. Calculation of wavelet packet coefficients related to \ˆ 0s 1 . . 302 6.4.3.7. Algorithm and 1-D reconstruction . . . 6.4.4. Extension of Meyer wavelet packets to 2-D 6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 6.6. Bibliography . . . . . . . . . . . . . . . . . . . . . .

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304 306 308 309

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315


Introduction

The scientist studies nature, not because it is useful; but because he takes pleasure in doing so and he derives pleasure because nature is beautiful. H. Poincaré, Science et Methode, 1905

For a long time, the signal processing (SP) community has limited itself to the field of 1-D signals, very often considering them to be an electronic quantity evolving over time. This approach was fundamentally associated with the type of acquisition which very often converted it in such a way as to display it on the screen of an oscilloscope and processed it with the help of electronic equipment. This occurred whatever the observation and its physical nature. This “engineering science” is strongly linked to a mathematical description and the tools that it uses in order to carry the information from a sender to a receiver. It was then necessary to allow the latter to extract the information with minimum errors or ambiguity. Finally, the field was enlarged to the understanding and modeling of our environment by adopting developed methods in a specific manner. All the initial processing of this type was directed to radar or sonar signals and the application was more military than environmental. It is necessary to note that starting from an electronic – and often mathematical – basis which underlines it, the discipline is strongly associated with computer science for the digital processing of a signal. Nevertheless, the methods continue to be based on the development of mathematical tools and the interpretation of results obtained is found principally in the frequency field. One of the principal components of SP continues to be spectral analysis (the subject of [CAS 06]).

Introduction written by René GARELLO.

14


Image processing (IP) has developed over the past two decades and is found, by its very nature, to be strongly linked to signal processing. Indeed, the French (ISIS) and international (IEEE) associations have included it in their signal processing communities. IP methods are essentially developed in the spatial domain and often propose a computer science approach (in the general sense of this discipline). Indeed, it is not rare to see the shape recognition, fuzzy logic, as well as artificial intelligence present in the field of IP in the same way as probabilistic approaches. These topics can be studied in more depth in [COC 01]. From its origin the spatial approach of processing has been guided by the object represented by the image, which in most cases happens to be a digital version of a photograph (or a series of photographs with regard to video). This image (in the visible as well as the infrared domain) is interpreted by the eye and the brain. What happens then to processing and analysis of 2-D signals which are not images but are no longer 1-D? We will see, as the chapters in this book unfold, that the processing appropriate to the extraction of information favors the spectral approach, but that the 2-D aspect is strongly present. Causality, for example, is not a very intuitive notion in this case. As a result, the position of 2-D processing and analysis is difficult compared to the processing of the 1-D signal (which is often very theoretical and does not allow us to tackle the problem of multi-dimensionality) and of image processing (2-D, but almost exclusively spatial). The study of the m-D case (particularly, 2-D) has generated valuable works, which are often identified in scientific publications in relation to signal processing. In spite of all this, the extension of 1-D techniques to 2-D is not swift, except in simple cases implying a separateness of the signals and the systems. The Fourier transform, which is a good example, illustrates a frequency approach of 2-D processing (and image processing) but quickly shows its limits concerning non-stationarities inherent in the majority of processed data. As a result, the multi-dimensional polynomials in z are not factorable in general and this does not allow an immediate extension of the developed methods in the 1-D cases. “Images” which are most often processed in the framework of 2-D analysis thus arise from the active sensors (sonar, radar, etc.) and are not necessarily directly interpreted by people. A very important part in the extraction of information – for example, in radar imaging – is linked to the traditional domain of image processing (detection of contours, segmentation, classification). Moreover, another work is dedicated to this [MAI 08]. However, the nature of the acquisition and the particularity of certain applications (or imaged scenes) make it one of the preferred supports for the implementation of analysis methods specifically developed within the 2-D framework. Indeed, in a majority of cases, the 2-D signal obtained is strongly non-stationary and, for certain applications, non-linear and non-Gaussian. At first, an articulation of this work concerning methods called “high resolution spectral analysis” had been envisaged (specifically Chapters 3 and 5). However,

Introduction

15

very quickly, it appeared that the characterization of 2-D signals also uses a modeling phase which is based as much on statistics as on the spectral behavior of the entire information. Would it be necessary to construct the analysis on a modeling a priori of the 2-D signal or to use the data as support for a processing model? The answer to this question leads to a division of the work into two parts, after this introduction: – a modeling approach (Chapters 2 to 4) which presents the 2-D extensions of the stochastic spectral analysis methods; – an approach for which the described methods relate more to the 2-D structure and are thus less sensitive to their random character (Chapters 5 and 6). In the first approach the model is stochastic, which allows us to reduce the uncertainty created by the finite and random character of the data. The book is representative of a traditional SP approach, which stresses a statistical approach of the concerned phenomenon. The methods appropriate to the processing of random 2-D signals are thus covered (Chapters 1 and 2), and non-stationary (Chapter 3) and non-linear fields (Chapter 4) are also tackled. The strong connection between these chapters based on the random character of 2-D signals is thus favored. In the second approach, data is a simple support of information. The uncertainty character present in the traditional processing methods (the Fourier transform, for example) is compensated for here, in some way, by the complementary dimensions (frequency, scale) introduced in Chapters 5 and 6. The objective of this work is to show that, at the conjunction of the fields of signal processing and image processing, the processing and analysis of 2-D signals have their own specificity. Thus, the objective consists of presenting all the methods of 2-D signal processing and their complementarities. The stochastic aspect and the structural aspect are not yet intimately linked, but we believe this is one of the keys to future developments in the field of signal and image processing and its applications. Bibliography [CAS 06] C ASTANI E´ F. (ed.), Spectral Analysis, ISTE Ltd, London, 2006. [COC 01] C OCQUEREZ J. (ed.), Traitement de l’image ; les bases, Hermes, IC2 Series, 2001. [MAI 08] M AITRE H. (ed.), Processing of Synthetic Aperture Radar Images, ISTE Ltd, London, forthcoming.


Chapter 1

Basic Elements of 2-D Signal Processing

1.1. Introduction Processing and analysis of 2-D signals raise many theoretical and practical questions that only apply to them and that we do not find in 1-D. Indeed, although certain notions or techniques like the Fourier transform find a natural extension to 2-D signals, this is not the same for the causality notion or rational 2-D transfer pole function. By way of introduction, we quote from the work of Dudgeon and Mersereau [DUD 84], which clearly states the subject and specificity of analysis and processing in 2-D signals: “is the processing of multi-dimensional signals different from that of the mono-dimensional signals? At a certain level, the answer is no. Several operations that we could carry out on multi-dimensional sequences are also carried out on monodimensional sequences – for example, sampling, filtering, calculation of transforms. However, at a more detailed level, we are obliged to note that the processing of the m-D signal can be very different. This is due to three factors: firstly, the 2-D problems generally bring into play more data than in 1-D; secondly the mathematics supporting the analysis of m-D systems are less complete than those concerning 1-D; thirdly, the m-D systems have many more degrees of freedom, which in turn gives the authors of the system a flexibility which we do not find in the case of 1-D”.

Chapter written by Claude CARIOU, Olivier ALATA and Jean-Marc LE CAILLEC.

18


This flexibility can be successfully exploited, but this largely depends on the nature of the problem. For example, the processing of radar signals, for which one of the dimensions of the signal corresponds to time, will take advantage of the large quantity of data in this dimension to use hybrid spectral analysis techniques. Another example is that of spatial dimensionality of acquisition systems and processing of signals: the implicit non-causality of the phenomena to observe in 2-D offers a wide geometric choice (rectangular, hexagonal, polar, etc.). A 2-D signal is a function of two variables which can be continuous, discrete or mixed. A photograph is one example of a continuous function of two variables, while a digital image is its discrete version. Another instance of a mixed 2-D signal is that of a network of electric sensors recording data in time: the time dimension is essentially continuous, while the number of sensors plays the role of discrete variables. In this chapter, we will focus on discrete 2-D signals. This restriction is obviously a consequence of the importance and current omnipresence of acquisition systems and digital processing of signals. Moreover, in order to overcome technological constraints, we study aspects linked to quantification and production of digital signals. When necessary, the emphasis will be on the specificities of signals and 2-D systems as compared to 1-D. We will first tackle the case of deterministic signals and their representation. After this, we will tackle random 2-D signals. For this, we will refer back to stationarity and ergodicity hypothesis, which are already well known in 1-D and which will allow us to extend the most traditional statistical characteristics to 2-D (statistics of the order 2 and of orders higher than 2). The third part will be devoted to systems, for which major questions pertaining to 2-D – and more generally to the m-D case – are raised (separateness, stability, causality). We will finally conclude with frequency characterization of these signals and systems by examining the 2-D Fourier and z transforms, as well as by introducing the general model of random stationary field of the second order, using the Wold decomposition.

1.2. Deterministic 2-D signals 1.2.1. Definition A discrete deterministic 2-D signal x is a function defined on set Z 2 (also called a 2-D network): x = {x(k1 , k2 )}(k1 ,k2 )∈Z 2

(1.1)


19

x(k1,k2)

1.5 1 0.5 0 6 5 4 3 2 k2

1 0

0

1

3

2

4

5

6

k1

Figure 1.1. Example of a 2-D signal

This discrete signal could be made up of complex or real values, depending on the case. We will note that this definition does not apply to practical cases, where the duration of signals is finite. In this case, instead of restricting the support of x to a compact sub-set S (generally rectangular) of Z 2 , we can consider that the 2-D signal is zero outside S. It then becomes imperative to point out that discrete 2-D signals, whatever they may be, are in practice generally acquired and sometimes manipulated in a matrix form with real or complex coefficients, i.e.: X = {xk1 ,k2 ; 0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1}

(1.2)

where K1 and K2 are size parameters of matrix X for each of the two dimensions and thus define a particular compact sub-set S. With an aim to generalize except in particular cases, we will avoid making reference to this practical aspect. An example of a 2-D signal representation with real values on a 2-D grid is shown in Figure 1.1.

1.2.2. Particular 2-D signals In this section, we describe some 2-D signals which are traditionally used.

20

Two-Dimensional Signal Analysis 1

0

0

0

k2 k1

Figure 1.2. 2-D unit impulse

The “unit impulse” or Kronecker sequence is an important signal in processing discrete signals and is defined in the 2-D case by: δ (k1 , k2 ) =

1 for k1 = k2 = 0 0 otherwise

(1.3)

We note that the 2-D unit impulse corresponds to the product of two 1-D Kronecker sequences: δ (k1 , k2 ) = δ (k1 ) δ (k2 )

(1.4)

A representation of the 2-D unit impulse is shown in Figure 1.2. Another signal which is used a lot in 1-D signal processing is the unit step. Its extension to 2-D is defined by: U (k1 , k2 ) =

1 for k1 ≥ 0 and k2 ≥ 0 0 otherwise

(1.5)

In this case too, we immediately see that the 2-D signal is the product of the two 1-D unit steps. However, we note that the 2-D unit step shown in Figure 1.3 is a nonzero function on only one quadrant of the Z 2 network.


21

1

0

0

0

k2 k1

Figure 1.3. 2-D unit step

The two preceding signals are particular instances of signals known as separable signals. This class of 2-D signal is defined by the following relation: x (k1 , k2 ) = x1 (k1 ) x2 (k2 )

(1.6)

We can show [DUD 84] that any 2-D signal of finite duration can be written in the form of a finite sum of separable signals. The separable signals, although rarely encountered in practice, can, however, be used during the development of 2-D systems (see section 1.3.6). Periodic sequences also constitute an important class of 2-D signals. However, generalization of the 1-D case is not immediate, as periodicity of a signal has to be verified in the two dimensions of the representation. A 2-D signal x (k1 , k2 ) is said to be periodic if: x (k1 + K11 , k2 + K21 ) = x (k1 , k2 ) x (k1 + K12 , k2 + K22 ) = x (k1 , k2 ) with: K11 K22 − K12 K21 = 0

(1.7)

22


1

0

k2

0

0

k1

Figure 1.4. Example of periodic 2-D signal, of periods (K11 = 4, K21 = 0) and (K12 = 0, K22 = 3)

Couples (K11 , K21 ) and (K12 , K22 ) constitute vectors (lines) of displacement of a coordinate in the plane (k1 , k2 ) in two periods. A particular case, often encountered in practice, is that of sequences for which K12 = K21 = 0. In this case, periodicity is defined by two displacement vectors, one horizontal and the other vertical, as shown in Figure 1.4. We then talk of rectangular periodicity and have: x (k1 + K11 , k2 ) = x (k1 , k2 ) (1.8) x (k1 , k2 + K22 ) = x (k1 , k2 ) The 2-D signal of infinite duration is thus only defined by data of K11 K22 independent samples, which can, but not necessarily, be chosen in rectangular support: {(k1 , k2 ) ; 0 ≤ k1 ≤ K11 − 1, 0 ≤ k2 ≤ K22 − 1} 1.3. Random 2-D signals 1.3.1. Definition In the first chapters of this work, we will consider a 2-D signal as having an outcome x = {x(k1 , k2 )}(k1 ,k2 )∈S⊂Z 2 , with S finite sub-set of Z 2 , of a random infinite sequence:


23

X = {X(k1 , k2 )}(k1 ,k2 )∈Z 2 called a stochastic 2-D process or also random field on the network, with a value in , or C in a more general case [GUY 95]. Traditionally, we write: X (k1 , k2 ) ∈ L2 (Ω, A, P )

X(k1 , k2 , ω) = x (k1 , k2 ) ∈ (or C)

(1.9)

with L2 (Ω, A, P ) the Hilbert space of square integrable random variables, which is a definition that allows us to give meaning to the notion of expected value E[.] of the product of two random variables in terms of scalar product [PAP 91]: X, Y = E[X Y ∗ ],

(1.10)

and to define the notions of distance and orthogonality of variables or of random vectors. We recall the expression of the expected value of a random variable X: E[X] = x fX (x) dx

(1.11)

where fX (x) denotes the probability density function of random variable X.

1.3.2. Characterization up to the second order A certain number of characterizations of random fields are required in order to be able to later define other concepts and manipulable entities. The study of a 2-D process, up to the second order, enables us to define four fundamental quantities: – the average of the process (or statistical moment of the first order): μX (k1 , k2 ) = E [X (k1 , k2 )]

(1.12)

– the variance:

2 2 σX (k1 , k2 ) = E |X (k1 , k2 ) − μX (k1 , k2 )|

(1.13)

– the second autocovariance function: RX (k1 , k2; l1 , l2 ) = E [X (k1 , k2 ) X ∗ (l1 , l2 )]

(1.14)

24


– the first autocovariance function: CX (k1 , k2; l1 , l2 ) = E [Xc (k1 , k2 ) Xc∗ (l1 , l2 )]

(1.15)

where: Xc (k1 , k2 ) = X (k1 , k2 ) − μX (k1 , k2 )

(1.16)

represents the centered process corresponding to X(k1 , k2 ). Relations (1.13) to (1.15) define the statistical moments of the second order.

1.3.3. Stationarity A random field is called strict sense stationary if all its statistical properties are invariable according to a translation of the spatial coordinates and it is termed stationary up to order K if all its statistical properties up to order K are invariant by translation. In practice, however, we often consider another type of stationarity, which is closely linked to physical and particularly energetic considerations: we say that a random field is wide sense stationary (WSS) if its average is a constant: μX (k1 , k2 ) = μ

(1.17)

and if its second autocovariance function is invariant for any change of origin: RX (k1 , k2 ; l1 , l2 ) = RX (k1 + m1 , k2 + m2 ; l1 + m1 , l2 + m2 ) = rX (k1 − l1 , k2 − l2 )

(1.18)

and that ∀ (m1 , m2 ) ∈ Z 2 . Note that wide sense stationarity is equivalent to stationarity up to the second order. Function rX (., .) is then called the second autocorrelation function of the 2-D process. This function is dependent only on a translation vector from one node to another of the Z 2 network or, to put it another way, from the distance and from the relative orientation of these nodes. We also define, under the WSS hypothesis, the first autocorrelation function: γX (m1 , m2 ) = E [Xc (k1 , k2 ) Xc∗ (k1 − m1 , k2 − m2 )] The following are the main properties of autocorrelation functions.

(1.19)


25

4

x 10 15 10 5 0 5 5

0

m2

5

0

5

5

m1

Figure 1.5. Example of first autocorrelation function γX

The autocorrelation functions are of Hermitian symmetry. Referring, for example, to the second autocorrelation function, we have: ∗ rX (−m1 , −m2 ) = rX (m1 , m2 )

(1.20)

Thanks to this property, the autocorrelation functions need to be estimated only on a half-plane and, as in the case of 1-D, rX (0, 0) ∈ + . In the specific case of a real random field, autocorrelation functions are real and even, i.e. symmetric as compared to node (0, 0). An example of the first autocorrelation function γX of a real WSS process is shown in Figure 1.5. The autocorrelation functions are semi-defined. Thus, they verify inequality: ∀ a (k1 , k2 ) ∈ C, k1

k2

l1

a (k1 , k2 ) γX (k1 − l1 , k2 − l2 ) a∗ (l1 , l2 ) ≥ 0

(1.21)

l2

As we will see later, this property guarantees non-negativity of the discrete Fourier transform, but also reversibility of autocorrelation matrices used in spectral analysis (see Chapter 3).

26


Let us note that, for the particular case of a random Gaussian field (see section 1.3.7.2), whose probability density function only expresses itself according to the average and to the second autocorrelation function, WSS implies stationarity in a strict sense. However, in many situations WSS of the 2-D process is not verified, especially in image processing. Thus, it can be useful, following the example of 1-D processing (in speech processing, for example) to formulate a short-term or piecewise stationarity.

1.3.4. Characterization of orders higher than two More precise definitions concerning 2-D signals and their characterization of higher orders will be detailed in Chapter 4. For a random real 2-D process stationary up to order P , we generalize the autocorrelation notion by defining the first 2-D poly-correlation (or moment) and the second poly-correlation (or cumulant) (see section 4.1.1). A particular case of poly-correlation is that corresponding to P = 3. The associated function, called the 2-D bicorrelation, consists of four indices, which makes both its estimation and manipulation very difficult, especially since it possesses relatively complex symmetry properties.

1.3.5. Ergodicity Similarly to 1-D signal processing, there is a problem with estimation of statistical characteristics of a 2-D process. Let us simply recall that ergodicity has to be, on the one hand, re-attached to a given order of characterization of a process and, on the other hand (even if it is less often referred to here), to the type of convergence of the estimator envisaged for the corresponding moment. The question rests in the ability to correctly estimate, in the given convergence criteria, a moment which is defined from an expected value of a set of outcomes of a random process only with the help of one outcome (more often a finite horizon) of this process. For example, if the outcome of a random 2-D field allows us, with the help of a spatial average calculation (temporal in the case of 1-D), to estimate in good measure the average μ of the 2-D process, then this process is said to be mean ergodic. In the same way, at order 2, we talk of the autocorrelation ergodic process. In general, convergence is envisaged in a quadratic average. In practice, to prove the ergodicity of a process of an order higher than 1 is difficult, and we often can only make this assumption. In the following, we make the hypothesis of autocorrelation ergodicity and from this fact average and autocorrelation functions can be approached by spatial averages from a finite outcome


27

{x = x(k1 , k2 )}, of size K1 K2 , of process X. These estimated spatial averages are written as:

μ ˆX =

K 1 −1 K 2 −1 1 x(k1 , k2 ) K1 K2

(1.22)

k1 =0 k2 =0

rˆX (m1 , m2 ) = ⎧ K 1 −1 K 2 −1 ⎪ 1 ⎪ ⎪ x (k1 , k2 ) x∗ (k1 − m1 , k2 − m2 ) ⎪ ⎪ ⎪ (K1 − m1 )(K2 − m2 ) ⎪ k1 =m1 k2 =m2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for 0 ≤ m1 ≤ K1 − 1; 0 ≤ m2 ≤ K2 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ K 1 −1 K2 −1+m (1.23) 2 1 x (k1 , k2 ) x∗ (k1 − m1 , k2 − m2 ) ⎪ (K1 − m1 )(K2 + m2 ) ⎪ ⎪ k1 =m1 k2 =0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for 0 ≤ m1 ≤ K1 − 1; −K2 + 1 ≤ m2 < 0, ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ rˆX (−m1 , −m2 ) for − K1 + 1 ≤ m1 < 0; −K2 + 1 ≤ m2 ≤ K2 − 1, ⎪ ⎪ ⎪ ⎩ 0 elsewhere. This expression denotes the non-biased estimation of the second autocorrelation function. The non-biased estimation of the first autocorrelation function is obtained by ˆX in the previous expression. replacing x(k1 , k2 ) by xc (k1 , k2 ) = x(k1 , k2 ) − μ Later on, X will be a process of zero average if there are no supplementary details.

1.3.6. Specificities of random 2-D signals As compared to the 1-D case, two particular cases of random processes can appear as soon as the representation dimension is more than or equal to 2. These particularities are directly dependent on the sensors used: – separable process: a separable process possesses an autocorrelation function which can be factorized under the form:

γX (m1 , m2 ) = γY (m1 ) γZ (m2 )

(1.24)

where γY (m1 ) and γZ (m2 ) are autocorrelation functions of the 1-D process. The knowledge (in general a priori) of separateness of a random 2-D field is very useful.

28

Two-Dimensional Signal Analysis 1 0.5 0 0.5 1 10 5 0 10

5 m2

5 0 10

5 10

m1

Figure 1.6. Autocorrelation function of an isotropic process

For example, in spectral analysis, the latter can be carried out using two 1-D spectral estimators. We find such processes in radar signal processing; – isotropic process: a WSS isotropic field possesses a description in which the autocorrelation function is invariant for any rotation of coordinates: it only depends on radius r = m21 + m22 . An example of autocorrelation function of the isotropic field is shown in Figure 1.6. These processes are often encountered in passive imaging.

1.3.7. Particular random signals As in 1-D, certain random processes are of major importance in processing and analyzing 2-D signals. We introduce them here, before tackling their frequency characterization in section 1.5.5. 1.3.7.1. White noise We say that a random 2-D WSS signal is a white noise (known to be of order 2) if it is formed from random variables {X(k1 , k2 )}(k1 ,k2 )∈Z 2 centered and non-correlated. Thus, we have: γX (m1 , m2 ) = E[X(k1 , k2 ) X ∗ (k1 − m1 , k2 − m2 )] 2 δ(m1 , m2 ) = σX 2 denotes the (constant) variance of the process. where σX

(1.25)


29

The extension of this notion is possible by relying on the characterization of higher orders together to define whiteness of order K of a random 2-D signal. In most cases, whiteness of order 2 does not lead to statistical independence of the random variables. This latter situation is called strong sense whiteness and the related processes are considered (formed of random variables) independently and identically distributed. 1.3.7.2. Gaussian process The definition of a random Gaussian signal primarily goes by the data of a probability density function (PDF) of a random Gaussian vector. If x is a random real Gaussian vector of dimension n, then its PDF is written as:

1 1 −1 T (1.26) exp − (x − μx ) Γx (x − μx ) fx (x) = 2 (2π)n/2 (det Γx )1/2 where: μx = E[x]

(1.27)

and Γx is the covariance matrix (n × n) of vector x: Γx = E (x − μx )(x − μx )T

(1.28)

In the case of a complex random Gaussian vector x = u + jv, of average μx and of covariance matrix: Γx = E (x − μx )(x − μx )†

(1.29)

x is said to be a circular complex Gaussian only if [DUV 91] u and v are jointly distributed following a Gaussian PDF with real values (relation (1.26)) and vector T T T u v has a covariance matrix of the form: 1 Γu v = 2

E −F F E

(1.30)

with FT = −F. In these conditions, if Γx = E + jF, then the PDF is given by: fx (x) =

πn

1 exp −(x − μx )† Γx −1 (x − μx ) det Γx

(1.31)

Due to the implicit non-causality on Z 2 , the extension to random 2-D signals requires “re-arranging” the nodes of a sub-set of a network in such a way as to form,

30


from the data of {X(k1 , k2 )}0≤k1 ≤K1 −1,0≤k2 ≤K2 −1 , a vector x, for example, by scanning the nodes line by line: x = [X(0, 0) X(0, 1) ... X(0, K2 − 1) X(1, 0) X(1, 1) ... ... X(K1 − 1, K2 − 1)]T

(1.32)

This type of construction is referred to as lexicographic (or path) scheduling. In such conditions, a random 2-D signal is said to be Gaussian if vector x constructed by the lexicographic scheduling follows a real or complex Gaussian PDF (relation (1.26) or (1.31)). If the signal is complex, WSS and centered, then the covariance matrix, renamed the autocorrelation matrix, takes the following form: ⎡ ⎢ ⎢ Γx = ⎢ ⎣

Γx (0) Γx (1) .. .

⎤ Γx (−1) · · · Γx (−K1 +1) Γx (0) · · · Γx (−K1 +2) ⎥ ⎥ ⎥ .. .. ⎦ . .

Γx (K1 −1) Γx (K1 −2) · · ·

(1.33)

Γx (0)

where each sub-matrix Γx (m) is a Toeplitz matrix of size (K2 × K2 ): ⎡ ⎢ ⎢ Γx (m) = ⎢ ⎣

γX (m, 0) γX (m, 1) .. .

γX (m, −1) γX (m, 0) .. .

⎤ · · · γX (m, −K2 + 1) · · · γX (m, −K2 + 2) ⎥ ⎥ ⎥ .. ⎦ .

γX (m, K2 − 1) γX (m, K2 − 2) · · ·

(1.34)

γX (m, 0)

The autocorrelation matrix is of the Toeplitz-block type, Hermitian and defined positive. It is of great importance in the processing of two dimensional signals and we will encounter it in Chapter 3 with regard to 2-D spectral estimation. A particularly important case of a random WSS signal is that of real centered Gaussian white noise for which we have: 2 Γx = σX IK1 K2

(1.35)

Let us note that all random variables making up a 2-D Gaussian white noise are statistically independent and allow PDF to be written in the form:


fx (x) =

K 1 −1 K 2 −1

k1 =0 k2 =0

1 2 2πσX

2

x (k1 , k2 ) exp − 2 2σX

31

(1.36)

1.4. 2-D systems Signal processing, in 1-D as in m-D, often consists of one or several transformations of a signal with the help of devices called systems. A system reacts to an input signal as an operator, by providing a modified output signal. We can distinguish two major approaches in the use of systems. First, let us consider a system which is completely defined and in which all parameters are specified; to a known input of such a system corresponds an output which can be explicitly calculated. The second approach for the use of systems consists of using properties (spectral, for example) in order to model any signal as being the response of this system to an excitation whose characteristics are more “simple”. In this case, the majority of the system parameters are unknown and it is necessary to determine them with the help of a deterministic 2-D sequence or from the outcome of a random field. In this section, we insist on 2-D systems which are linear and invariant by translation: these form the most important class of systems and are the basis of major applications such as filtering or analysis and 2-D spectral parametric estimation. 1.4.1. Definition First of all a system is an operator on signals: to any collection of N input sequences x1 , x2 , ..., xN corresponds an output sequence y such that: y = T [x1 , x2 , ..., xN ]

(1.37)

where T denotes the operator associated with the system. Formally, the latter can be equipped with any rule allowing us to explicitly determine the output by knowing the inputs. 1.4.2. Main 2-D operators In practice, only some operators, which we will analyze below, are useful for the description of 2-D systems: – addition: y(k1 , k2 ) = x1 (k1 , k2 ) + x2 (k1 , k2 )

(1.38)

32


– multiplication by a constant: y(k1 , k2 ) = cx(k1 , k2 )

(1.39)

– multiplication: y(k1 , k2 ) = x1 (k1 , k2 )x2 (k1 , k2 )

(1.40)

– spatial translation: y(k1 , k2 ) = x(k1 − m1 , k2 − m2 )

(1.41)

– memoryless non-linearity, an example of the quadration: y(k1 , k2 ) = [x(k1 , k2 )]2

(1.42)

1.4.3. Main properties Here we present two sought properties of 2-D systems, which make their description much simpler. A system represented by an operator T is said to be linear only if: ∀x1 , x2 ,

∀(c1 , c2 ) ∈ C 2: T [c1 x1 + c2 x2 ] = c1 T [x1 ] + c2 T [x2 ]

(1.43)

This linearity property allows the introduction of the impulse response notion of a system. By using previously defined operations like addition, multiplication by a constant and translation, any sequence x can be re-written in the following form: x(k1 , k2 ) =

+∞

+∞

x(m1 , m2 )δ(k1 − m1 , k2 − m2 )

(1.44)

m1 =−∞ m2 =−∞

where quantities {x(m1 , m2 )} play the role of constants, weighting an infinity of translated Kronecker sequences. The application of a linear operator on this sequence makes it possible to write: y(k1 , k2 ) = T = =

+∞

+∞

x(m1 , m2 )δ(k1 − m1 , k2 − m2 )

m1 =−∞ m2 =−∞ +∞ +∞

x(m1 , m2 )T [δ(k1 − m1 , k2 − m2 )]

m1 =−∞ m2 =−∞ +∞ +∞ m1 =−∞ m2 =−∞

x(m1 , m2 )h(m1 , m2; k1 , k2 )

(1.45)


33

where h(m1 , m2; k1 , k2 ) is the response of the system in (k1 , k2 ) to an unit impulse localized in (m1 , m2 ): it is the impulse response of the linear system. In addition, a system of input x(k1 , k2 ) and output y(k1 , k2 ) is said to be invariant by translation only if: ∀x

∀(m1 , m2 ) ∈ Z 2: T [x(k1 − m1 , k2 − m2 )] = y(k1 − m1 , k2 − m2 )

(1.46)

These two properties of linearity and of invariance by spatial translation of systems are formally independent of each other.

1.4.4. Linear time-invariant (LTI) system The most frequently studied and used class of systems in signal processing, even though it suffers from loss of generality, is that of systems said to be linear time-invariant (LTI). For these systems, the impulse response no longer depends on localization in (m1 , m2 ) of the unit impulse, but only on the translation vector between this impulse and position (k1 , k2 ) of the node where we want to calculate it. Thus, relation (1.45), which expresses the output of the system, becomes:

y(k1 , k2 ) =

+∞

+∞

x(m1 , m2 ) h(k1 − m1 , k2 − m2 )

(1.47)

m1 =−∞ m2 =−∞

This relation, known as the 2-D convolution product, emphasizes the fact that an LTI system is completely characterized by its impulse response which is invariant by translation h(m1 , m2 ). Later, we will note the 2-D convolution operator by symbol ⊗⊗ (symbol ⊗ traditionally represents the 1-D convolution operator). The output of an LTI system is written as: y(k1 , k2 ) = [x ⊗ ⊗h](k1 , k2 )

(1.48)

or more simply: y = x ⊗ ⊗h

(1.49)

34


Moreover, we can show that the convolution operator, in 2-D as in 1-D, is: – commutative: x ⊗ ⊗h = h ⊗ ⊗x

(1.50)

– associative: (x ⊗ ⊗h) ⊗ ⊗g = x ⊗ ⊗(h ⊗ ⊗g)

(1.51)

– distributive as compared to the addition: x ⊗ ⊗(h + g) = (x ⊗ ⊗h) + (x ⊗ ⊗g)

(1.52)

1.4.5. Example In order to illustrate the 2-D convolution operation of a signal by an LTI system, let us consider the simple example of an impulse response h equal to a unit step U shown above (see section 1.2.2). Let us assume that sequence x is such that x(m1 , m2 ) = 1 for {0 ≤ m1 ≤ M1 − 1, 0 ≤ m2 ≤ M2 − 1}, and x(m1 , m2 ) = 0 elsewhere. Examining relation (1.47) indicates that in order to obtain the convolution product at point (k1 , k2 ), it is necessary to calculate a total of products point to point of the input sequence x by a version, which is spatially “reversed” and of translated origin in (k1 , k2 ) of the sequence h, as shown in Figure 1.7a. When k1 and k2 vary, the position of this sequence evolves simultaneously and leads to a different convolution product (Figure 1.7b and 1.7c). Figure 1.8 shows the 2-D signal y arising from such a system.

1.4.6. Separable system An LTI system, whose impulse response is a separable 2-D sequence, is called a separable system. The impulse response then takes the following form: h(m1 , m2 ) = h1 (m1 ) h2 (m2 )

(1.53)

Under this condition, output y of a separable system in response to an input x is written as: y (k1 , k2 ) = =

+∞

+∞

m1 =−∞ m2 =−∞ +∞

h1 (m1 )

m1 =−∞

x (k1 − m1 , k2 − m2 ) h1 (m1 ) h2 (m2 ) +∞ m2 =−∞

(1.54) x (k1 − m1 , k2 − m2 ) h2 (m2 )


35

m2

x(m1, m2)

k1

m1

k2 h(k1

m1, k2

m2)

(a) m2 k1 k2 m1

(b) m2

k1 k2

m1

(c)

Figure 1.7. 2-D convolution operation: (a) input signal x and impulse response h reversed; (b-c) calculation of convolution for two values of (k1 , k2 )

By assuming that: g (k1 , k2 ) =

+∞

x (k1 , k2 − m2 ) h2 (m2 )

(1.55)

h1 (m1 ) g (k1 − m1 , k2 )

(1.56)

m2 =−∞

relation (1.54) becomes: y (k1 , k2 ) =

+∞ m1 =−∞

36


30 20 10 0 15 10 15

5

10 5

0 k2

0 5

5

k1

Figure 1.8. Output from system for M1 = M2 = 5

The output can thus be obtained by 1-D convolution of the set of lines of x by impulse response h1 , followed by 1-D convolution of the set of columns of the preceding result by impulse response h2 . Of course, the order of these processes can be reversed. Let us note that the use of separable systems has nothing reserved to separable signals described previously.

1.4.7. Stability of 2-D systems The stability of LTI systems in 1-D, as in m-D, is a necessary condition for their practical use. In practice, there are two types of stability: – “strong” stability, which requires that at any bounded input x of the system corresponds a bounded output y. Hence, we talk of BIBO (bounded input, bounded output) stability. For a BIBO system, if |x(k1 , k2 )| ≤ B ∀(k1 , k2 ), then there exists B such that |y(k1 , k2 )| ≤ B ∀(k1 , k2 ). As in the case of 1-D, we can show that a necessary and sufficient condition of stability is that impulse response h of the system can be absolutely integrable: +∞

+∞

m1 =−∞ m2 =−∞

|h (m1 , m2 )| < +∞

(1.57)


37

BIBO stability is the most employed in practice; – stability in quadratic mean is a stability that is more “weak” than the previous one. We say that an LTI system is stable in quadratic mean if: +∞

+∞

2

|h (m1 , m2 )| < +∞

(1.58)

m1 =−∞ m2 =−∞

A BIBO system is stable in quadratic mean, but the converse is false. We will tackle later on (see section 1.5.4.6) an approach which enables the characterization of stability of LTI systems. 1.4.8. Support of the impulse response – causality The study of 1-D systems leads naturally to a simple definition and in the correct sense of causality: a 1-D system is causal if its output does not precede its input. In addition, since the support dimension of 1-D signals is generally the time, the connection with the common sense of causality is clearly established. When the dimension of signals and systems is more than or equal to 2, any notion of explicit causality disappears. This is because Z n , n ≥ 2, is not provided with a “natural” total order relation contrary to Z. Thus, we find ourselves in a situation where the notion of causality has to be established, by imagining, for example in 2-D, non-trivial relations of partial order between network nodes, allowing us to define notions of the past, present and future of a 2-D data (see section 1.6.1). The impulse response of a causal 1-D system is characterized by the fact that any negative index is zero. In the case of 2-D, the extension of this property depends on considering that the impulse response is zero outside a support which remains to be specified. A particular case of 2-D support is the quarter plane or quadrant which we have seen above in relation to the impulse response of a 2-D unit step. An extension of the quadrant notion to the supports defined by the cones in the discrete Z 2 network is possible. Indeed, we can show, under certain hypotheses, that there is a linear application which makes it possible to transform such a support into a quarter plane support ([DUD 84] and see Chapter 2).

38

Two-Dimensional Signal Analysis m2

m2

m1

m1

(a)

(b)

m2

m2

m1

(c)

m1

(d)

Figure 1.9. Supports for impulse response of a 2-D system: a) quarter plane support; b) non-symmetric half plane; c) symmetric half plane support; d) full plane support

The following are the 2-D supports which are mostly used (see Figure 1.9): – the quarter plane (QP) support which we have just mentioned, for which h(m1 , m2 ) = 0 if m1 ≥ 0, m2 ≥ 0; – the non-symmetric half plane (NSHP) support for which the associated causality notion is similar to the scanning of the 2-D network, line by line. In this case, h(m1 , m2 ) = 0 for m1 = 0, m2 ≥ 0 and m1 ≥ 1, −∞ ≤ m2 ≤ +∞; – the symmetric half plane (SHP) support for which one of the two dimensions is causal and the other is non-causal. Indeed, we have h(m1 , m2 ) = 0 for m1 ≥ 0, −∞ ≤ m2 ≤ +∞; – finally, the full plane (FP) support where h(m1 , m2 ) = 0 for m1 ∈ Z, m2 ∈ Z. The first two supports correspond to the causal systems; the other two are associated with non-causal systems. A 2-D causal system is thus a system whose impulse response is the support in the NSHP.


39

These supports characterize the systems with infinite impulse response (IIR). By truncating these supports to make them finite, the corresponding systems are said to be finite impulse response (FIR).

1.5. Characterization of 2-D signals and systems We now examine dual representation of 2-D signal and LTI systems, i.e. their frequency representation.

1.5.1. Frequency response of an LTI system The frequency response of a 1-D system such as m-D represents the generally complex “gain” of the output of the system for the application at its input of a complex harmonic component. Let h(m1 , m2 ) be the impulse response of an LTI system and x(m1 , m2 ) a harmonic component of the complex exponential type: x(m1 , m2 ) = ej2π(ν1 m1 +ν2 m2 )

(1.59)

where ν1 and ν2 represent normalized frequencies (or reduced frequencies), with continuous variations, of complex sinusoid in each of the dimensions of the 2-D field origin. The output of the system can be determined by: +∞

y(k1 , k2 ) =

+∞

h(m1 , m2 ) ej2π(ν1 [k1 −m1 ]+ν2 [k2 −m2 ])

m1 =−∞ m2 =−∞

1 k1 +ν2 k2 ) = ej2π(ν +∞ +∞ .

−j2π(ν1 m1 +ν2 m2 )

(1.60)

h(m1 , m2 ) e

m1 =−∞ m2 =−∞ j2π(ν1 k1 +ν2 k2 )

=e

h(ν1 , ν2 )

By assuming that:

h(ν1 , ν2 ) =

+∞

+∞

h(m1 , m2 ) e−j2π(ν1 m1 +ν2 m2 )

(1.61)

m1 =−∞ m2 =−∞

and by defining this quantity as frequency response of the system with 2-D frequency (ν1 , ν2 ), we see that the output of an LTI system of complex sinusoidal input is also a complex sinusoid weighted by a term, which is precisely this frequency response

h(ν1 , ν2 ).

40


We can show that the frequency response is periodic in two frequency dimensions, i.e.:

h(ν1 + 1, ν2 ) = h(ν1 , ν2 )

(1.62)

h(ν1 , ν2 + 1) = h(ν1 , ν2 )

The data h(ν1 , ν2 ) on a frequency field (ν1 .ν2 ) ∈ ξ 2 = [−0.5 ; 0.5[ ×[−0.5 ; 0.5[ is thus sufficient to completely define the frequency response of a 2-D system. For example, let us then consider an LTI system whose impulse response is the following: h (m1 , m2 ) =

1 for − M ≤ m1 ≤ M; −M ≤ m2 ≤ M 0 elsewhere

The corresponding frequency response is thus equal to:

h(ν1 , ν2 ) = = =

+∞

m1 =−∞ m2 =−∞ M M

h(m1 , m2 ) e−j2π(ν1 m1 +ν2 m2 ) e−j2π(ν1 m1 +ν2 m2 )

m1 =−M m2 =−M M −j2πν1 m1

e

m 1 =−M

=

+∞

1+2

M

M

e−j2πν2 m2

m2 =−M

cos(2πν1 m1 )

1+2

m1 =1

M

cos(2πν2 m2 )

m2 =1

(1.63) The frequency and impulse responses of this system are represented in Figure 1.10. We note that the central part of the frequency response, corresponding to the region where |ν1 | and |ν2 | are small, shows a significant lobe which, for the considered system, translates into a 2-D low-pass filter behavior. By generalizing this example, we note that separateness of a 2-D system leads to separateness of the expression of frequency response. Indeed:

h(m1 , m2 ) = h1 (m1 ) h2 (m2 ) ⇒ h(ν1 , ν2 ) = h 1 (ν1 ) h 2 (ν2 )

(1.64)

where h 1 (resp. h 2 ) denotes the frequency response of 1-D system of impulse response h1 (resp. h2 ).


41

Inversely, we show that impulse response h(m1 , m2 ) can be determined from

frequency response h(ν1 , ν2 ) by the following relation: h(m1 , m2 ) = h(ν1 , ν2 ) ej2π(ν1 m1 +ν2 m2 ) dν1 dν2 ξ2

(1.65)

1.5.2. 2-D Fourier transform 1.5.2.1. Definition By once again taking relation (1.65), we note that the impulse response of a 2-D system can be created as a superposition of complex harmonic components. For any 2-D sequence x = {x(k1 , k2 )}(k1 ,k2 )∈Z 2 and by adopting the same method, we obtain: x(k1 , k2 ) = x(ν1 , ν2 ) ej2π(ν1 k1 +ν2 k2 ) dν1 dν2 (1.66) ξ2

In this relation, x(ν1 , ν2 ) represents the 2-D Fourier transform (2-D FT) of the sequence x defined by: FT [x(k1 , k2 )] x(ν1 , ν2 )

+∞

+∞

x(k1 , k2 ) e−j2π(ν1 k1 +ν2 k2 )

(1.67)

k1 =−∞ k2 =−∞

The Fourier transform of a sequence is a periodic function of frequency variables, i.e.:

x(ν1 + 1, ν2 ) = x(ν1 , ν2 )

(1.68)

x(ν1 , ν2 + 1) = x(ν1 , ν2 )

Let us note that in the m-D case, the Fourier transform is written as:

x(ν) =

T

x(k) e−j2π k

k

with: k = [k1 , k2 , · · · , km ]

T

ν = [ν1 , ν2 , · · · , νm ]

T

(1.69)

42

Two-Dimensional Signal Analysis 1 0.5 0 5 0 5

m2

5

0

5

m1

20 10 0 10

nu2

nu1

Figure 1.10. Impulse and frequency responses of LTI system for M = 2 and a frequency support (ν1 , ν2 ) ∈ ξ 2 = [−0, 5; 0, 5[ ×[−0, 5; 0, 5[

As for relation (1.66), it defines the 2-D inverse Fourier transform (2-D IFT). We can easily show that the response y(k1 , k2 ) of a 2-D system of frequency

response h(ν1 , ν2 ), with an input sequence of Fourier transform x(ν1 , ν2 ), is expressed in the following form: y(k1 , k2 ) = y (ν1 , ν2 ) ej2π(ν1 k1 +ν2 k2 ) dν1 dν2 (1.70) ξ2

where:

y (ν1 , ν2 ) = h(ν1 , ν2 ) x(ν1 , ν2 )

(1.71)

denotes the Fourier transform of y(k1 , k2 ). As in 1-D, the Fourier transform of 2-D convolution product y = h ⊗ ⊗x is the traditional product of impulse response of

Fourier transform h(ν1 , ν2 ) and the Fourier transform of input x(ν1 , ν2 ). 1.5.2.2. Properties We now briefly recall the properties of the 2-D Fourier transform: – linearity: if FT [x] = x and FT [y] = y , then:

FT [a x + b y] = a x + b y , ∀(a, b) ∈ C 2

(1.72)


43

– delay: if FT [x(k1 , k2 )] = x(ν1 , ν2 ), then:

FT [x(k1 − m1 , k2 − m2 )] = e−j2π(ν1 m1 +ν2 m2 ) x(ν1 , ν2 )

– modulation: FT x(k1 , k2 ) ej2π(f1 k1 +f2 k2 ) = x(ν1 − f1 , ν2 − f2 )

(1.73)

(1.74)

– multiplication: ! " FT [x(k1 , k2 ) y(k1 , k2 )] = x ⊗ ⊗ y (ν1 , ν2 )

(1.75)

where the convolution product is applied here to continuous and periodic functions which are Fourier transforms, i.e.: " ! x ⊗ ⊗ y (ν1 , ν2 ) = x(f1 , f2 ) y (ν1 − f1 , ν2 − f2 ) df1 df2 (1.76) ξ2

– conjugation: ∗

FT [x∗ (k1 , k2 )] = x (−ν1 , −ν2 )

(1.77)

– Parseval’s theorem: if FT [x] = x and FT [y] = y , then:

+∞

+∞

x(k1 , k2 ) y ∗ (k1 , k2 ) =

k1 =−∞ k2 =−∞

ξ2

∗

x(ν1 , ν2 ) y (ν1 , ν2 ) dν1 dν2

(1.78)

In particular, if y = x, we obtain the following relation: +∞

+∞

k1 =−∞ k2 =−∞

2

|x(k1 , k2 )| =

# #2 # # # x(ν1 , ν2 )# dν1 dν2 ξ2

(1.79)

translating the conservation of energy of a signal into two spatial and frequency representations. 1.5.3. Discrete 2-D Fourier transform 1.5.3.1. Definition The Fourier transform takes a particular form when the 2-D signal is periodic. Particularly, if the periodicity is rectangular (see equation (1.8)), and by assuming that

44


K1 = K11 , K2 = K22 ), we define development coefficients in the Fourier series by the following relation:

x(l1 , l2 ) =

K 1 −1 K 2 −1

−j2π

x(k1 , k2 ) e

l1 k1 K1

+

l2 k2 K2

(1.80)

k1 =0 k2 =0

We show that coefficients x(l1 , l2 ) are periodic and of the same periodicity as the sequence x(k1 , k2 ), which is written as: K 1 −1 K 2 −1 l1 k1 l2 k2 1 j2π K + K 1 2 x(l1 , l2 ) e x(k1 , k2 ) = K1 K2

(1.81)

l1 =0 l2 =0

for 0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1. This latter relation translates the decomposition in Fourier series of the 2-D sequence of the origin. In practice, 2-D sequences are rarely exactly periodic, but rather with finite spatial (or spatio-temporal) support; x(k1 , k2 ) is only available for 0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1. By supposing that the observed 2-D sequence is made periodic we define the 2-D discrete Fourier transform (2-D DFT) as the coefficients of the development in Fourier series and we write: DFT [x(k1 , k2 )] x(l1 , l2 )

K1 −1 K2 −1

−j2π

x(k1 , k2 ) e

l1 k1 K1

+

l2 k2 K2

(1.82)

k1 =0 k2 =0

Relation (1.81) defines the 2-D inverse discrete Fourier transform (2-D IDFT). 1.5.3.2. Properties The properties of the 2-D DFT are globally the same as those of the Fourier transform of an infinite sequence seen above: – linearity: if DFT [x] = x, DFT [y] = y , and x and y have the same spatial support, then:

DFT [a x + b y] = a x + b y , ∀(a, b) ∈ C 2

(1.83)

– convolution of two sequences: if x(k1 , k2 ) and y(k1 , k2 ) are two finite sequences, with 0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1, and DFT [x] = x, DFT [y] = y , we have: DFT [x ⊗c ⊗c y] = x y

(1.84)


45

where the symbol ⊗c represents the 1-D circular convolution operator and the 2-D circular convolution is given by: [x ⊗c ⊗c y] (m1 , m2 ) =

K 1 −1 K 2 −1

x(k1 , k2 ) y(m1 − k1 (mod K1 ), m2 − k2 (mod K2 ))

k1 =0 k2 =0

=

K 1 −1 K 2 −1

y(k1 , k2 ) x(m1 − k1 (mod K1 ), m2 − k2 (mod K2 )) (1.85)

k1 =0 k2 =0

– delay: if DFT [x(k1 , k2 )] = x(l1 , l2 ), then:

DFT [x(k1 − m1 (mod K1 ), k2 − m2 (mod K2 ))] =e

−j2π

m1 k1 K1

+

m2 k2 K2

x(l1 , l2 )

(1.86)

– modulation: m1 k1 m2 k2 j2π K + K 1 2 DFT x(k1 , k2 ) e = x(l1 − m1 (mod K1 ), l2 − m2 (mod K2 )) (1.87)

– multiplication: DFT [x(k1 , k2 )y(k1 , k2 )] =

1 x ⊗c ⊗c y (l1 , l2 ) K1 K2

(1.88)

– conjugation: ∗

DFT [x∗ (k1 , k2 )] = x (K1 − l1 (mod K1 ), K2 − l2 (mod K2 ))

(1.89)

– Parseval’s theorem: if DFT [x] = x and DFT [y] = y , then:

K 1 −1 K 2 −1

x(k1 , k2 ) y ∗ (k1 , k2 ) =

k1 =0 k2 =0

K 1 −1 K 2 −1 ∗ 1 x(l1 , l2 ) y (l1 , l2 ) K1 K2 l1 =0 l2 =0

(1.90) 1.5.3.3. Calculation of the 2-D DFT The analysis of relation (1.82) shows that the direct calculation of the 2-D discrete Fourier transform requires K12 K22 multiplications and additions. This number, which

46


can be very big when the data handled is, for example, digital images of large size is reduced drastically by using the algorithm of the 1-D fast Fourier transform (FFT) [COO 65], which is first made of rows and then columns. We can show that to carry out a complex 2-D DFT with the help of FFT the number of multiplications is thus only (K1 K2 /2) log2 (K1 K2 ) (and twice as much for the addition). For example, to calculate the 2-D DTF of a piece of data of size 512 × 512, the number of multiplications necessary as compared to direct calculation is about 28,000 times smaller [LIM 90]. We will not go into the details of this algorithm; excellent descriptions of it are given in [DUD 84] and [LIM 90].

1.5.4. 2-D z transform The Fourier transform of a sequence with any type of support (relation (1.67)) only converges uniformly for stable sequences in the BIBO sense. However, several classes of signals do not admit a representation by 2-D FT. That is the reason why another transform has been introduced: the 2-D z transform (2-D ZT). In a dual field, this allows us to represent a much larger variety of signals and systems. Moreover, 2-D ZT also allows us to study differential equations and especially to analyze the stability of 2-D systems. The ZT is known to possess characteristics which, contrary to FT, are quite different in 2-D (and m-D) as compared to the 1-D case. 1.5.4.1. Definition The 2-D ZT of a sequence x(k1 , k2 ) is defined by: ZT [x(k1 , k2 )] X(z1 , z2 ) +∞ +∞ x(k1 , k2 ) z1−k1 z2−k2

(1.91)

k1 =−∞ k2 =−∞

where (z1 , z2 ) ∈ C 2 . The first difficulty in 1-D cases, where only one single complex variable z intervenes, is found here. Indeed, X(z1 , z2 ) is defined on a support of four dimensions. The 2-D z transform is a generalization of the 2-D Fourier transform, as: X(z1 , z2 )|(z1 =ej2πν1 ,z2 =ej2πν2 ) +∞ +∞ = x(k1 , k2 ) e−j2πν1 k1 e−j2πν2 k2 k1 =−∞ k2 =−∞

= x(ν1 , ν2 )

(1.92)


47

Thus, by evaluating X(z1 , z2 ) in the region of C 2 such that (z1 = ej2πν1 , z2 = ej2πν2 ), which is called the surface, bi-circle or unit torus, we access the 2-D FT of the sequence. 1.5.4.2. Region of convergence By representing complex variables under the form z1 = r1 ej2πν1 and z2 = r2 ej2πν2 , with r1 = |z1 |, r2 = |z2 |, the 2-D ZT is thus written as: +∞

X(z1 , z2 ) =

+∞

x(k1 , k2 ) r1−k1 r2−k2 e−j2πν1 k1 e−j2πν2 k2

k1 =−∞ k2 =−∞

= FT x(k1 , k2 ) r1−k1 r2−k2

(1.93)

The 2-D FT convergence requires the signal to be absolutely integrable to be specific: +∞

+∞

|x(k1 , k2 )| r1−k1 r2−k2 < +∞

(1.94)

k1 =−∞ k2 =−∞

The 2-D ZT convergence thus depends on the values of r1 and r2 . Thus, region R of C 2 for which X(z1 , z2 ) converges is called the region of convergence or Reinhardt domain. For 1-D signals, it is known that the region of convergence is, generally, a ring defined on C and centered in z = 0. In 2-D, the representation of the region of convergence is more complex, as it is defined on C 2 . However, if we take into account the previous comment, it is possible to represent the region of convergence on a 2-D domain by using axes (r1 , r2 ) (or (log r1 , log r2 )). Figure 1.11 represents an example of the region of convergence. In this representation, for example, the unit torus corresponds to coordinate nodes (r1 = 1, r2 = 1). As in the 1-D case, writing a 2-D ZT is not complete without data of its region of convergence, in the sense where the corresponding 2-D sequence cannot be determined without ambiguity. For example, the two following sequences: x(k1 , k2 ) = ak1 bk2 U (k1 , k2 ) and: x(k1 , k2 ) = −ak1 bk2 U (−k1 − 1, k2 ) both have for 2-D ZT the function: X(z1 , z2 ) =

1 1 −1 1 − az1 1 − bz2−1

48

Two-Dimensional Signal Analysis r2

0

r1

Figure 1.11. Representation of the region of convergence of a 2-D ZT

However, they have for region of convergence the fields defined respectively by: r1 > |a| ; r2 > |b| and: r1 < |a| ; r2 > |b| In practice, we often consider some specific cases of 2-D sequences leading to specific properties of associated regions of convergence. Support on a quadrant: as already stated, the class of sequences with support on a quadrant occupies an important place in 2-D processing because of its simplicity. In this case, the 2-D ZT is written as: X(z1 , z2 ) =

+∞ +∞

x(k1 , k2 ) z1−k1 z2−k2

(1.95)

k1 =0 k2 =0

We easily show that if (z01 , z02 ) ∈ R, then (z1 , z2 ) ∈ R if r1 ≥ |z01 |, r2 ≥ |z02 |. As a consequence the limit separating region of convergence and region of divergence in the representation r2 = f (r1 ) is a decreasing monotonous function, as shown in Figure 1.12a. Support on a symmetric half plane: in this case, we have x(k1 , k2 ) = 0 for k2 < 0, and: X(z1 , z2 ) =

+∞

+∞

k1 =−∞ k2 =0

x(k1 , k2 ) z1−k1 z2−k2

(1.96)

Basic Elements of 2-D Signal Processing r2

r2

|z02|

|z02|

0

|z01|

r1

0

(a)

|z01|

49

r1

(b)

Figure 1.12. Properties of the regions of convergence: a) sequence with quarter plane support; b) sequence with a half plane support

If (z01 , z02 ) ∈ R, then any (z1 , z2 ) such that r1 = |z01 |, r2 ≥ |z02 |, belongs to the region of convergence. The limit R thus only depends on variable r1 , as shown in Figure 1.12b. Full plane support: there is no general result in the case of a sequence of infinite extension on Z 2 , since convergence depends directly on the sequence whose 2-D ZT is evaluated. However, let us note that 2-D ZT of a sequence often converges in practice on a finite region R. 1.5.4.3. Properties Just as FT and DFT, 2-D ZT possesses numerous properties which we recall here. We note that these properties are the basic tools for numerous calculation processes, particularly for the synthesis of recursive 2-D filters and examination of their stability, to which we will return later: – linearity: if ZT [x] = X with region of convergence Rx and ZT [y] = Y with region of convergence Ry , then: ZT [a x + b y] = a X + b Y, ∀(a, b) ∈ C 2

(1.97)

with region of convergence R Rx ∩ Ry; – convolution: if y(k1 , k2 ) =

m1 m2

x(k1 − m1 , k2 − m2 ) h(m1 , m2 )

(1.98)

50


where X has Rx as region of convergence, H has Rh as region of convergence and hence: ZT [y(k1 , k2 )] = Y = X H

(1.99)

where Ry Rx ∩ Rh; – delay: if ZT [x(k1 , k2 )] = X(z1 , z2 ), then: ZT [x(k1 − m1 , k2 − m2 )] = z1−m1 z2−m2 X(z1 , z2 )

(1.100)

The associated region of convergence remains identical between the 2-D ZT of the original sequence and of its translation, except perhaps for |z1 | = 0 or ∞, |z2 | = 0 or ∞; – modulation: ZT ak1 bk2 x(k1 , k2 ) = X(a−1 z1 , b−1 z2 )

(1.101)

The associated region of convergence remains identical to that of ZT of the original sequence, averaging scale factors |a| according to variable z1 and |b| according to variable z2; – multiplication: ZT [x(k1 , k2 )y(k1 , k2 )] =

$ $ dv1 dv2 z 1 z2 1 , Y (v1 , v2 ) X 2 (j2π) C1 C2 v1 v2 v1 v2

(1.102)

where C1 and C2 are two closed contours oriented in the trigonometric sense, both totally belonging to the integrated region of convergence and surrounding the origin of each variable v1 and v2; – conjugation: ZT [x∗ (k1 , k2 )] = X ∗ (z1∗ , z2∗ )

(1.103)

with the same region of convergence as X; – differentiation: ZT [k1 k2 x(k1 , k2 )] = z1 z2

∂2 X(z1 , z2 ) ∂z1 ∂z2

with the same region of convergence as X;

(1.104)


51

– reflections: ZT [x(−k1 , k2 )] = X(z1−1 , z2 )

(1.105)

ZT [x(k1 , −k2 )] = X(z1 , z2−1 )

(1.106)

ZT [x(−k1 , −k2 )] = X(z1−1 , z2−1 )

(1.107)

– Parseval’s theorem: if ZT [x] = X and ZT [y] = Y , then: +∞

+∞

x(k1 , k2 ) y ∗ (k1 , k2 ) =

k1 =−∞ k2 =−∞

1 (j2π)2

$

$ X(z1 , z2 ) Y

C1

C2

∗

1 1 , z1∗ z2∗

dz1 dz2 z1 z2

(1.108)

where C1 and C2 are two closed contours, oriented in the trigonometric sense, both totally belonging to the integrated region of convergence and surrounding the origin of each variable z1 and z2 . 1.5.4.4. Transfer function of a 2-D system ZT is a tool adapted to the description (analysis and synthesis) of mono- or multidimensional LTI systems, and in particular of linear filters (recursive or not) with two indices. First, let us consider an LTI system with sequence x(k1 , k2 ) = z1k1 z2k2 as input; and using relation (1.47), the output of the system is written as: y(k1 , k2 ) = H(z1 , z2 )z1k1 z2k2

(1.109)

where H(z1 , z2 ) is the 2-D ZT of the impulse response of the system and is defined as the transfer function of an LTI system. Relation (1.109) shows that complex exponential sequences z1k1 z2k2 are proper functions for LTI systems. Let us note that the transfer function which is evaluated on the unit torus (|z1 | = 1 and |z2 | = 1) leads to

the expression of the frequency response of the system h(ν1 , ν2 ). Let us now consider a system having as input sequence x(k1 , k2 ) and as output sequence y(k1 , k2 ), and then let us suppose that this system is written by the following differential equation:

(m1 , m2 )∈Da

am1 ,m2 y(k1 − m1 , k2 − m2 ) =

(m1 , m2 )∈Db

bm1 ,m2 x(k1 − m1 , k2 − m2 )

(1.110)

52


where {a(m1 , m2 )} and {b(m1 , m2 )} are sets of constant coefficients (real or complex) not all zero on the respective supports Da and Db (supposed as finite) and equal to zero outside these supports1. Applied to this differential equation, relation (1.99), which gives the 2-D ZT of the convolution product, is written as: A(z1 , z2 ) Y (z1 , z2 ) = B(z1 , z2 ) X(z1 , z2 )

(1.111)

from where we draw the transfer function in z of the system: H(z1 , z2 ) =

B(z1 , z2 ) Y (z1 , z2 ) = X(z1 , z2 ) A(z1 , z2 )

(1.112)

The transfer function H(z1 , z2 ) associated with the differential equation of relation (1.110) is thus the quotient of two polynomials with two variables, which are respectively 2-D ZT finite duration sequences am1 ,m2 and bm1 ,m2 . This variety of transfer function, called rational, plays an important role in processing 2-D signals, as much from the point of view of synthesis of filters (for example, for applications in image processing) as from the point of view of analysis and characterization of 2-D signals (for example, in the usage of linear prediction; see Chapter 2). As in 1-D processing, we define in the present case the notions of zero and poles of a 2-D rational transfer function. Let us recall that zeros of H(z1 , z2 ) (resp. its poles) are sets {(z1 , z2 )}, such that B(z1 , z2 ) = 0 and A(z1 , z2 ) = 0 (resp. A(z1 , z2 ) = 0). However, these notions must be precise; indeed, contrary to the 1-D case, roots of 2-D polynomials are not isolated nodes in C, but surfaces in C 2 and are thus difficult to represent. For example, let us consider the following 2-D polynomial: A(z1 , z2 ) = 1 − a1,0 z1−1 − a0,1 z2−1 − a1,1 z1−1 z2−1 where a1,0 , a0,1 and a1,1 are complex constants. The roots of this polynomial are such that: z2 =

a0,1 + a1,1 z1−1 1 − a1,0 z1−1

thus forming a continuous surface in a four-dimensional space.

1. Here we will not discuss conditions of obtaining a general or particular solution y(k1 , k2 ) to this differential equation or the calculation aspect of recursive systems in accordance with limit conditions. These problems, which continue to interest us from the point of view of specificity of 2-D (and m-D) systems as compared to 1-D, are detailed in [LIM 90] and [DUD 84].


53

1.5.4.5. 2-D inverse ZT The 2-D inverse z transform (2-D IZT), which makes it possible to retrieve sequence x(k1 , k2 ), is defined by the following relation, which is a simple extension of the 1-D case:

x(k1 , k2 ) =

1 (j2π)2

$ C1

$ C2

X(z1 , z2 )z1k1 −1 z2k2 −1 dz1 dz2

(1.113)

where C1 and C2 are two closed contours, oriented in the trigonometric sense, belonging to region of convergence X(z1 , z2 ) and surrounding the origin of each variable z1 and z2 . In the 1-D case, if function X(z) is expressed under the form of a rational fraction of polynomials in z, which is often the case – especially for recursive filters – the inverse z transform can be calculated by using, after finding roots and poles, the development in the simple total of rational fractions and then by adding up the corresponding IZT components. This is generally not possible in 2-D, as multi-dimensional polynomials cannot be factorized. The total development of simple rational fractions cannot be carried out. As regards to the transfer function of a 2-D system which can be recursively calculated, one solution consists of numerically analyzing its impulse response by observing the output for an input constituted of a unit impulse δ(k1 , k2 ). 1.5.4.6. Application to the study of stability of LTI systems In section 1.4.7 we have seen general stability conditions of the LTI system. Relation (1.57) defining the BIBO stability is not very usable in practice and this is due to two reasons. First, it requires calculation of an infinite sum based on an impulse response and, second, this same impulse response is often not available, as an LTI system is generally specified by a finite differential equation shown by relation (1.110). That is why it is useful to characterize the stability of a system by the analysis of its transfer function. Since the absolute sum of a 2-D sequence is strictly equivalent to the analysis of its 2-D ZT on the unit torus, the impulse response h(m1 , m2 ) of a system defined by a differential equation will be absolutely summable if the transfer function H(z1 , z2 ) of relation (1.112) does not present any singularity for |z1 | = 1 and |z2 | = 1, i.e., if the polynomial denominator A(z1 , z2 ) does not contain any root on the unit torus. This requires the unit torus to be entirely included in the region of convergence of the transfer function. Since this is not generally available, we look for stability conditions from the expression A(z1 , z2 ).

54


By assuming that the transfer function is rational, of the form: H(z1 , z2 ) = 1/A(z1 , z2 ) with: A(z1 , z2 ) = 1 +

am1 ,m2 z1−m1 z1−m2

(m1 , m2 )∈D

we can thus state several stability conditions of the system, in accordance with the type of support D envisaged for coefficients: – non-causal support [RAN 85]: A(z1 , z2 ) = 0, |z1 | = 1, |z2 | = 1 – semi-causal support (causal following the first dimension and non-causal following the second) [RAN 85]: A(z1 , z2 ) = 0, |z1 | ≥ 1, |z2 | = 1 – causal support: two situations exist: – NSHP support [RAN 85]: A(z1 , z2 ) = 0, |z1 | ≥ 1, z2 = ∞ A(z1 , z2 ) = 0, |z1 | = 1, |z2 | ≥ 1 – QP support [HUA 72]: A(z1 , z2 ) = 0, |z1 | ≥ 1, |z2 | = 1 A(u, z2 ) = 0, |z2 | ≥ 1, ∀u such that |u| ≥ 1 Let us note that if the impulse response of the system is at a quadrant support, several results concerning stability are available [BEN 99, DEC 77, DUD 84, JUS 73, STR 77]. 1.5.4.7. Minimum or non-minimum phase LTI system By definition, in the 1-D case, a filter is said to be a minimum phase filter if its direct transfer function H(z) and its reciprocal transfer function 1/H(z) are both causal and of finite energy, i.e. stable. As we have already mentioned, in the case of 1-D signals the notion of causality is, on the one hand, defined without ambiguity and, on the other hand, is often linked to physical phenomena unfolding over the course of time (thus implying causal transfer functions), which implies that the study of the non-minimum phase systems presents a rather theoretical interest. In the 2-D case, since the notion of causality is more arbitrary (as it is linked to a chosen support) and generally not induced by an irreversible physical phenomenon,


55

non-minimum phase systems are more scattered and their study presents a primordial interest in the case of 2-D signal processing. We will see this aspect in Chapter 2 for the description of stochastic 2-D models (AR, ARMA, etc.) or the identification of 2-D non-minimum phase LTI systems with the help of higher statistical orders (see Chapter 4).

1.5.5. Frequency characterization of a random 2-D signal Here we define the 2-D power spectral density (2-D PSD) of a random field, which is an essential quantity that we will analyze in the following chapters. We will not dwell here on the physical sense attributed to frequency (or spectral) representation of stochastic processes, or on the purely mathematical justification of the existence of the spectral measurement of a process. We will content ourselves by recalling that the PSD of a discrete random 2-D signal represents the distribution, in the spectral field, of the power of the 2-D process. A random WSS field possesses the remarkable property which can be represented in the form of a superposition of complex exponentials (for the fields of complex values) or even sinusoids (for real fields), each one having a particular spatial frequency and all having random and de-correlated amplitudes. In the first place, we define the PSD of a random WSS field as variance of the random amplitude of complex exponentials (or sinusoids) according to spatial frequency. The main interest in spectral representation is focussed mainly on the characterization of phenomena such as periodicity of the processes. As we will see in Chapter 3, the (deterministic or random) periodic or quasi-periodic phenomena find, in the PSD, a representation mode, which is more synthetic than in the spatial or 2-D spatiotemporary domain. Let X(k1 , k2 ) be a complex random WSS field, of the second autocorrelation function rX (m1 , m2 ). Its PSD is the 2-D Fourier transform of this autocorrelation function: SX (ν1 , ν2 ) r X (ν1 , ν2 )

=

+∞

+∞

rX (m1 , m2 ) e−j2π(ν1 m1 +ν2 m2 )

(1.114)

m1 =−∞ m2 =−∞

for (ν1 , ν2 ) ∈ 2 . This definition is an extension of the Wiener-Khinchine theorem to 2-D processes.

56


The properties of the 2-D PSD are the following: – the PSD is a continuous function of variables ν1 and ν2 (normalized frequencies), defined on 2; – it is a periodic function of a unit period in each dimension. Its description on the set ξ 2 = [−0.5; 0.5[ ×[−0.5 ; 0.5[ is thus sufficient; – the PSD is a non-negative function of normalized 2-D frequency because of semi-defined positivism of the autocorrelation function; – if the random field is real, then the 2-D PSD is a pair function, i.e., a pointwise symmetry as compared to the origin of frequencies (ν1 = 0, ν2 = 0); – the PSD can been seen as the 2-D ZT of the second autocorrelation function, which is evaluated on the unit torus: SX (ν1 , ν2 ) = RX (z1 , z2 )|z1 =ej2πν1 ,z2 =ej2πν2

(1.115)

– by supposing that the autocorrelation function is such that: −−−−−−−−→ 0 |rX (m1 , m2 )| − √

(1.116)

m21 +m22 →∞

and if we have an outcome x(k1 , k2 ) of the random signal on a rectangular 2-D window (0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1), we show that the 2-D PSD is equal to: SX (ν1 , ν2 ) = lim

K1 ,K2 →∞

⎡

1 E⎣ K1 K2

#K −1 K −1 #2 ⎤ 1 2 # # # # x(m1 , m2 ) e−j2π(ν1 k1 +ν2 k2 ) # ⎦ # # #

(1.117)

k1 =0 k2 =0

Later on, we will often impose that signals must be centered, i.e. at zero average. In this case, the PSD of a random field will be expressed as the Fourier transform of the first autocorrelation function: SX (ν1 , ν2 ) FT [γX (m1 , m2 )] =

+∞ m1 =−∞

+∞ m2 =−∞

γX (m1 , m2 ) e−j2π(ν1 m1 +ν2 m2 )

(1.118)

The PSD is a function which “summarizes” certain statistical properties of the second order of a random WSS field. However, in general, it does not allow access to the complete law of the field, except in the case of a centered Gaussian field for which the autocorrelation function is a sufficient statistic. In this case, there is a bijection between FDP and PSD.


X(k1, k2)

h(k1, k2)

57

Y(k1, k2)

Figure 1.13. LTI system at random input

2 Let us note that for a centered white noise X of variance σX and according to relation (1.35), the PSD is equal to: 2 SX (ν1 , ν2 ) = σX

(1.119)

1.5.6. Output of a 2-D system with random input Let us consider a 2-D system of deterministic impulse response h(k1 , k2 ) whose input is a random WSS signal X(k1 , k2 ) of average μX and autocorrelation function rX (m1 , m2 ) (see Figure 1.13). We place ourselves in the general case of complex signals and systems. The output of the system is also a random signal Y (k1 , k2 ) and, using relation (1.47): Y (k1 , k2 ) =

+∞

+∞

X(m1 , m2 ) h(k1 − m1 , k2 − m2 )

(1.120)

m1 =−∞ m2 =−∞

We easily show, with the help of the properties already stated, that statistics (up to the order 2) of the output are the following: – average:

E[Y (k1 , k2 )] = μX h(0, 0) = μY

(1.121)

– autocorrelation function: rY (m1 , m2 ) = rX (m1 , m2 ) ⊗ ⊗h(m1 , m2 ) ⊗ ⊗h∗ (−m1 , −m2 )

(1.122)

– PSD: # #2 # # SY (ν1 , ν2 ) = SX (ν1 , ν2 ) ## h(ν1 , ν2 )##

(1.123)

which is a particularly important relation which we will once again find in Chapters 2 and 3 and which is also characteristic of higher order linear systems, as we will see in Chapter 4.

58


1.6. 2-D Wold decomposition The 2-D Wold decomposition [HEL 58, HEL 62] is inspired from the 1-D case [THE 92, WOL 54]. In 1-D, this decomposition can be summarized as follows: any random WSS 1-D signal is the sum of two signals, i.e. a purely deterministic signal and another purely random signal. For random 2-D fields, the situation is a little more complex. We will see in what follows that a random 2-D deterministic field can be divided into two fields: the evanescent field and the purely deterministic field. This decomposition into a purely deterministic field, evanescent field and purely random field constitutes the 2-D Wold decomposition of a random and regular WSS field. The random fields will supposedly be centered. 1.6.1. Innovation, determinism and regularity in the 2-D case In 1-D signal processing, the order of a sequence according to an index is immediate, as well as the definition of the past and the future of a sample. In the case of a 2-D sequence, this is not the case anymore (see section 1.4.8). By using the example of a straight line in a plane, Whittle [WHI 54] has emphasized the arbitrary choice of a past as compared to any node on this line. However, it is possible to schedule coordinates of a 2-D field S, sub-set of Z 2 [DUD 84, FRA 93]. The column-wise and line-wise lexicographic arrangements are most commonly used. Generally, a lexicographic path of S followed by a straight line of any rational slope can be characterized by a NSHP D of Z 2 , whose definition can be generalized as compared to section 1.4.8 as follows [LOU 89]: 1) D is stable by addition, 2) D ∪ −D = Z 2 , 3) D ∩ −D = {(0, 0)}. Such a set is of (if α = 0) type: & % D = (k1 , k2 ) ∈ Z 2 /k2 α + k1 β > 0 & % ∪ (k1 , k2 ) ∈ Z 2 /k2 α + k1 β = 0, k1 ≥ 0

(1.124)

with (α, β) ∈ Z and α, β first among them. The couples (α, β) allow us to define the sequencing of the plane. Thus, with any NSHP D of couple (α, β), we can associate the relation of total order on Z 2 defined by: 2

(k3 , k4 ) ≺ (k1 , k2 )

if

(k1 − k3 , k2 − k4 ) ∈ D

The NSHP related to the column-wise lexicographic path will be noted by Dh . The associated integer couple is (1, 0) according to equation (1.126). For the particular case


59

of NSHP Dv associated with the line-wise lexicographic path, it is necessary to use the following definition: % & % & (1.125) Dv = (k1 , k2 ) ∈ Z 2 /k1 > 0 ∪ (k1 , k2 ) ∈ Z 2 /k1 = 0, k2 ≥ 0 The order relation associated with a NSHP allows us to define the smallest subspace HkD1 ,k2 (Y ) of Hilbert space L2 (Ω, A, P ), which can be assimilated to the strict past (sr) of the random process Y to the node (k1 , k2 ): HkD1 ,k2 (Y ) = sp {Y (k3 , k4 )/ (k3 , k4 ) ≺ (k1 , k2 ) , (k3 , k4 ) = (k1 , k2 )} (1.126) We can define the process of innovation I Y,D = I Y,D (k1 , k2 )(k1 ,k2 )∈Z 2 of Y as compared to NSHP D: I Y,D (k1 , k2 ) = Y (k1 , k2 ) − Y (k1 , k2 )/HkD1 ,k2 (Y ) where Y

(k1 , k2 )/HkD1 ,k2

(Y ) is the projection of Y (k1 , k2 ) on

(1.127) HkD1 ,k2

(Y ).

The norm2 of this innovation has been highlighted in [HEL 58]. For any NSHP D, the innovation variance is expressed in the following manner: ⎞ ⎛ ' Y,D '2 ⎟ ⎜ 'I (1.128) (k1 , k2 )' = exp ⎝ ln SY (ν1 , ν2 ) dν1 dν2 ⎠ ξ2

It is necessary to note that this norm does not depend on NSHP D, but only on the spectral properties of Y . The preceding consideration allows us to characterize two main classes of the 2-D process [LOU 89]: – if I Y,D (k1 , k2 ) = 0, process Y is said to be deterministic; – otherwise, the process is said to be regular. Moreover, it is said that Y is purely random if H (Y ) = H I Y,D , where H(X) is the Hilbert space caused by {X(k1 , k2 )}(k1 ,k2 )∈Z 2 . In other words, Y is completely expressed according to its innovation. Thus, when Y is regular, we define the normalized innovation of a NSHP D, which is a white noise of unit variance: I Y,D (k1 , k2 ) (1.129) v Y,D (k1 , k2 ) = Y,D I (k1 , k2 ) 2. We recall (see section 1.3.1) that space L2 (Ω, A, P ) of random variables of integrable square becomes a Hilbert space when it posseses scalar product X, Y = E [X Y ∗ ], and whose associated norm will be noted by . .

60


It is possible to characterize the determinism and regularity notions from spectral representations of the random 2-D field [HEL 62]: – process Y is deterministic only if: ln SY (ν1 , ν2 ) dν1 dν2 → −∞

(1.130)

ξ2

– process Y is regular only if: ln SY (ν1 , ν2 ) dν1 dν2 > −∞

(1.131)

ξ2

The two main consequences of these results are that process Y is purely random, if it can be entirely expressed according to its innovation and that the spectrum of such a process is absolutely continuous (analytical function).

1.6.2. Total decomposition of three fields As we have explained in section 1.3.1, a 2-D signal is considered an outcome of a family of random variables Y spread over a regular grid whose nodes (also called sites) are represented by integer couples (k1 , k2 ), with (k1 , k2 ) ∈ Z 2 . This family forms a random 2-D field which we arrange according to NSHP Dh . The 2-D Wold decomposition which we tackled above helps us to obtain:

Y (k1 , k2 ) = W (k1 , k2 ) + V (k1 , k2 )

(1.132)

where: W = {W (k1 , k2 )}(k1 ,k2 )∈Z 2 is a purely random field, and:

V = {V (k1 , k2 )}(k1 ,k2 )∈Z 2 is a deterministic field.

This decomposition is justified by the study of spectral distribution of Y [FRA 95, HEL 62]: for spectral distribution, the V field admits the singular part of the spectral distribution of Y .


61

V in turn is divided in the following manner: V (k1 , k2 ) = P (k1 , k2 ) + Ev (k1 , k2 )

(1.133)

with: P = {P (k1 , k2 )}(k1 ,k2 )∈Z 2 the purely deterministic field, and: Ev = {Ev (k1 , k2 )}(k1 ,k2 )∈Z 2 the evanescent field. Moreover, the general writing of an evanescent field is also done in the form of an infinite and countable orthogonal decomposition into “elementary” evanescent fields, i.e. associated with a NSHP characterized by an integer couple (α, β) (relation (1.124)), which is expressed as: Ev (k1 , k2 ) =

E(α,β) (k1 , k2 )

(1.134)

(α,β)∈Z 2

% & The field E(α,β) (k1 , k2 ) (α,β)∈Z 2 is the evanescent component generated by column-wise innovations of the deterministic field respecting the order defined by (α, β) ∈ Z 2 , α, β first among them. Thus, if Y is a random 2-D which is field regular and WSS, then Y (k1 , k2 ), ∀ (k1 , k2 ) ∈ Z 2 can be expressed in a unique way by orthogonal decomposition: Y (k1 , k2 ) = W (k1 , k2 ) + P (k1 , k2 ) +

E(α,β) (k1 , k2 )

(1.135)

(α,β)∈Z 2

1.6.3. Example of an outcome It is thus possible to synthesize – into digital image form – the outcomes of random “2-D Wold” fields [FRA 93]. An example of an outcome is shown in Figure 1.14. This outcome of a random field is the sum of three spatial outcomes; Figures 1.15a, 1.15b and 1.15c respectively represent outcomes of a purely random field, of a purely deterministic field and an evanescent field whose decomposition is limited to a single couple (α, β). It is necessary to note that the outcome of a purely random field (Figure 1.15a) emerges from a 2-D autoregressive Gaussian model and a finite causal order. The representation of the PSD (Figure 1.15b) does not emerge from an estimation; it is obtained from analytical expression of the PSD of such a field, as we will see in Chapter 2. Here, the purely deterministic field corresponds to a harmonic real component constituted from a single 2-D frequency.

62


(a)

(b)

Figure 1.14. Example of Wold field representation: a) spatial, b) spectral (PSD estimated by periodogram; see Chapter 3)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1.15. Example of Wold decomposition: (a-c) spatial representation, (d-f) spectral representation


63

1.7. Conclusion In this chapter, we have introduced the main basic concepts for processing and analysis of 2-D discrete signals. After first reviewing deterministic signals, then random signals and their characterization in their definition sphere, we have presented 2-D systems as well as traditional tools which enable the description of signals and 2-D systems (Fourier transform, discrete Fourier transform, z transform). Finally, we have finished with the most general description of a 2-D field. Here we will recall that, apart from a natural increased complexity due to the dimensionality of representation, all differences noticed, as compared to the 1-D case, arise from the non-existence of an explicit causality for signals and 2-D systems. For many years, some key problems remained unresolved, among which we find that of the search for an extension on valid Z 2 (i.e., semi-definite positive) of self-correlation function of a 2-D process, or the problem of stability of 2-D systems. We now have to go much further in the description of 2-D signals as well as tools which enable their analysis.

1.8. Bibliography [BEN 99] B ENIDIR M., BARRET M., Stabilité des filtres et des systèmes linéaires, Dunod, 1999. [COO 65] C OOLEY J. W., T UKEY J. W., “An algorithm for the machine calculation of complex Fourier series”, Math. Comput., vol. 19, p. 297–301, 1965. [DEC 77] D E C ARLO R., M URRAY J., S AEKS R., “Multivariate Nyquist theory”, Int. J. Control, vol. 25, p. 657–675, 1977. [DUD 84] D UDGEON D. E., M ERSEREAU R. M., Multidimensional Digital Signal Processing, Englewood Cliffs, NJ, Prentice-Hall, Signal Processing Series, 1984. [DUV 91] D UVAUT P., Traitement du signal, Hermès, 1991. [FRA 93] F RANCOS J. M., M EIRI A. Z., P ORAT B., “A unified texture model based on a 2-D Wold-like decomposition”, IEEE Transactions on Signal Processing, vol. 41, no. 8, p. 2665–2678, August 1993. [FRA 95] F RANCOS J. M., M EIRI A. Z., P ORAT B., “A Wold-like decomposition of twodimensional discrete homogeneous random fields”, Ann. Applied Prob., vol. 5, no. 1, p. 248–260, 1995. [GUY 95] G UYON X., Random Fields on a Network. Modeling, Statistics, and Applications, Springer-Verlag, 1995. [HEL 58] H ELSON H., L OWDENSLAGER D., “Prediction theory and Fourier series in several variables”, Acta Mathematica, vol. 99, p. 165–202, 1958.

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[HEL 62] H ELSON H., L OWDENSLAGER D., “Prediction theory and Fourier series in several variables II”, Acta Mathematica, vol. 106, p. 175–213, 1962. [HUA 72] H UANG T. S., “Stability of two-dimensional recursive filters”, IEEE Transactions on Audio Electroacoustics, vol. 20, no. 2, p. 158–163, June 1972. [JUS 73] J USTICE J. H., S HANKS J. L., “Stability criterion for N-dimensional digital filters”, IEEE Transactions on Automatic Control, vol. 18, no. 3, p. 284–286, June 1973. [LIM 90] L IM J. S., Two-Dimensional Signal and Image Processing, Englewood Cliffs, NJ, Prentice-Hall, Signal Processing Series, 1990. [LOU 89] L OUBATON P., “Champs stationaires au sens large sur Z 2: propriétés structurelles, et modèles paramétriques”, Traitement du signal, vol. 6, no. 4, 1989. [PAP 91] PAPOULIS A., Probability, Random Variables, and Stochastic Processes, 3rd edition, McGraw-Hill, 1991. [RAN 85] R ANGANATH S., JAIN A.-K., “Two-dimensional linear prediction models – Part I: spectral factorization and realization”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-33, no. 1, p. 280–299, February 1985. [STR 77] S TRINTZIS M. G., “Test of stability of multidimensional filters”, IEEE Transactions on Circuits and Systems, vol. 24, no. 8, p. 432–437, August 1977. [THE 92] T HERRIEN C. W., Discrete Random Signals and Statistical Signal Processing, Englewood Cliffs, NJ, Prentice-Hall, Signal Processing Series, 1992. [WHI 54] W HITTLE P., “On stationary processes in the plane”, Biometrika, vol. 41, p. 434–449, 1954. [WOL 54] W OLD H., The Analysis of Stationary Time Series, Almquist and Wicksell, Uppsala, Sweden (first published in 1938 entitled A Study in the Analysis of Stationary Time Series), 1954.

Chapter 2

2-D Linear Stochastic Modeling

2.1. Introduction In the previous chapter, a certain number of definitions linked to random signals and spectral analysis were recalled in the 2-D case. In this chapter, we will make an effort to extend these definitions to propose some purely random stationary signal models to the second order so as to carry out processing of 2-D signals supposedly stationary to the second order at least by zone (such as, for example, “textured” images). The following 2-D linear stochastic models will be presented: ARMA (autoregressive moving average), AR (autoregressive) and MA (moving average) models, as well as the “L-Markovian” field model, also called the “Gauss-Markov” (GM) model in the “Gaussian” case. We will explain why the causal models in their representation are subject to restrictions due to the geometry of their support. We will also see that all these models constitute specifications of random 2-D fields, which are regular and stationary to the second order, in accordance with the estimation of the power spectral density (PSD) that they allow us to obtain (this aspect will be covered in Chapter 3). After this presentation, we will tackle the implementation of the models by describing the estimation methods of their parameters. Furthermore, this description will lead us to introduce the minimum variance representations (MVR) of random 2-D fields, which are regular (see section 1.6) and stationary to the second order (or stationary in general; see section 1.3.3). We will see that, irrespective of the geometric

Chapter written by Olivier A LATA and Claude C ARIOU.

66


support, it is possible to approach any absolutely rational continuous PSD in a precise manner using this representation and the associated 2-D estimation method called the “normal equation” or “Yule-Walker” (YW) method. We will also consider global and adaptive methods of estimation as well as 2-D lattice representations (arising from 1-D lattice representation). They have been developed with the objective of finding in 2-D cases the properties of digital stability, of representation of covariance sequences and modularity on the order obtained in 1-D cases.

2.2. 2-D ARMA models 2.2.1. Definition A random field Y , which is stationary to the second order, will have a representation called 2-D ARMA if it can be seen as the output of a 2-D LIT filter (see section 1.4.3), which is probably non-causal, excited by a noise E independently and identically distributed, with zero mean (centered), of variance 1. Thus, the general description of this model is a differential equation: am1 ,m2 Y (k1 − m1 , k1 − m2 ) =

Y (k1 , k2 ) + (m1 ,m2 )∈Ds

bm1 ,m2 E(k1 − m1 , k2 − m2 )

b0,0 E(k1 , k2 ) +

(2.1)

(m1 ,m2 )∈De

where De and Ds are prediction supports associated with the input and output, respectively. Later on, we will suppose that the input and output supports coincide and we will denote this using De = Ds = D. It is said that Y is an AR process (resp. MA) if bm1 ,m2 = 0, ∀ (m1 , m2 ) ∈ D (resp. am1 ,m2 = 0, ∀ (m1 , m2 ) ∈ D). The set of AR coefficients (resp. MA) is the set ({am1 ,m2 }(m1 ,m2 )∈D ) (resp. ({bm1 ,m2 }(m1 ,m2 )∈D )). If we apply the 2-D z transform (see section 1.5.4) to equation (2.1), the transfer function of the filter is expressed in the form of a rational fraction of 2-D polynomials in z1 and z2: H (z1 , z2 ) =

B (z1 , z2 ) A (z1 , z2 )

(2.2)

with: B (z1 , z2 ) = b0,0 + (m1 ,m2 )∈D

and: A (z1 , z2 ) = 1 + (m1 ,m2 )∈D

bm1 ,m2 z1−m1 z2−m2

am1 ,m2 z1−m1 z2−m2

The PSD of Y (see section 1.5.5) is written as: # j2πν j2πν #2 1 2 # #B e ,e SY (ν1 , ν2 ) = 2 |A (ej2πν1 , ej2πν2 )|

(2.3)


67

For models whose prediction support is finite, polynomials B and A are of finite degree. It is necessary to mention that contrary to the 1-D case, a PSD of this form does not admit a priori the factorization of a minimum rational phase1, which signifies that a process Y whose PSD is a rational fraction of polynomials of finite degree does not necessarily have a causal finite representation. Having said that, as in 1-D, the main advantage of such a modeling is that any 2-D stationary process can be approximated by a 2-D AR model of finite order [GUY 93, RAN 85] as we will see in section 2.4.2. We now present the different geometries of support D which have been most frequently used in the ARMA model (resp. AR, MA). 2.2.2. 2-D ARMA models and prediction supports 2.2.2.1. Causal models The two main advantages of ARMA, AR and MA 2-D causal models are to offer the possibilities, on the one hand, of using the numerous algorithms of estimation parameters of models whose implementation is relatively easy and, on the other hand, the ability to realize the recursive processes following a lexicographic path, which is associated with any non-symmetric half plane (NSHP). Indeed, certain applications of these models, such as localization of faults in a texture [FRE 97, MEY 96] or reduction of noise in an image [LIU 93], can be based on this property. We will explain the characteristics of a process Y of the 2-D causal ARMA (resp. AR, MA) type of finite order [KOR 85, KOR 86, LOU 89]. The following is the definition of a 2-D causal ARMA model (resp. AR, MA) of finite order. D EFINITION 2.1. According to [LOU 89], let D0 be an NSHP allowing us to define an arrangement of the plane Z 2 (see section 1.6.1). We note that Y is a causal ARMA process for D0 if the causal factorization (in the sense of the order relation associated with D0 ) to a minimum phase of the PSD SY (ν1 , ν2 ) is the quotient of two polynomials whose support D is included in D0: am1 ,m2 Y (k1 − m1 , k2 − m2 ) = (m1 ,m2 )∈D b0,0 v Y,D0 (k1 , k2 ) + bm1 ,m2 v Y,D0 (k1 (m1 ,m2 )∈D

Y (k1 , k2 ) +

− m1 , k2 − m2 )

(2.4)

1. The causal factorization of a minimum phase enables the construction of a recursively stable filter of a recursively stable inverse.

68


where v Y,D0 (k1 , k2 ) is the normalized innovation of Y as compared to the NSHP D0 (see equation (1.129) in section 1.6.1). m2

M2

M1

m1

Sites of the finite causal support zone of cone D*

Figure 2.1. NSHP support and associated cone D∗

Let D∗ be the smallest cone of Z 2 of peak (0,0) containing D (hatched zone in Figure 2.1). Then, H(z1 , z2 ) admits the following development (proposition 2.2 in [LOU 89]): ⎧ ⎪ ⎨ H (z1 , z2 ) =

ϕm1 ,m2 z1−m1 z2−m2 , ϕm1 ,m2 ∈ , ∀ (m1 , m2 ) ∈ D∗

⎪ ⎩

∞

(m1 ,m2 )∈D ∗

(m1 ,m2 )∈D ∗ 2 |ϕm1 ,m2 |
0 ∪ & a1 k&2 − b1 k1 ≥ 0 >0 ∪ & b 2 k1 − a 2 k2 ≥ 0

with a1 b2 −a2 b1 > 0 and (a1 , b1 ) (resp. (a2 , b2 )) first among them, and Y as a regular stationary 2-D process. Then, v Y,D1 (k1 , k2 ) = v Y,D2 (k1 , k2 ) if and only if: . ξ2

∀

ln Sy (ν1 , ν2 ) ej2π(k1 ν1 +k2 ν2 ) dν1 dν2 = 0 (k1 , k2 ) ∈ (D1 − D1 ∩ D2 ) ∪ (D2 − D1 ∩ D2 )

A particular case of this model is, of course, the causal NSHP model of finite order (M1 , M2 ) (see Figure 2.5) defined by the prediction support: % & N SHP = %(m1 , m2 ) ∈ Z 2 /0 < m1 ≤ M1 , m2 = 0 ∪ DM 1 M2 & (m1 , m2 ) ∈ Z 2 / − M1 ≤ m1 ≤ M1 , 0 < m2 ≤ M2

(2.8)

Later on, we will only use the causal model with finite QP or NSHP supports.

72


k2

k2

m1

m1

M1

M2

M2 m2 Non-causal support

M1

m2 k1

Semi-causal support

k1

Figure 2.6. Non-causal and semi-causal prediction supports

2.2.2.4. Semi-causal model The 2-D semi-causal ARMA model (resp. AR, MA) is causal in one direction and non-causal in the other. In the case of the path following the NSHP Dh , the support of such a finite order model (M1 , M2 ) corresponds to the support presented in the right-hand side of Figure 2.6: % & SC DM = (m1 , m2 ) ∈ Z 2 /0 < m1 ≤ M1 , 0 ≤ m2 ≤ M2 , (m1 , m2 ) = 0 (2.9) 1 M2 This model was the object of few studies [JAI 81, RAN 85] as compared to other models presented in this chapter and there are few parametric estimation methods. This can be explained in several ways. First of all, it is less general than the 2-D noncausal AR model. Due to its semi-causality, it allows only 1-D recursive processing, i.e. row by row or column by column. There is no correspondence with the 1-D AR model. Indeed the causal and non-causal 1-D AR models have been studied for the past 30 years. From this fact, the extension to the 2-D case could be realized, particularly regarding the parametric estimation methods for the 2-D causal AR model. We will not study this further in this chapter. 2.2.2.5. Non-causal model The prediction support of the 2-D non-causal ARMA model (resp. AR, MA) of finite order (M1 , M2 ) is shown in Figure 2.6 and is defined by the following equation: NC = DM 1 M2 & % (m1 , m2 ) ∈ Z 2 / − M1 ≤ m1 ≤ M1 , −M2 ≤ m2 ≤ M2 , (m1 , m2 ) = 0

(2.10)


73

The non-causal AR model was sometimes referred to as a simultaneous autoregressive (SAR) model [KAS 83, MAO 92]. This model was the object of several studies as, in the cases where we suppose that the noise is Gaussian, it offers the possibility of using the parameter estimation algorithm of relatively easy implementation [KAS 83, ZHA 93]. It is necessary to add that, contrary to the 1-D case, where any ARMA process (resp. AR, MA) of finite order can be written equally in causal or non-causal finite form, in the 2-D case, a 2-D non-causal ARMA process (resp. AR, MA) of finite order can in certain cases admit only one 2-D causal ARMA representation (resp. AR, MA) of infinite order [GUY 93]. Moreover, this model does not present the limitations of the causal model which were presented in section 2.2.2.1. It refers to the most general 2-D ARMA model. 2.3. L-Markovian fields 2.3.1. 2-D Markov fields and L-Markovian fields Before introducing 2-D Markov fields (MFs), it is necessary to define the concept of neighborhood [CRO 83, GEM 84, WAN 94]. Let S0 be a finite sub-set of S for which we arrange the sites by using the lexicographic index (s = k2 × K1 + k1 which corresponds to the couple (k1 , k2 )): S0 = {s1 , s2 , ..., sK } with K = K1 × K2 . V = {Vs , s ∈ S0 } is a neighboring system on S0 if: 1) s ∈ / Vs , ∀s ∈ S0 2) ∀(s, t) ∈ S02 , s ∈ Vt ⇔ t ∈ Vs Thus, the couple (S0 , V ) represents a graph and Vs is called the neighboring set of s. In the case where S0 is the support of a digital image, the different outcomes in each pixel of the gray level can be perceived as a family of random variables indexed on S0 , which we denote by Y = {Ys , s ∈ S0 }, each Ys being a value in Δ2. For the moment, we leave out the hypotheses about the neighborhood at the limits% of the image. We will denote the & possible set of configurations for Y by: Ω = y = {ys }s∈S0 / ∀s ∈ S0 , ys ∈ Δ . D EFINITION 2.2. According to [GEM 84] Y is a 2-D Markov field on (S0 , V ) if: 1) positivity condition: P r(Y = y) > 0 for any y ∈ Ω; 2. Δ will be equal to R or C in the case of the L-Markovian field or to [1, . . . , R] in the case of the region field (see section 2.6.1).

74


2) Markov property: P (Ys = ys |Yr = yr , r = s ) = P (Ys = ys |Yr = yr , r ∈ Vs ); 3) homogenity condition: p (ys |Vs ) does not depend on the neighboring configuration and is translation-invariant. Later on, we bring back the notion of neighbors around a site s to the set of NC NC where DM is the non-causal prediction support defined by equa{Ys−r }r∈DM 1 M2 1 M2 ∗ tion (2.10). If we define Hs (Y ) = sp {Yr /r = s}, the space created by the random variables Yr , r = (k1 , k2 ) ∈ Z 2 , other than Ys , Y , which is the stationary field of the second order, is a L-Markovian field [GUY 93, LOU 89, WAN 94, WOO 72] if and only if for any outcome y = {ys }s∈I of Y we can write: ⎧ cr Ys−r , cr = c−r ⎨ Es = Ys − Ys /Hs∗ (Y ) = Ys + NC r∈DM (2.11) M 1 2 ⎩ Es ⊥ Yr if r = s From equation (2.11), the PSD of Y can be obtained [GUY 93]: SY (ν1 , ν2 ) =

σe2 1+ NC (m1 ,m2 )∈DM 1 M2

cm1 ,m2 e−j2π(m1 ν1 +m2 ν2 )

(2.12)

with σe2 as the variance of E. The random process Es is thus a colored noise of PSD: ⎛ ⎞ ⎜ ⎟ cm1 ,m2 e−j2π(m1 ν1 +m2 ν2 ) ⎠ SE (ν1 , ν2 ) = σe2 ⎝1 + NC (m1 ,m2 )∈DM M 1

(2.13)

2

It is necessary to note that a 2-D (causal, semi-causal or non-causal) AR model can be written in the form of a Markovian L-field ([GUY 93] or [WAN 94] for the equations, making it possible to go from the 2-D AR model to the Markovian L-field model). However, contrary to the 1-D case, the opposite is not true, which shows that this model of random 2-D fields is more general than the 2-D AR model. Moreover, when E is a Gaussian process, then Y is a Gauss-Markov process.

2.3.2. 2-D L-Markovian fields and Gibbs fields The use of Gibbs field (GF) in image processing [GEM 84] has allowed us to tackle different applications such as the restoration of images [GEM 84] or even the segmentation of textured images [BOU 91, KER 95] with Markovian approaches.


75

Indeed, they represent an alternative to estimate or synthesize a Markovian field as they are not specified by local statistical characteristics but by global potentials. From the formalism based on the MF and the GF, methods such as simulated annealing (SA) or the iterative conditional mode (ICM) could be applied to the image in order to optimize criteria such as the maximum a posteriori (MAP) [BOU 91] or the maximum posterior mode (MPM) [GUY 93]. We will consider these notions later on (see section 2.6). Before defining a GF, it is necessary to introduce the clique notion: a sub-set C of S0 (C ⊂ S0 ) is a clique if any pair of sites of C is constituted of neighboring sites. We denote the set of cliques of S0 by S. Order 1

4-connected

Order 2

Order 3

Order 4

8-connected

Figure 2.7. Different neighboring systems on an associated regular and clique grid

Figure 2.7 gives two examples of neighboring systems on a regular grid with cliques that we can define from these neighborhoods. The order of a clique is equal to the number of sites which it brings into play. Now, we can define a Gibbs distribution. D EFINITION 2.3. According to [GEM 84] a Gibbs distribution on (S0 , V ) is a probabilistic measure π on Ω which is presented in the following manner: U (y)

e− T (2.14) π (y) = Z where Z is a normalization constant, and T and U can been seen respectively as the temperature and the energy function. U and Z are respectively written as U (y) VC (y) and Z = e− T . The set of functions VC , C ∈ S0 , which are U (y) = C

y

all functions on Ω, is called a potential.

76


Now we can give the following theorem, which is a simplified version of the Hamersley-Clifford theorem [BES 86], which renders MF and GF equivalent. T HEOREM 2.3. According to [GEM 84] let V be a neighboring system. Y is a MF on V if and only if p(y) = P (Y = y) is a Gibbs distribution on V . An example of the utilization of Gibbs distributions in segmentation of textured images will be given in section 2.6.

2.4. “Global” estimation methods There are three big families of parametric estimation methods for 2-D models: first of all, methods based on the writing of the probability density (PD) of the process to be studied, such as those based on the criteria of maximum likelihood (ML); then, methods said to be global, such as the Yule-Walker equations or normal equations for which the parametric estimation is performed by taking into account all the data of the image; and finally, the adaptive methods, for which the parametric estimation is recursively realized, from site to site by following the route of the image. Later on, we will see that the various estimation methods arising from the 1-D signal theory and adapted to 2-D are those which made the 2-D AR model a great success. Moreover, certain algorithms like the Levinson algorithm (at the root of a particular resolution method of Yule-Walker equations), whether it is in 1-D or in 2-D, allows us not only to reduce the complexity of the calculations, but also to obtain other existence conditions and of stability of 2-D filters than those proposed in section 1.5.4.6.

2.4.1. Maximum likelihood 2.4.1.1. Estimation criteria by supposing the fixed order The estimation methods based on the expression of ML [COH 91] often present three disadvantages. First, their development is based on the hypothesis that the process to be analyzed possesses a PD which is known a priori. We can thus express a conditional probability for this process whose maximization provides a parametric estimation of the model associated with the process. Secondly, these methods require a “good” initialization of the parameters of the model before the optimization procedure. Indeed, one of the major drawbacks of these methods is the risk of obtaining a set of parameters which corresponds to a local maximum of the conditional probability. Thirdly, the more the order of the models increases, the more difficult the estimation becomes. One of the main advantages is that these methods are applicable to all the models which have been previously presented.


77

Let Y = {Ys , s%∈ S0 } be a 2-D random field and & Ω the set of possible configurations for Y : Ω = y = {ys }s∈S0 /∀s ∈ S0 , ys ∈ Δ . We suppose that we have one or several outcomes of Y . If the nature of the PD of Y is/ known and depends on an 0 unknown set of parameters noted by θ (for example, θ = {am1 ,m2 }(m1 ,m2 )∈D , σe2 for a 2-D AR model by supposing the order of D as known), the law of conditional probability of y can be noted by f (y |θ ). The estimation of set θˆml in the sense of the ML criteria is the set which maximizes density f (y |θ ): θˆml (y) = arg max f (y |θ )

(2.15)

θ

In the case of a 2-D causal AR model whose excitation is a white noise, we will see that the writing of this criterion is easily simplified. By supposing that D is one of the causal supports of finite dimensions previously presented (equations (2.7) and (2.8)), it is possible to write the law of conditional probability of the observations as the product of the laws of conditional probability of the estimated prediction errors eˆs on any of the sites s of the image:

f (y |θ ) =

f (ˆ es |θ )

(2.16)

s∈S0

where we define the outcome of the field of prediction errors from an estimation of θˆ in the following manner: a ˆm1 ,m2 y (k1 − m1 , k2 − m2 ) (2.17) eˆD (k1 , k2 ) = y (k1 , k2 ) + (m1 ,m2 )∈D

Indeed, if we suppose that Yˆθ (k1 , k2 ) = − a ˆm1 ,m2 Y (k1 − m1 , k2 − m2 ), " (m1 ,m2 )∈D ! we will have P (Y |θ ) = P Y − Yˆθ |θ = P (Eθ |θ ) where Eθ is the excitation of the 2-D AR causal model. Thus, according to equation (2.17), set {ˆ es }s∈S0 can be considered as an outcome of it. If this excitation is assumed to be Gaussian in nature for any site s, it is possible to write: ! # " # f y #θˆ =

(k1 ,k2 )∈S0

2

e ˆ (k1 ,k2 ) 1 − D 2ˆ 2 σe e 2 2πˆ σe

(2.18)

where eˆD (k1 , k2 ) depends on the observed field y and the set of parameters θˆ on the estimated model. σ ê2 is the estimation of the variance of the prediction error. Equation (2.18) can be re-written by taking the opposite of the Napier logarithm for the probability function; we thus talk of log-probability: 2 ! ! # "" 1 2 eˆD (k1 , k2 ) #ˆ + ln σ ê + ln (2π) (2.19) = − ln f y #θ 2 σ ê2 (k1 ,k2 )∈S0

78

Two-Dimensional Signal Analysis 2

We can write: (k1 ,k2 )∈S0

eˆD (k1 , k2 ) ≈ K1 K2 σ ê2 . For the minimization of the

criteria, it is not necessary to take into account the constant terms of equation (2.19). We obtain: 2 ê θˆml = arg min σ (2.20) θ

Thus, it is necessary to estimate, in the case of the 2-D AR causal Gaussian model, the parameters {am1 ,m2 }(m1 ,m2 )∈D which will allow us to minimize the estimation of the variance of the prediction error (we will return to this problem in section 2.4.2). It is necessary to emphasize that until now the order (M1 , M2 ) of causal D was assumed to be known. If we try to use the criterion proposed in equation (2.20) to estimate this order, we will, in the majority of cases, obtain a considerable order, as the variance of the prediction error has a tendency to decrease as soon as the prediction support increases. In the following section, we discuss the main approach that allows us to resolve the problem of the order estimation of the prediction support. 2.4.1.2. Probability criteria “penalized” to estimate the order of the model The information criteria (IC) are the tools which have become very common to estimate the order of the parameter models (AR, Markovian), and also for the selection of the models, or for the classification and non-supervised segmentation [BOU 91, OLI 99]. Initially proposed to obtain the order of the 1-D AR model [AKA 74], IC gave substance to numerous works with the objective of overcoming the main drawback of the Akaike criteria, i.e. its non-convergence. In general, IC differ only by their “penalty” factor C(Q) (equation (2.21)), where Q is the number of observations. We can also write these criteria in a general manner, which is said to be penalized (or probability) entropy, as follows: ! # " # kˆ = arg max [IC(k)] = arg max −2 ln f y #θˆk,ml + C(Q)k k

k

(2.21)

with θˆk,ml as the !set# of "k free parameters which minimize the expression of log# probability − ln f y #θˆk . Here, y is the sequence of Q observations, Q = K1 × K2 , of the process that we wish to model with the help of parametric law θˆk with k free parameters. The retained number kˆ is that which minimizes IC(k) (equation (2.21)). Later on, there is no question of justifying the writing of the different penalties which have been proposed but rather to give a non-exhaustive enumeration of these penalties. The Akaike criterion [AKA 74], obtained from the Kullback-Leibler distance, is written in the following manner C(Q) = 2, where even C(Q) = l (for example, l = 3 in [BOZ 94]), since l can sometimes be proposed without any justification. The most frequently used criterion in other recent works is the Schwarz


79

BIC (Bayesian Information Criterion) [SCH 78]: C(Q) = ln(Q). This criterion is used to estimate the order of the 2-D AR non-causal model [KAS 83] and a comparable writing is obtained [RIS 86] by the MDL (Minimum Description Length) approach which is based on the notion of stochastic complexity. The formulation C(Q) = ln(ln(Q)) has been proposed [HAN 79] but converges only in probability. Arising from the minimization between θ and θˆk from a Kullback-Leibler measurement and from Rissanen’s stochastic complexity [RIS 86], the consistent criterion φβ [MAT 96, JOU 98], C(Q) = Qβ ln(ln(Q)), 0 < β < 1, is a justified compromise between BIC [SCH 78] and [HAN 79]. In addition, its expression generalizes that of the BIC [ALA 00]. The expression of the BIC for a 2-D AR model of order (M1 , M2 ), studied on an image of size (K1 , K2 ), gives: ! " ! # " ˆ 1, M ˆ 2 = arg min −2 ln f y ##θˆk,ml + ln (K1 K2 ) g (M1 , M2 ) (2.22) M with θˆk,ml =

(M1 ,M2 )

/

{ˆ am1 ,m2 }(m1 ,m2 )∈D , σ ˆε

0 and g(M1 , M2 ) = k, as the number of

QP the free parameters of the model. For a QP causal model, we have: D = DM 1 M2 (equation (2.7)), and g(M1 , M2 ) = (M1 + 1)(M2 + 1); for an NSHP causal model, N SHP (equation (2.8)) and g(M1 , M2 ) = (2M1 + 1)M2 + M1 + 1. D = DM 1 M2

The extension of the criterion φβ to the 2-D AR model gives [ALA 00]: " ! ˆ 1, M ˆ2 = M ! # " # β arg min −2 ln f y #θˆk,min + (K1 K2 ) ln (ln (K1 K2 )) g (M1 , M2 )

(2.23)

(M1 ,M2 )

with

ln(ln(Q)) ln(Q)

≤β ≤1−

ln(ln(Q)) ln(Q) .

2.4.2. Yule-Walker equations The parametric estimation methods of 2-D linear stochastic models using YuleWalker equations (also called 2-D normal equations) is the method most commonly employed by users. These equations, as in the 1-D case, create a link between the first autocorrelation function (see section 1.3.3) and the parameters of the model. Relying on this estimation method, a set of theorems which determines the validity of this method for the representation of any random 2-D process has been developed [RAN 85]. Indeed, the representation obtained by its use is what we call a minimum variance representation (MVR) of finite order, which, as we will see, coincides according to the geometry of the support with certain previously presented 2-D stochastic linear models. These MVRs allow us, for a given precision, to approximate any rational PSD on condition that the order of the prediction support is correctly chosen.

80


2.4.2.1. Representation of minimum variance and formulation We call the linear predictor of Y (k1 , k2 ), YˆD (k1 , k2 ), according to a neighborhood D of Y (k1 , k2 ) called prediction support, the random variable: am1 ,m2 Y (k1 − m1 , k2 − m2 ) (2.24) YˆD (k1 , k2 ) = − (m1 ,m2 )∈D

The field ED = Y − YˆD , of prediction errors associated with D is thus written in conformity with equation (2.17). The set {am1 ,m2 }(m1 ,m2 )∈D is called the set of prediction coefficients associated with D. The filter which uses Y as an input and ED as an output will be called the analysis filter in the following. As a transfer function it has: A (z1 , z2 ) = 1 + am1 ,m2 z1−m1 z2−m2 (2.25) (m1 ,m2 )∈D

and the z transform of Y is written as: Y (z1 , z2 ) =

σe2 A (z1 , z2 )

(2.26)

D EFINITION 2.4. According to [RAN 85] the predictor YˆDmin (k1 , k2 ) with minimum variance (later noted by YˆD (k1 , k2 )) is that which minimizes the variance σe2D (noted later by σe2 of the prediction error). The random sequence Y thus admits a representation of minimum variance as compared to the prediction support D: Y (k1 , k2 ) = YˆD (k1 , k2 ) + ED (k1 , k2 ) where ED is a random 2-D sequence which allows us to adjust the properties of the sequence of the first autocorrelation function, γY (m1 , m2 ), (m1 , m2 ) ∈ Z 2 , to the sequence of predictors YˆD . ED satisfies the following orthogonal condition: E [ED (k1 , k2 ) Y (k3 , k4 )] = σe2 δ (k1 − k3 , k2 − k4 )

(2.27)

The orthogonal condition of equation (2.27) associated with the MVR allows us to write the equation systems known as the Yule-Walker 2-D equations [RAN 85, YOU 94]: " E

Y (k1 , k2 ) +

P (m1 ,m2 )∈D

# am1 ,m2 Y (k1 − m1 , k2 − m2 ) Y (k1 − k3 , k2 − k4 )

(2.28)

= σe2 δ (k1 − k3 , k2 − k4 )

for (k3 , k4 ) ∈ D ∪ (0, 0), from where: γY (k3 , k4 ) +

(m1 ,m2 )∈D

am1 ,m2 γY (k3 − m1 , k4 − m2 ) = σe2 δ (k3 , k4 ) (2.29)


81

D is equally the non-causal, semi-causal and causal prediction support of finite or infinite order. This system of equations can also be obtained by minimizing the variance σe2 of the prediction error as compared to the coefficients {am1 ,m2 }(m1 ,m2 )∈D . This parametric estimation criterion is called the least squares (LS) criterion and corresponds to the minimization of the quadratic mean error: 2 E ED arg min {ˆ am1 ,m2 }(m1 ,m2 )∈D = (2.30) {am1 ,m2 }(m1 ,m2 )∈D The set {ˆ a } 1 ,m2 )∈D is calculated with the system of equations (2.29) as m1 ,m2 (m ! " 2 ∂E ∂ED D ˆ = E 2ED ∂a∂E = E 2 Y − Y and the set E ∂am D,m ∂am1 ,m2 m1 ,m2 1 2 {ˆ am1 ,m2 }(m1 ,m2 )∈D nullifies the different partial derivatives. 2.4.2.2. Non-causal support and L-Markovian fields Here we suppose that D = DN C , the non-causal infinite prediction support, A (z1 , z2 ) = 1 + am1 ,m2 z1−m1 z2−m2 and a−m1 ,−m2 = am1 ,m2 . (m1 ,m2 )∈DN C

The multiplication by E(k1 − k3 , k2 − k4 ) in equation (2.28), instead of Y (k1 − k3 , k2 − k4 ), allows us to obtain the first autocorrelation function of the prediction error: ⎞ ⎛ am1 ,m2 δ (n1 − m1 , n2 − m2 )⎠ γE (n1 , n2 ) = σe2 ⎝δ (n1 , n2 ) + (m1 ,m2 )∈DN C

(2.31) Thus, the sequence of prediction errors {Es }s∈S is not white and its PSD is equal to σe2 A (z1 , z2 ). Here the sequence {Es }s∈S corresponds to a type of 2-D MA model with a non-causal support. The variance of the prediction error is given by:

−1 −1 2 σe = (2.32) SY (ν1 , ν2 ) dν1 dν2 ξ2

In the case where we limit the size of the prediction support to an order (M1 , M2 ), the orthogonality condition allows us to obtain the system: 2 1 ,M2 γY (n1 , n2 ) + aM m1 ,m2 γY (n1 − m1 , n2 − m2 ) = σM1 ,M2 δ (n1 , n2 ) NC (m1 ,m2 )∈DM M 1

2

(2.33) NC ∪ (0, 0) (see equation (2.10)), which can be re-written in the for (n1 , n2 ) ∈ DM 1 M2 matrix form as: NC 2 NC ΓY,NC M1 ,M2 aM1 ,M2 = σM1 ,M2 1M1 ,M2

(2.34)

82


where: – ΓY,NC M1 ,M2 is a Toeplitz block matrix called the 2-D Yule-Walker matrix; T – 1NC M1 ,M2 = [0, . . . , 0, 1, 0, . . . , 0] is a column vector of dimension (2M1 + 1)× (2M2 + 1); NC – aNC M1 ,M2 is a column vector of the same dimension as 1M1 ,M2 containing the % M ,M & 1 ,M2 = 1, arranged line by line, parametric set am11,m22 (m ,m )∈DN C , with aM 0,0 1

2

column by column or conversely.

M1 M2

Matrix ΓY,NC M1 ,M2 is defined as positive [ALA 98] and the system of equations (2.34) allows a single solution for the set of coefficients: NC θM 1 ,M2

=

%

1

&

1 ,M2 aM m1 ,m2 (m1 ,m2 )∈D N C M M 1

2

2 , σM 1 ,M2

The number of different elements of matrix ΓY,NC M1 ,M2 is 2(2M1 + 1)(2M2 + 1) − NC (2M1 + 1) − (2M2 + 1) + 1 while θM1 ,M2 is a set of M1 (2M2 + 1) + M2 + 1 free M1 ,M2 NC 1 ,M2 parameters as aM m1 ,m2 = a−m1 ,−m2 , ∀ (m1 , m2 ) ∈ DM1 ,M2 . NC Thus, the set of coefficients θM corresponds to an infinity of the first possible 1 ,M2 autocorrelation functions. This result will also be true in the case where the prediction support is semi-causal or causal. Nevertheless, it is possible to formulate a theorem for each representation which assures the validity of the approach in each case [RAN 85]. According to equations (2.26) and (2.31), a non-causal MVR of order (M1 , M2 ) allows us to exactly establish the PSD of an L-Markovian field whose neighborhood in a NC . In the case of modeling of any 2-D pixel would exactly cover the support DM 1 ,M2 random process, the z transform of the prediction error of order (M1 , M2 ) will have the following form: 2 EM1 ,M2 (z1 , z2 ) = σM + EYM1 ,M2 (z1 , z2 ) AM1 ,M2 (z1 , z2 ) (2.35) 1 ,M2

where EYM1 ,M2 (z1 , z2 ) expresses the z transform associated with the first intercorrelation function between {Ys }s∈S and {EM1 ,M2 (s)}s∈S . Thus, the following definition for a non-causal minimum variance model associated with any random process Y is proposed. D EFINITION 2.5. According to [RAN 85] a non-causal model type, of minimum variance, of order (M1 , M2 ), for a process Y is a non-causal MVR of order (M1 , M2 ) for another process Y˜ where:


Y˜ (k1 , k2 ) = NC (m1 ,m2 )∈DM M 1

where

% M ,M & am11,m22 (m

1 ,M2 ˜ ˜ aM m1 ,m2 Y (k1 − m1 , k2 − m2 ) + EM1 ,M2 (k1 , k2 )

83

(2.36)

2

NC 1 ,m2 )∈DM M 1 2

is calculated by solving the system of equations

˜M ,M is a 2-D MA process whose PSD is written as (2.34) and process E 1 2 σe˜2 A (z1 , z2 ). Here we note the properties of such a model: 2 M1 ,M2 1 ,M2 aM m1 ,m2 = a−m1 ,−m2 NC 1) M1 ,M2 (m1 , m2 ) ∈ DM 1 ,M2 1 ,M2 a−m1 ,m2 = aM m1 ,−m2 2 2 2 2 2) σM ≥ σM and σM ≥ σM 1 ,M2 1 +1,M2 1 ,M2 1 ,M2 +1

3) if Y is L-Markovian with a PSD, which is expressed in the following manner: SY (ν1 , ν2 ) =

σ2 cm1 ,m2 e−j2π(m1 ν1 +m2 ν2 )

1+ NC (m1 ,m2 )∈DM M 1

(2.37)

2

then the non-causal model with minimum variance of order (M1 , M2 ) allows the exact representation of this PSD. In addition, the theorem [RAN 85] assures the convergence of a sequence of increasing order of non-causal models towards any rational PSD. This theorem allows the use of a minimum variance non-causal model of finite order (equivalent to a type of L-Markovian model) for analysis. Indeed, a sequence of increasing order of such a model will converge towards the modeling of a process whose PSD has as an asymptotic value that of the 2-D random field to be analyzed on condition that it is absolutely continuous and rational. 2.4.2.3. Causal support and 2-D AR model The causal factorization with minimum phase has been elaborately discussed [EKS 76] as the problem of its existence is more difficult to establish in 2-D than in 1-D, due to the fact that there is no theorem assuring that any 2-D polynomial is factorable. However, these authors have shown that there was an infinity of possible causal factorizations (QP or NSHP) which depended on the order of the route of the image. Thus, the causal factorization of a holomorphic PSD associated with a causal representation of minimum variance of infinite order is written as: S (z1 , z2 ) =

σe2 A (z1 , z2 ) A z1−1 , z2−1

(2.38)

84


with A (z1 , z2 ) = 1 + (m1 ,m2 )∈DN SHP

am1 ,m2 z1−m1 z2−m2 .

For any holomorphic PSD, there is a factorization such as that of equation (2.38) for which the filter associated with A(z1 , z2 ) is stable and at a minimum phase, [RAN 85], i.e., it allows a stable and causal inverse. In addition, A(z1 , z2 ) can be factorized into a product of two polynomials, whose supports are two adjacent QPs [EKS 76]. In the case of QP or NSHP supports of finite order, we obtain a matrix similar to that of equation (2.34). the Yule-Walker matrix obtained with NSHP support and We call ΓY,NSHP M1 ,M2 Y,QP ΓM1 ,M2 that which is associated with a QP support [ALA 98]. QP We also note by 1NSHP M1 ,M2 and 1M1 ,M2 the vector columns of dimensions M2 (2M1 + 1) + M1 + 1 and (M1 + 1)(M2 + 1) respectively. QP Since ΓNSHP M1 ,M2 (resp. ΓM1 ,M2 ) is defined as positive, this assures, as before, the existence and the unity of the set:

C θM 1 ,M2

=

%

1

&

1 ,M2 aM m1 ,m2 (m1 ,m2 )∈D C M

1 M2

2 , σD C M

1 M2

which contains M2 (2M1 + 1) + M1 + 1 elements when C = N SHP (resp. (M1 + 1)(M2 + 1) elements when C = QP ). Since the number of different elements in ΓY,NSHP (resp. ΓY,QP M1 ,M2 M1 ,M2 ) is 2(M2 + 1)(2M1 +1)−(M2 +1)−(3M1 +1)+1 (resp. 2(M2 +1)(M1 +1)−(M2 +1)−(M1 + 1) + 1), this procedure does not assure a priori a correspondence of the representation N SHP SHP and ΓN between the sets θM M1 ,M2 (it is the same for the QP support). 1 ,M2 Similarly to the non-causal model, a theorem concerning the use of the NSHP causal models for the modeling of the random fields to holomorphic PSD guaranteeing the convergence of the covariance sequences has been proposed [RAN 85]. Finally, if Y is a 2-D AR causal NSHP process (resp. QP) of order (M1 , M2 ), the causal NSHP MVR (resp. QP) of order (M1 , M2 ) allows us to exactly represent its PSD.


85

2.4.2.4. Extension to the 2-D AR non-causal model We have seen (see definition 2.5) that the use of the Yule-Walker equations for the parametric estimations of a model at non-causal minimum variance of order (M1 , M2 ) is equal to the parametric estimation of a model of the L-Markovian field type which has an identical support. In the case of a 2-D AR non-causal process (also called an SAR model or 2-D AR NC), the parametric estimation with the help of the YuleWalker equations is biased [WAN 94]. In order to solve this problem, two estimation methods have been developed. First, the Yule-Walker estimations were used [KAS 83] in order to obtain a set of initial parameters of an iterative method based on an approximation of the log-probability (see section 2.4.1), then, more recently, a separate estimation of the causal and non-causal parts of the SAR model from a reformulation of the Yule-Walker equations was proposed [ZHA 93, ZHA 95]. These authors have thus compared their new method to that of the ML approached by [KAS 83]. They have shown that from an estimation point of view the two methods are equivalent, while from a calculative viewpoint, it presents them with a lower complexity of calculation. 2.4.2.5. Extension to the 2-D ARMA model Since the application of the ARMA models (equation (2.1)) to the 2-D data is relatively underdeveloped in other works, the problem of the parametric estimation of this model remains largely open even today. In 1-D, an extension of the YW equations [KAY 88] allows us to provide a parametric estimation of the AR and MA parts. A first phase provides the parameters of the AR part, then in the course of the second phase the parameters of the MA part. The bi-dimensional implementation of this algorithm was proposed for the 2-D spectral estimation by 2-D ARMA modeling [CAR 91] with a QP support (see Chapter 3). However, it is not the only existing approach to estimate the parameters of this model [AKA 73, BUZ 96].

2.4.3. 2-D Levinson algorithm (for the parametric 2-D AR estimation) The Levinson algorithm [LEV 47] helps us to solve an equation system such as that in (2.34) without making a direct calculation of the inverse of the covariance matrix which is, in the stationary case, a Toeplitz matrix. Applied to the parametric estimation of a 1-D AR causal model, this algorithm enables us to obtain a particular structure of the linear filter composed by a succession of identical cells called a lattice structure [LEV 84, MAR 76]. The main properties of this approach in 1-D are to provide a higher digital robustness as compared to the direct use of the YW equations, to provide a recursive and modular structure (i.e., the calculation of a filter structure of order P + 1 does not imply a recalculation of all the cells up to the order P ), as well as

86


the possibility of directly testing from the lattice parameters the BIBO stability of an estimated filter (see section 1.4.7). The extension of the 1-D lattice representation to the 2-D case is not immediate. Nevertheless, a certain number of works have been carried out with the objective of obtaining the properties previously quoted in 2-D and resolving the theoretical problem of the unicity of the representation between the 2-D YW matrix and the parameters of the 2-D AR model (see section 2.4.2). In what follows (sections on global estimation and adaptive estimation), we will only explain the most interesting solutions obtained from the set of these properties. 2.4.3.1. Recalling the 1-D case The 1-D AR causal model is described by the following equation: Y (k) = −

P

am Y (k − m) + E (k)

(2.39)

m=1

where the set {Y (k)}k∈Z is a family of random variables (see section 1.3.7.1), which belongs to L2 (Ω, A, P ) and constitutes a 1-D AR causal, homogenous and stationary to the second order process and the set {E (k)}k∈Z is a white noise of zero mean. The order of the model is an integer noted by P . This order allows us to define a 1-D prediction support. Thus, we call a prediction error of order P of any process the process: EP (k) = Y (k) +

P

am Y (k − m)

(2.40)

m=1

Henceforth, we will call process EP , now noted by EPf , a direct prediction error of order P and the set of coefficients {am }1≤m≤P , henceforth noted by / 0 afm,P , the direct transverse coefficients of order P . The direct and back1≤m≤P

ward prediction errors of order P (see Figure 2.8) can thus be defined as follows: EPf (k) = Y (k) +

P

afm,P Y (k − m)

(2.41)

m=1

EPb (k) = Y (k − P ) +

P

abm,P Y (k − P + m)

(2.42)

m=1

& & % % The parametric estimations afm,p 1≤m≤p and abm,p 1≤m≤p , for 1 ≤ p ≤ P , by the minimization of the LS criterion lead to the recurrence relation on the order


87

between the prediction errors of order p and p-1 for 1 ≤ p ≤ P [STR 90] if Y is a stationary value in : f b Epf (k) = Ep−1 (k) − kp Ep−1 (k − 1)

(2.43)

f b Epb (k) = Ep−1 (k) (k − 1) − kp Ep−1

(2.44)

with E0f (k) = E0b (k) = Y (k). The coefficients {kp }, 1 ≤ p ≤ P , are called reflection coefficients (because of a possible analogy with the propagation of waves in tubes of identical length but with different sections [MAR 76]) or partial correlation (PARCOR) coefficients. In01-D, the / f unicity of the representation exists between the sets {kp }1≤p≤P and am,P 1≤m≤P

the continuous covariances in the matrix which comes into play in the 1-D YuleWalker equations. By taking the z transform in equations (2.43) and (2.44), we obtain the following recursive polynomial 1 ≤ p ≤ P: f b (z) − kp z −1 Ep−1 Epf (z) = Ep−1 (z)

(2.45)

f b Epb (z) = z −1 Ep−1 (z) (z) − kp Ep−1

(2.46)

with E0f (z) = E0b (z) = Y (z). It is necessary to note that it is possible to find a “+” in equations (2.43) and (2.44), instead of “-” as in [MAR 76] and other works. We have chosen the convention which is most often used in 2-D [CAS 97, LIU 93, LIU 96].

Direct prediction to order p to t Backward prediction to order p to t−1 Direct prediction to order p+1 to t

t

Figure 2.8. Structure of the recursion

Figure 2.8 summarizes the structure of the recursion in which we express a polynomial of order p+1 according to the two polynomials of order p. These polynomials are called Szegö orthogonal polynomials or orthogonal polynomials [GAM 79] on the unit circle considering the scalar product between z polynomials $ P (z) , Q (z) =

dz P (z) Q (z) Y (z) Y z −1 2πjz

88

Two-Dimensional Signal Analysis P

where Y (z) = 1/A(z) and A (z) = 1 + p=1

afp z −p , A (z) having all the roots inside

the unit circle. Indeed, this scalar product calculated on the unit circle (z = ej2πν ) gives [MAR 76]: P (z) , Q (z) =

0,5

P ej2πν Q ej2πν 2

|A (ej2πν )|

−0,5

dν

(2.47)

and the following properties can be verified: 3

4 A (z) , z −l = 0, l ∈ [1, P ] 3 f 4 3 4 Ep (z) , z −l = Epb (z) , z −l = 0, ∀l ∈ [1, p] from where we have the naming of 3the orthogonal polynomials on the unit circle. In 4 addition, it is necessary to note that z −k , z −l = γY (k − l) is close to the definition of the scalar product of L2 (Ω, A, P ) applied to a stationary 1-D random process of zero mean {Y (k)}k∈Z : Yk , Yl = E [Yk Yl∗ ] = γY (k − l). From these definitions, the reflection coefficients are obtained in the following manner, 1 ≤ p ≤ P: 6 5 f b (z) , Ep−1 (z) Ep−1 kp = ' ' 'E b (z)'2

(2.48)

p−1

Different expressions can be found in other works according to the chosen approach [LEV 84]. In 2-D, the recursion expression given by equations (2.41) and (2.42) does not appear in a direct manner. Thus, it is possible to find different propositions in other works for its extension to the 2-D case.

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

+

+

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

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

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

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

+

+

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

Figure 2.9. Block diagram of the 1-D lattice analysis filter

The structure of the lattice analysis filter is given in Figure 2.9. It is a succession of cells which possess an identical structure. In addition, the lattice filter can be considered as a result of the orthogonalization procedure such as that by GramSchmidt on the set of data at the input of the filter. One of the consequences of this


89

orthogonalization is that obtaining the filter of order P +1 is performed by simply adding a supplementary cell to the filter of order P . Thus, the parametric estimation of the filter of order P results in the parametric estimation of all the filters of orders less than P while, for the transverse form, all the calculations are redone for each order. The lattice synthesis filter (see Figure 2.10) of a 1-D process is obtained from the following equations: f b (k) = Epf (k) + kp Ep−1 (k − 1) Ep−1

(2.49)

f b Epb (k) = Ep−1 (k) (k − 1) − kp Ep−1

(2.50)

always with E0f (k) = E0b (k) = Y (k). y (k)

.. .. .. .. .. .. .. .. .. 21 .. z .. .

. e f (k) ... e f1 (k) .. ..................... P21 .. . k1 .. .. k1 .. .. .. ..................... + .. .. b e bP21 (k) . e1 (k) +

.. .. .. .. .. .. .. .. z21 ..

.. f .. e P (k) .. .. kP ... . kP ... .. .. + .. .. e bP (k) +

Figure 2.10. Block diagram of the 1-D lattice synthesis filter

The conditions necessary for the 1-D causal filter estimated by the Levinson algorithm to be recursively stable and with inverse recursively stable (i.e. minimum phase) are simply expressed by the use of the lattice structure. It is enough that all the reflection coefficients are strictly less than 1 in order for the conditions to be verified [MAR 76, STR 90]. In the 2-D case, the extension of this property from different approaches in the existing lattice is still the subject of research. In addition, it constitutes one of the main advantages in the development of these approaches, whose objective is to simplify the conditions of existence of the 2-D analysis and synthesis filters which are difficult to verify with the help of the transverse form and the writing of the corresponding transfer function in the form of an inverse of a 2-D polynomial in z1 and z2 . 2.4.3.2. Approach for 2-D causal and non-causal prediction models From a 1-D recursion at each stage, the 2-D Levinson algorithm was extended to the causal and non-causal support types [CAS 97]. The recursion is made by following a path of the sites of the prediction support (see Figure 2.11) and these authors propose an approach based on the Hilbert space L2 (Ω, A, P ) of random variables of summable squares equipped with the scalar product Y, X = E [Yc Xc∗ ].

90


m2

m2

M2

M2

M1

M1

m1

(a)

m1

(b)

Figure 2.11. Possible routes for the polynomial recursion: a) NSHP support; b) non-causal support

Process Y is defined on the probabilistic space (Ω, A, P ) (see section 1.3.6). Let S0 be a sub-set of Z 2 , card(S0 ) = N + 1, enumeration N being possibly equal to infinity. We define the application Υ : {0, 1, . . . , N } → S0 which allows us to create an order of the path (an enumeration) of S0 , {Υ (0) , Υ (1) , ..., Υ (N )}. In the example of Figure 2.11 a, the set S0 is enumerated in the following manner: {Υ (0) = (0, 0) , Υ (1) = (1, 0) , ..., Υ (N ) = (3, 2)} We thus note that% Υk,l = {Υ (k) , Υ (k & + 1) , . . . , Υ (l)} where 0 ≤ k < l < N and Hk,l (Ys ) = sp Ys−Υ(r) /k ≤ r ≤ l as Hilbert sub-space of L2 (Ω, A, P ). The following residual of direct and backward prediction residuals can thus be defined for any (k, l), 0 ≤ k < l < N: f = Ys−Υ(k) − Ys−Υ(k) /Hk+1,l (Ys ) Ek,l

(2.51)

b Ek,l = Ys−Υ(l) − Ys−Υ(l) /Hk,l−1 (Ys )

(2.52)

where Yk /Hk,l (Ys ) is the projection sub-space Hk,l (Ys ). From 0 / of0 Yk on/ the Hilbert f b this fact, the residual sequences Ej,l and Ej,l , for k ≤ j ≤ l, are orthogonal to the sub-space Hk,l (Ys ). The multidimensional Levinson recursion for any (k, l), f b = Ek,k = Yk is written as: 0 ≤ k < l < N and Ek,k f f f b Ek+1,l − kk,l = Ek,l−1 Ek,l

(2.53)

f b b b Ek,l = Ek+1,l − kk,l Ek,l−1

(2.54)

f = with kk,l

f f b ,Ek+1,l E b ,Ek,l−1 Ek,l−1 b and kk,l = k+1,l . 2 2 f b Ek+1,l Ek,l−1


91

In addition, we can show that the set of reflection coefficients allows us to reconstruct the first autocorrelation function of process Y and contrary to the transverse coefficients (see section 2.4.2). 6 5 ' b ' f ' k + Yk − E f (2.55) , Y γY (Υ (k) − Υ (l)) = 'Ek+1,l l k,l−1 k,l This structure results in the minimization of norms for any of the direct and backward prediction errors [CAS 97]. This result is very important as it allows us to say that, from any Y process, the determination of the reflection coefficients amounts to constructing the closest minimum variance model to Y for the set of supports brought into play in the recursion. 2.4.3.3. Multichannel approach and 2-D QP AR model The multichannel approach of the 2-D linear prediction [ELS 87, GAM 79, THE 81] is based on the existing equivalence between the covariances contained in matrix ΓY,QP M1 ,M2 (see section 2.4.2.3) and those of the matrix which allow us to obtain the multichannel transverse coefficients of a multichannel constructed on the QP support: Ys =

M1

(Am1 )T Ys−m1

(2.56)

m1 =1

with: T

Ys−m1 = [Y (k1 − m1 , k2 ) , Y (k1 − m1 , k2 − 1) , . . . , Y (k1 − m1 , k2 − M2 )] and Am1 matrix (M2 + 1) (M2 + 1).

Thus, the direct usage of the multichannel Levinson algorithm [WIG 65] can be derived [THE 81] in order to obtain the 2-D transverse prediction coefficients from the multichannel coefficient matrices. The recursion structure uses the multichannel reflection coefficient matrices for relations between the multichannel direct and backward prediction error vectors. Figure 2.12 represents the multichannel extension of Figure 2.8. From this multichannel approach of the 2-D linear prediction, the previous recursion can be expressed [GAM 79] in the form of an orthogonal polynomial recursion on the unit circle. This writing has allowed us to produce the existence conditions for the analysis filters and the multichannel synthesis which is expressed from the direct and backward matrices of the reflection coefficients which we note here by Kfm1 and f b Kb m1 : 0 ≤ λm2 (Km1 Km1 ) < 1 for 1 ≤ m1 ≤ M1 and 1 ≤ m2 ≤ M2 + 1, where λm2 (A) denotes the mth 2 eigenvalue of A. It has also shown that the multichannel Levinson algorithm assures the minimization of the norms for all the direct and backward prediction errors as well as a reduction in the value of these norms from the order m1 to m1 + 1.

92

Two-Dimensional Signal Analysis m2

m2 M2

M2

m1

m1

m 1 +1

m1

m2 M2

m 1 +1

m1

Figure 2.12. Multichannel structure of the recursion associated with the QP support

2.4.3.4. Other approaches Regarding the “global” algorithms based on the 2-D lattice representations, the first consists of [MAR 80] in a calculated recursion on the 1-D approach but infinite in one direction. This approach possesses interesting properties but presents a major disadvantage: the support of the reflection coefficients is not identical to that of the obtained 2-D transverse filter, which makes its usage difficult. The second approach [PAR 84], developed for the QP support, is a 2-D polynomial recursion in four directions; in addition, it is known by the name of 2-D lattice approach to four fields. This structure presents a major disadvantage: even though it was developed from a QP support, it does not enable the representation of all the filters associated with the 2-D QP AR models. In order to overcome this disadvantage, an improvement of the recursion was proposed [ERT 95] by introducing intermediary backward prediction errors. However, this improvement did not allow us to completely solve the problem. More recently, an optimal lattice structure in four fields [KAY 97] was developed. This approach was generalized for different types of prediction supports [KAB 96], i.e. QP, NSHP and semi-causal, and a recent approach [KAY 96] has proposed a 2-D extension of the Burg algorithm for causal models based on a 1-D recursion of the kind used in [CAS 97] (see section 2.4.3.2).


93

2.5. “Adaptive” or “recursive” estimation methods The 2-D adaptive estimation algorithms are extensions of 1-D adaptive algorithms which have been developed to estimate the transverse coefficients or the reflection coefficients associated with a causal 1-D AR model. The denomination “adaptive” signifies that the parametric estimation of the model is made recursively by routing the signal from its first sample to its last, which implies the use of causal models. The parameters are all supposed to converge towards an asymptotic value in the case where the 1-D signal is stationary in the broad sense. These methods make it possible to reduce, according to the problem approached, the calculative complexity as compared to the Yule-Walker equations (see section 2.4.2) and robust implantations on specialized processors, especially in their lattice form. The 2-D adaptive algorithms have been developed according to different estimation criteria. They will be introduced here with a quick presentation of the 1-D adaptive estimation algorithms for transverse coefficients. We will then recall the existing 2-D adaptive algorithms for each criterion. 2.5.1. Connectivity hypotheses for adaptive or recursive algorithms For the parametric estimation of causal models, a “cylindrical” connectivity has been proposed [LIU 96, SEQ 90, YOU 86]. This connectivity is used to maintain “continuity” for the filling of a QP or an NSHP support during a lexicographic path of the image S0 (see Figure 2.13 for a QP support). The limit conditions of the image are expressed as ∀ (k1 , k2 ) ∈ S0 , ∀m0 > 0, such that −K1 < k1 − m0 < 0, ∀n0 > 0, such that k2 − n0 < 0: Y (K1 + k1 − m0 , k2 − 1) , k2 > 0 (2.57) Y (k1 − m0 , k2 ) = Y (k1 − m0 , 0) = 0 Y (k1 , k2 − n0 ) = 0, k2 − n0 < 0

(2.58)

2.5.2. Algorithms 2.5.2.1. Methods based on a stochastic gradient The expression of the causal 1-D AR model (equation (2.39)) can be written in a matrix form. The prediction error of order P at instant k is expressed in the following manner: EP (k) = Yk + aP T (k) YP (k)

(2.59) T

T

with aP (k) = [a1 (k), a2 (k), . . ., aP (k)] and YP (k) = [Yk−1, Yk−2, . . . , Yk−P ] , where ap (k), 1 ≤ p ≤ P are the transverse coefficient values estimated at

94


s

s+s 1 Vertical path direction

s+s 1+s2

Figure 2.13. Lexicographic path of the image by a QP support

instant k along the path of the signal following increasing k during an adaptive estimation procedure. The parametric estimation by using the LS criterion (see section 2.4.2) 2 means minimizing the partial derivatives of E EP (k) as compared to the transverse coefficients: ∂ E EP2 (k) = 2E [EP (k) YP (k)] = 0 ∂ aP (k) The gradient vector is expressed by: ∇ (k) = 2E [EP (k) YP (k)]

(2.60)

The calculation of the transverse coefficients at instant k+1 can be done according to the coefficients at instant k in the following manner: aP (k + 1) = aP (k) + μE [EP (k) YP (k)]

(2.61)

Direct estimation of the gradient from the set of data requires a lot of calculations. A practical method [WID 76] – named least mean squares (LMS) – estimates the ˆ (k) = 2eP (k) yP (k). It refers to an algorithm gradient by its instantaneous value: ∇ which offers a simple implementation and a good numerical stability concerning the rounding errors. It has seen so many extensions that it would not be appropriate to


95

mention it here. 2-D transverse and lattice adaptive algorithms have been developed with the help of a similar approximation (because of this, they possess all the LMS denominations). Among them, it is necessary to mention the 2-D LMS algorithm [HAD 88] which follows exactly, irrespective of the causal support used, the same development as in 1-D and thus the two-dimensional adaptive lattice LMS (TDALLMS) algorithm was developed [YOU 86, YOU 92] from the lattice structure in four fields [PAR 84]. 2.5.2.2. Methods based on the recursive least squares criterion The recursive least squares (RLS) algorithm [HAY 86, KUN 91, NAJ 88, STR 90] are based on the LS criterion. These optimization methods consist of minimizing, at each iteration (on k for the 1-D signal, on s for the image S0 ), a cost function J(k) (or J(s) for the image) which is chosen as the sum of the squares of errors from the beginning of the recursion (instant k (resp. site s) = 0): J (k) =

k

yi + aP T (k)yP (i)

(2.62)

i=0

By nullifying the derivatives of J(k) as compared to the transverse coefficients, we obtain the following matrix equation: −1

aP (k) = (ΓyP (k)) with ΓyP (k) =

k i=0

y(k)

(2.63)

γP

y(k)

T

yP (i) yP (i) and γP

k

= i=0

yi yP (i). Equation (2.63) is the

YW equation estimated at instant k. Due to this, in the 2-D case, this approach could be used by taking into account the properties concerning the causal models and the YW estimation type (see section 2.4.2.3). y(k)

If the signal is stationary, ΓyP (k) and γP contain the estimations of the first autocorrelation function of Y . The optimal values of the transverse coefficients are obtained for k tending towards infinity. The calculation of the coefficients at instant k+1 from those estimated at instant k can be made with the help of the following equations: gP (k + 1) =

ΓyP (k) yP (k + 1)

(2.64)

T

1 + yP (k + 1) ΓyP (k) yP (k + 1) T

ΓyP (k + 1) = ΓyP (k) − gP (k + 1) yP (k + 1) ΓyP (k) aP (k + 1) = aP (k) + gP (k + 1)

!

T

yk+1 + ˆ aP (k) yP (k + 1)

(2.65) " (2.66)

96


QP support of order (M1,M2) Prediction vector direct to S Backward prediction vector associated with the QP support of order (M1,M2) to S Figure 2.14. QP support and vectorial approach

For more details on the development of this approach, see [ALE 86, CAR 83, CIO 84, HAY 86]. The 2-D extensions of this algorithm are based on a multichannel approach of the 2-D linear prediction [BOU 89] by lightly modifying the NSHP support. The algorithms based on the RLS criterion (see equation (2.62)) have the advantage of a faster convergence of the parameters towards their asymptotic values, which leaves us to assume that they will be better adapted when a few samples of the process to analyze are available. In addition, the introduction of a forgetting factor in the development [ALE 86, LIU 93] allows us to use them in the case of a non-stationary signal. 2.5.2.3. Methods based on the geometric approach of the RLS criterion The concepts presented in this section use the minimization of the norm of the orthogonal projection on the prediction sub-space residuals [LEV 84, LIU 93, SEQ 89]. Because of this, we will find expressions close to those used in formulae (2.51) and (2.52). Two algorithms have been developed for the estimation of the QP causal and NSHP AR models: the 2-D FRLS (2-D fast recursive least squares) [SEQ 89, SEQ 90] and the 2-D FLRLS (2-D fast lattice recursive least squares) [LIU 93, LIU 96]. The second is the “lattice” version of the first. They are based on the matrix of the RLS criterion as well as on a vectorial approach of the causal supports (see Figure 2.14 for the QP case – we will content ourselves with the presentation of this case later on).


97

Figure 2.15. Non-zero content of y m (s) 2

T

Let ym2 (s) = [ym2 (s) , ym2 (s − 1) , . . . , y (0) , 0, . . . , 0] be the vector (see Figure 2.15 by following the arrows), with s = k1 + k2 × K1 as the index of the lexicographic path of the image (thus we have ym2 (s) = y (k1 , k2 − m2 )). This vector is size K1 × K2 , i.e. the size of the image studied. Broadly speaking it contains the past of “ym2 ” following the path defined by s completed by zeros. T Thus, we will have ym2 (s + 1) = [ym2 (s + 1) , ym2 (s) , . . . , y (0) , 0, . . . , 0] in the same dimension as ym2 (s) but with at least a zero at the end. The vector y0 (K1 × K2 − 1) will be the only one to contain all the intensities of the covered image. Thus, we define the following quantities (0 ≤ m1 ≤ M1 ): Yf (s) = [y1 (s + 1), . . . , yM2 (s + 1), y0 (s)]

(2.67)

f ,s = [y1 (s), . . . , yM2 (s) , y0 (s − 1), y1 (s − 1), . . . , yM2 (s − 1), . . . Ym 1 y0 (s − m1 ), y1 (s − m1 ), . . . , yM2 (s − m1 )]

(2.68) b,s = [y1 (s + 1), . . . , yM2 (s + 1) , y0 (s), y1 (s), . . . , yM2 (s), . . . Ym 1 (2.69) y0 (s − m1 + 1), y1 (s − m1 + 1), . . . , yM2 (s − m1 + 1)] b Ym (s) = [y0 (s − m1 ), y1 (s − m1 ), . . . , yM2 (s − 1)] 1

(2.70)

The matrix Yf (s) is associated with the space generated by the direct prediction f ,s , 0 ≤ m1 ≤ M1 , represent the direct prevector since the beginning of the path. Ym 1 diction spaces following increasing orders and are formed by the vectors contained

98


in the QP supports of increasing order (m1 , M2 ) at index s. Similarly, we define b (s), which is the concatenation of the vectors contained in the backward predicYm 1 b,s , 0 ≤ m1 ≤ M 1 , tion vectors of order (m1 , M2 ), 0 ≤ m1 ≤ M1 , as well as Ym 1 for the spaces associated with the backward prediction vector. For the 2-D FRLS f ,s b,s b , YM (s) and YM are used. In addition, algorithm, only the matrices Yf (s), YM 1 1 1 it will be the same for the matrices for prediction errors, which will now be presented. The objective of the two algorithms is to minimize the RLS criterion by minimizing the traces of the constructed matrices with the help of the estimations of elements of the first autocorrelation functions of backward and direct prediction errors. The latter is obtained in the following manner: f f f f Em1 (s) = Π⊥ f ,s Y (s) = Y (s) − Π f ,s Y (s) Ym Ym1 1 (2.71) b b b ⊥ Eb m1 (s) = ΠY b,s Ym1 (s) = Ym1 (s) − ΠY b,s Ym1 (s) m1

T where Π⊥ X = Id − ΠX and ΠX = X X X

−1

m1

XT .

The X matrix is composed of more lines than columns and the vector columns are linearly independent (a condition necessary and sufficient in order for XT X to be reversible). Matrices constructed with the help of estimations of elements of first autocorrelation functions of backward and direct prediction errors are hence written T T b as [Efm1 (s) Efm1 (s)] and [Eb m1 (s) Em1 (s)]. ! " ! " T T b The minimization of tr Efm1 (s) Efm1 (s) and tr Eb (s) E (s) allows m1 m1 us to obtain the parameters of the model for m1 = M1 in the case of the 2-D FRLS and for 0 ≤ m1 ≤ M1 in the case of the 2-D FLRLS. We now give the formulae of the lattice for 1 ≤ m1 ≤ M1 which we can obtain by means of orthogonalities between the different vectorial spaces [ALA 98, LIU 96]: f f Em1 (s) = Efm1 −1 (s) − Eb m1 −1 (s − 1) Km1 (s) (2.72) b b f Em1 (s) = Em1 −1 (s − 1) − Em1 −1 (s) Kb m1 (s) with: ⎧ ! "−1 ⎪ T T f b ⎨ Kfm (s) = Eb Eb (s − 1) E (s − 1) m1 −1 (s − 1) Em1 −1 (s) −1 −1 m m 1 1 1 ! "−1 T T ⎪ f f ⎩ Kb Efm1 −1 (s) Eb m1 (s) = Em1 −1 (s) Em1 −1 (s) m1 −1 (s − 1) (2.73) For m1 = 0, we have: f E0 (s) = Yf (s) − Y0f ,s Kf0 (s) b,s b b Eb 0 (s) = Y0 (s) − Y0 K0 (s)

(2.74)


99

Figure 2.16. Structure of the recursion

with: ⎧

−1 ! ! "T "T ⎪ f ,s f ,s f ⎪ Y0f ,s Yf (s) Y0 Y0 ⎨ K0 (s) =

−1 ! ! "T "T ⎪ b,s b,s ⎪ ⎩ Kb Y0b,s Y0b (s) Y Y (s) = 0 0 0

(2.75)

It is necessary to note that the matrices of the reflection coefficients Kfm1 (s) and Kb m1 (s), 1 ≤ m1 ≤ M1 are of dimension (M2 + 1) × (M2 + 1), while the matrices Kf0 (s) and Kb 0 (s) are of dimension (M2 ) × (M2 + 1). This recursion can be represented in Figure 2.16. It is possible to obtain in s the current values of all the vectors previously presented by using the vector π = T [1, 0, . . . , 0] of length K1 × K2: f y (s) = [y1 (s + 1) , . . . , yM2 (s + 1) , y0 (s)] = π T Yf (s) (2.76) b b (s) = [y0 (s − m1 ) , . . . , yM2 (s − m1 )] = π T Ym (s) ym 1 1 f f ˆm (s) em1 (s) = π T Efm1 (s) = yf (s) − y 1 (2.77) b T b b b ˆm (s) em1 (s) = π Em1 (s) = ym1 (s) − y 1 for: 0 ≤ m1 ≤ M1 .

100


Obtaining the transverse parameters of the 2-D AR model is performed by the intermediary of the 2-D z transform [ALA 98] (see section 1.5.4). Among the properties of this lattice approach, for example, is the possibility of reconstructing the first autocorrelation function of the random field from the matrices of the reflection coefficients. These algorithms function very reasonably with the calculation time and great calculating robustness, even for high orders of (M1 , M2 ) (for example, (30, 30)). 2.6. Application: segmentation of textured images Image segmentation consists of establishing a partition of the image support in homogenous regions, each one having different characteristics. Recently, numerous methods have been developed from the Bayesian estimation techniques (see, for example, [BOU 91, BOU 94, CAR 96, COH 92, THE 86, ZHA 94]). Some of these methods are based on the usage of the 2-D AR model: the image is considered as the mix of R random 2-D fields, stationary in the broad sense (see section 1.3.3), each one of which can be characterized by a 2-D AR model. In practice, we will talk of a 2-D AR model, characterizing one of the R textures presented in the image. Causal models (NSHP and QP, respectively) are used because of the balance [THE 86, BOU 91] between the MVR and these models when their parameters are estimated by the 2-D Yule-Walker equations (see section 2.4.2). Indeed, the main advantage of this estimation method is simple implementation and a fast calculation time for the supports of weak orders (for example, (1,1)). Let us recall that the parameters of the 2-D NC AR model cannot be obtained directly from the writing of the same equations in the non-causal case. On the contrary, a GM model can be used [KRI 97]. The image of the regions is modulated in the form of a hidden MF. In [THE 86], different textured fields are modeled by the 2-D NSHP AR models and an algorithm optimizing a criteria of the MAP type is used for the segmentation. From this work, a multiscale structure type has been introduced [BOU 91] for the original image and the image of the regions in order to improve the optimization of the same MAP criterion. These authors have used the QP support instead of the NSHP support and called their segmentation algorithm Multiple Resolution Segmentation (MRS). We have chosen to present this method, which is enriched by the use of the information criterion φβ (see section 2.4.1.2), as it gives highly satisfactory results both on images containing synthetic textures and natural Brodatz textures.

2.6.1. Textured field and segmented field As previously mentioned, the digitized image is denoted by Y , and is supposed to contain R different textures. In order to separate Y into its R regions, the MRS


101

algorithm is based on the following three main points: the textures are modeled by the stochastic 2-D QP AR Gaussian models3, the field of the regions is modeled by a hidden MF and a Gaussian pyramid structure is used for the original textured field and the region field at the same time. A texture r, 1 ≤ r ≤ R, modeled bya 2-D QP AR model will be characterized by 1 % r & 2 am1 ,m2 (m ,m )∈DQP , σr , μr , where the following parametric model: θr = 1

2

M1 M2

μr is the mean of the gray levels of the texture r and σr2 , is the variance of the excitation QP noise of the model4. We recall that DM is the finite prediction support of order 1 M2 (M1 , M2 ) (equation (2.7)). In order to write the following, we assume that the different characterization sets associated with the textures are known. We will also see the solution proposed by Bouman to estimate R and the parameters of the models in a non-supervised manner. Let X be the field of the regions, of the same dimensions as Y , which contain the attributed class in each site of the image. The sites of X thus take their values in the set Δ = {1, ..., R}. The conditional probability density function of Y knowing X, f (y/x), thus exists and we note by p (x) the a priori probability, P (X = x) x being an outcome of X. The MAP type estimation of X is calculated in the following manner: x ˆM AP = arg max {P (X = x/Y = y)} x

(2.78)

which can, by using the Bayes formula and the opposite of the logarithm, take us to: x ˆM AP = arg min {U (x/y)} x

(2.79)

where U (x/y) ∝ − ln (f (y/x)) − ln (p (x)). f (y/x) can be expressed as the product of the probability density function of the prediction errors (see equation (2.17)), supposedly white and Gaussian, in each site s to which the class r is attributed by means of the algorithm, x ˆs = r: 2 e2s 1 7 exp − 2 (2.80) f (y/x) = 2σxˆs 2πσ 2 s∈S

x ˆs

3. An order (0, 0) corresponds to modeling the textures by the Gaussian white noise to which a mean is added. 4. In the case of an order (0, 0), θr = σr2 , μr .

102


The distribution of X, p (x), is modeled by a Gibbs distribution whose equivalence with MF we have seen (see section 2.3): p (x) =

1 exp {−β1 t1 (x) − β2 t2 (x)} Z

(2.81)

We use the neighborhood of type 8-connectivity (Figure 2.7) to define the neighborhood system to the hidden MF of the regions and we choose to use only cliques of order 2. t1 (x) = Vh (x) + Vv (x) is thus the number of the sites in the image having different values from their direct neighbors, horizontally or vertically and t2 (x) = Vd1 (x)+Vd2 (x) the number of sites having different values with their direct neighbors diagonally. Vh , Vv , Vd1 and Vd2 respectively are the horizontal, vertical and diagonal cliques (see section 2.3). β1 and β2 respectively are chosen to be equal to 0.414 λ and 0.293 λ [BOU 91], where λ is considered as the unit cost of length for the boundaries between different regions. From what follows, for U (x/y) we get: U (x/y) =

ls (y/xs ) + β1 t1 (x) + β2 t2 (x)

(2.82)

s∈S

where ls (y/xs ) =

1 2

/

e2s 2 σx s

0 + ln σx2s + ln (2π) .

It was possible to eliminate the normalization constant Z appearing in equation (2.81) from the Gibbs distribution as we suppose, in this part, that the number of R textures is fixed. To estimate this number R and the parameters of the model, the Gibbs distribution will be replaced by a multinominal distribution (see section 2.6.3). For the minimization of the U (x/y) function, we propose using the method employed in [BOU 91]: an initialization of the field of the regions is obtained by allocating to each site the texture r, 1 ≤ r ≤ R, which minimizes ls (y/xs ). It refers to the ML type of estimation of the segmented field. Other types of initialization could be envisaged. Next, the final segmentation is calculated by using the Iterative Conditional Mode (ICM) algorithm. This algorithm consists of a local maximization of U (x/y): x ˆs = arg min {ls (y/xs ) + β1 v1 (xs ) + β2 v2 (xs )}

(2.83)

xs

where v1 (xs ) (resp. v2 (xs )) is the number of sites of different labels of s for the horizontal and vertical cliques (resp. diagonal cliques). As longas there are changes


103

in labels, the image is covered and the segmentation is optimized, site by site. It is necessary to note that a lexicographic route of the image would inevitably result in an anisotropy in the final result. A simple solution to rectify this defect is to separate the sites of the image into sub-sets where each element of the sub-set does not belong to any neighborhood of another element. The MAP criterion is then optimized by the ICM by taking the sub-sets one after the other or in parallel. For a neighborhood 8-connectivity, the image is split into four sub-sets. There are other algorithms to minimize the MAP criterion (equation (2.79)). Among others, we can mention the stochastic iterative algorithms: the SA (or Gibbs sampler) [GEM 84] or the Metropolis algorithm in a recent version [BAR 98], the reversible jump Monte Carlo Markov Chain (MCMC). Contrary to the ICM, these algorithms perform a convergence towards the global minimum, on the condition that they are parametrized correctly, which is not always possible. Moreover, in general, they have considerable convergence duration as compared to the ICM. Later on, we will see a method to improve the convergence and the calculation time of this algorithm simultaneously.

2.6.2. Multiscale or hierarchical approach The improved results that we can obtain with the algorithm previously presented is given in [BOU 91] by using a multiscale structure (quadtree) (see Figure 2.17). Besides, other works have dealt with image segmentation regarding the usage of a hierarchical representation of the image [BOU 94, KRI 97]. (0)

As mentioned before, the R different values of the function ls (y/xs ) (see equation (2.82)) are calculated at each site s of the image, where level 0 signifies the level of the highest resolution and the resolution decreases with ν (ν being the (ν) level in the multiscale structure). ls (y/xs ) are thus calculated for level 1 then 2 for the different textures by using the pyramid structure of Figure 2.17. For a top-down type of calculation, the minimization of the MAP criteria is thus achieved, at each level, with the help of the ICM logarithm, since the final segmentation is obtained at level 0. The initialization to the level of the lowest resolution (here ν = 2) of the region fields is calculated in the same way as previously (i.e. by using the (2) = arg min ls (y/xs ), 1 ≤ xs ≤ R). Thus, an ICM algorithm is applied ML: x ˆν=2 s xs

at level 2. Up to level 0, the region fields obtained by the ICM at level ν provides the initial region fields at level ν − 1 by the intermediary of the pyramid structure. Now we are going to see how to estimate the number of classes and the model parameters.

104


Y

(2)

Y (1)

Y=Y (0) Figure 2.17. Multiscale structure (quadtree)

2.6.3. Non-supervised estimation of the parameters Simplifying the Gibbs distribution by a multinominal distribution, i.e. by initially neglecting the influence of the neighborhood in the segmentation, will enable us to write an information criterion (of maximum likelihood “penalized”; see section 2.4.1.2) associated with the image. The minimization of this criterion will provide us with the first segmentation by blocks to the lowest resolution level. From this, we will be able to estimate the number of R textures and the parameters of the models of these textures. For other non-supervised approaches, see the theses by Barker [BAR 98] and Mô [MO 98]. The latter work presents different approaches to the EM (expectation maximization) algorithm. 2.6.3.1. Multinominal distribution The image, at the lowest resolution level, is first divided into blocks. The hypothesis of a multinominal distribution of the textures associated with these blocks means that we assume that their real order is not taken into account and thus they have been randomly re-arranged. In addition, we suppose that we have a texture for each block. This assures a number of R maximum textures in the image. From this, the algorithm will carry out the fusion of the regions and verify, for all iterations, the validity of the attribution of a block to a region. Let S (ν) be the grid associated with the splitting of the image into blocks, Ar ⊂ S the set of blocks associated with the texture r, 1 ≤ r ≤ R and Wr the cardinal (ν)

of Ar . The cardinal of S (ν) is thus W =

R

Wr . The probability a priori in the r=1


105

r image of the texture r is thus estimated by ρˆr = W W . In the multinominal distribution, the texture r is characterized by this/parameter. The parameter model of a 0texture in the image is thus written as θr = {am1 ,m2 }(m1 ,m2 )∈DQP , σr2 , μr , ρr and the M1 M2

R

parametrized model of the image defined by the partition {Ar }r=1 is thus expressed R by θR = {θr }r=1 , where R is part of the parameters to estimate. For a given R, the model contains PR = R [(M1 + 1) (M2 + 1) + 2] − 1 parameR ters. The joint conditional distribution of y and of the partition {Ar }r=1 knowing θ is written as [BOU 91]: pθR (y, A1 , . . . , AR ) =

R

exp {−L (Ar , θr ) + Wr ln (ρr )}

(2.84)

r=1

with L (Ar , θr ) =

ls (y/r). s∈Ar

2.6.3.2. Information criterion ˆ we can use an information criterion. As we have In order to estimate θˆR and R, seen earlier, these criteria are written as the ML criterion to which a penalty term dependent on the number of the free parameters in the model is added (see section 2.4.1.2). In [BOU 91], the Akaike criterion has been used. It has been proved since that this criterion is not consistent. That is why it has been replaced in the simulations with the φβ criterion [MAT 96]: φβ = −2 ln (pθR (y)) + Cφβ (Q) .PR

(2.85)

where pθR (y) is the conditional distribution of y knowing θR , PR is the number of the free parameters of the model, Q is the number of pixels and Cφβ (Q) = Qβ ln (ln (Q)) with β = ln(ln(Q)) ln(Q) [CAP 00]. As the use of pθR (y) is “difficult”, Bouman proposed to replace it in the criterion (equation (2.85)) with pθR (y, A1 , . . . , AR ) (equation (2.84)). Thus, when we merge the two regions Ac = Ak ∪Al , we obtain a variation in the criterion which is expressed in the following manner: ) − 2L (A"k , θk ) − 2L!(Al , θl ) " c (k, l) = 2L (Ac , θc! k + 2Wk ln WkW+W + 2Wl ln l − 2 [(M1 + 1) (M2 + 1) + 2]

Wl Wk +Wl

(2.86)

Here we suppose that the orders (M1 , M2 ) of the R 2-D QP AR models used are equal and do not vary. Any fusion which gives a negative value will result in the

106


Average = 121

(a)

(b)

(c)

(d)

Figure 2.18. a) Original image containing two synthesis textures; b) histogram of the image; c) region fields and d) estimated region fields

minimization of the value of the criterion. Each fusion that is realized will be one which results in the biggest minimization of the criterion. ˆ textures, the partition {Ar }Rˆ and the parameters θˆ ˆ will be The number of R r=1 R estimated on the basis of three procedures, each leading to a minimization of the criteˆ {Ar }Rˆ rion. Each of these procedures can be seen as the minimization following R, r=1 or θˆRˆ by fixing two other quantities.


(a)

(b)

(c) Figure 2.19. a) Original image containing five Brodatz textures; b) histogram of the image and c) segmented image

107

108


(a)

(b) Figure 2.20. a) Extract of an SAR image and b) result of the segmentation

2.6.4. Examples of segmentation 2.6.4.1. Synthesis textures The image presented in Figure 2.18a, of size 128 × 128, contains two synthesis textures obtained from the following 2-D QP AR models of order (1, 1): % & {{am1 ,m2 } = {−0.423 ; −0.567; 0.12}} σ 2 = 22.551, μ = 127 and

% & {{am1 ,m2 } = {−0.359 ; −0.925; 0.295}} σ 2 = 10.118, μ = 127

λ has been taken as equal to 1.5 for this image.


109

We can note on the histogram of Figure 2.18b that the separation of the two textures cannot be made with an image segmentation alogrithm by thresholding the histogram as the distributions of the two textures are “superposed”. Figure 2.18d gives the image segmentation of the synthesis textures. We note that the number of textures has been correctly estimated. We also note that the algorithm made it possible, at the same time, to detect that one of the two textures was contained in the non-connected region, which is split into four parts as is shown in the region fields to be obtained (Figure 2.18c). 2.6.4.2. Brodatz textures Now, we give an example with natural textures taken from [BRO 66]. The image of size 256 × 256 (Figure 2.19a) contains five textures which, we can easily see, are not perfectly stationary. As before, we provide the histogram (Figure 2.19b) of this image. Here again, we note that it contains very little information to be able to separate the image into regions. The algorithm, with λ = 2.5 gave the image Figure 2.19c which contains six regions. Indeed, the “wooden” texture has been separated into two. This result, however, is not the opposite of the visual observation. 2.6.4.3. SAR image Figure 2.20 shows an image made by synthetic aperture radar (SAR) from the ERS-1 satellite (for more information, see [MAI 08]). The imaged scene represents an ice zone floating on the ocean close to the Antarctic. The textures here are representative of the difference in the roughness perceived by the radar. The nonsupervised algorithm automatically detects two classes and gives a faithful enough segmentation of the considered zone.

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Chapter 3

2-D Spectral Analysis

3.1. Introduction The spectral analysis of signals occurs in many application fields, both in the 1-D case [CAS 06] and in the general m-D case. Multidimensional and 2-D signals in geophysics, radioastronomy, radar, sonar and other applications use spectral analysis systematically for the visualization and interpretation of phenomena underlying the signals. As seen in earlier chapters, the processing, and particularly spectral analysis, of 2-D signals presents more difficulties both in theory and practice compared with the 1-D case. In the early 1980s 2-D spectral analysis reached its peak with the development and use of “modern” spectral estimation techniques. The advent of the maximum entropy technique, similar to linear prediction in 1-D but different in m-D, stimulated considerable research on powerful non-linear algorithms. However, since the mid-1990s, no significant research has been done in this area. Nevertheless, 2-D spectral analysis is an effective tool in the new fields of application, bringing into play increasingly complex sensors (passive or active imagery, “intelligent” antennae, etc.).

Chapter written by Claude CARIOU, Stéphanie ROUQUETTE and Olivier ALATA.

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In this chapter, after having located the problem, we will review the techniques now widely used in 2-D spectral analysis, distinguishing them from the traditional techniques: the parametric and semi-parametric approaches and high-resolution methods. We will then try to describe and compare the effectiveness of these techniques based on some essential criteria in spectral analysis, such as the ability to detect and estimate harmonic components and adequacy of PSD (power spectral density) theory for broadband 2-D processes. Finally, we will illustrate the use of a parametric technique in the image analysis of spatial remote sensing.

3.2. General concepts As seen in Chapter 1, a 2-D digital signal is an outcome (or a trajectory) in a finite set of a process defined in network Z 2 , which signifies that: – the set means are inaccessible (we often have one trajectory); – the exact calculation of autocorrelation function (ACF) γX (m1 , m2 ) is impossible, assuming a large stationary process, since the signal is of finite duration. Of course, these problems are not specific to 2-D. On the other hand, if we have an estimation of ACF on a finite subset of Z 2 , nothing guarantees the existence of its valid (positive semi-definitive) duration, i.e. leading to a non-negative PSD. This problem is specific to multidimensional signals. We clearly see the emergence of a major problem presented by the determination of the PSD of a random field (see Chapter 1) from a finite outcome: this must be the objective of an estimation. The quality of a statistical estimator is essentially judged by its characteristics of bias and variance, and in the case of spectral estimator SˆX , its concepts must be specified. First, a PSD is a function defined in finite spectral support ξ 2 (see Chapter 1). Therefore, we measure the bias by the difference in each spatial frequency between estimation and the true value of the PSD: B[SˆX (ν1 , ν2 )] = E[SˆX (ν1 , ν2 )] − SX (ν1 , ν2 )

(3.1)

and the estimation variance is defined by: Var[SˆX (ν1 , ν2 )] = E[(SˆX (ν1 , ν2 ) − E[SˆX (ν1 , ν2 )])2 ]

(3.2)

Of course, these two quantities are important for the spectral estimation of any signal, but there are other characteristics which allow us to evaluate the quality of a spectral estimator. Indeed, we often consider spectral estimation as a tool for the analysis of periodic or quasi-periodic processes. In this case, we know that PSD theory consists of Dirac impulses or lobes (or peaks) which are generally broadly based on the


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signal considered and whose positions in the spectral field characterize the harmonic components or highlight the dominant spatial frequencies of the 2-D random field. For example, if the signal originates from one or more sinusoids or complex exponentials (harmonic signals), the problem of spectral estimation boils down to that of the estimation of frequency parameters and the power of these harmonic components, which are highlighted in the theoretical PSD by Dirac impulses, to the corresponding 2-D frequencies. When we apply a spectral estimation method to such signals, the resulting PSD, if it effectively contain peaks, must be studied at least under the two following aspects. The quality of the spatial frequency estimation and the power of the periodic signal must be examined. Through the application of spectral estimation methods, we should be able to evaluate the resolution quality, i.e. the ability to discriminate its periodicities, by simultaneously detecting several peaks in corresponding frequencies by assuming that the signal contains several periodicities whose spatial frequencies are close. As we will see later, the answer to these two questions – as elsewhere in the 1-D case – is not immediate and depends on various parameters, which are simultaneously relative to the signal considered (signal-to-noise ratio, relative position of peaks in spectral field, observation size) and methods chosen (method type and/or PSD modeling, order of the model for parametric methods). In this chapter, we will describe the most well-known 2-D spectral estimation methods. It will not be necessary to explain each method, but mainly to indicate the modeling or the principle on which it is based as well as its known performances and its relation to methods used in 1-D signal processing. Three classes of methods will be examined: first, the traditional or non-parametric methods which are derived from the definition of the PSD of a process;second, a more extended class, known as parametric methods. This class itself comprises several families of techniques emphasizing either a linear modeling of a random 2-D field (autoregressive (AR) model, autoregressive moving average (ARMA) model, GaussMarkov (GM) model; see Chapter 2), or a set of constraints applicable to the signal (maximum entropy (ME), minimum variance (MV) estimator). These two classes of spectral estimators allow us to deal with problems such as simple or multiple detection of real or complex harmonic noisy components, or even the analysis of the broadband WSS process. There exists a third class, consisting of high-resolution (HR) techniques1, whose MUSIC estimator is a representative. These techniques solely address the problem of spectral analysis of signals which can be modeled as a sum of harmonic components.

1. This qualifier was used in other works regarding certain parametric methods. See the collective work [KOP 98] for a formal definition of high-resolution estimators.

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3.3. Traditional 2-D spectral estimation The traditional methods of spectral estimation are also known as non-parametric methods because they do not represent any modeling of the signal or its PSD. The two common traditional methods are periodogram and correlogram. These techniques are direct durations of the 1-D case. They are based on the 2-D Fourier transform, about which we know (see Chapter 1) that the kernel is separable. Thus, to each dimension of the signal corresponds an axis in the frequency domain. 3.3.1. Periodogram technique The periodogram is the widely used spectral estimator due to the simplicity of its implementation, which relies on algorithm resources of the 2-D rapid Fourier transform (2-D RFT; see Chapter 1). For 2-D signals, the generalized periodogram is defined as follows:

P (ν1 , ν2 SˆX

1 = K1 K2 =

#K −1 K −1 #2 1 2 # # # −j2π(ν1 k1 +ν2 k2 ) # w(k1 , k2 ) x(k1 , k2 ) e # # # # k1 =0 k2 =0

1 2 |FT [w x]| K1 K2

(3.3)

where x(k1 , k2 ) is the 2-D signal observed in a rectangular window 0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1.

Signal w(k1 , k2 ), defined on the same support, is a deterministic function called the window function or smoothing function. By using this function, it is possible during the analysis of harmonic signals to reduce the power of sidelobes, but often at the cost of widening the main lobe, which causes a reduction in the resolution quality of the estimator. Some of the well-known windows are Bartlett, Hamming, Hanning and Parzen windows. They are mostly separable, i.e. obtained from direct product of equivalent windows in 1-D processing, for which a relatively abundant number of works are available (see, for example, [HAR 78]). We will see that there are very few comparative results available relating to the use of 2-D window function [HUA 72]. We show that the periodogram estimator is not consistent. In fact, for a finite number of signal samples, it is a biased estimator. However, it is asymptotically non-biased [BAR 50]. In addition, whatever the number of samples, its variance is approximately proportional to the exact square of the PSD at the frequency considered. This variance


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can be reduced by an N factor by taking the mean of estimations obtained from N windows of the 2-D signal whose covering can go up to 50% without any notable loss of information [JEN 68, WEL 67]. This technique, called the Welch periodogram, allows us to attain better precision in the estimation of 2-D frequencies, but this goes hand in hand with a reduction in the frequency resolution of the estimator.

3.3.2. Correlogram technique The second type of traditional method is the correlogram or Blackman-Tukey spectral estimator [BLA 58]. By directly taking the original definition of the PSD of a process (Wiener-Kinchine theorem; see Chapter 1) and by replacing the ACF theory by one of the possible estimates, possibly observed by a window, the 2-D correlogram is given by: C (ν1 , ν2 ) = SˆX

M1

M2

w(m1 , m2 ) γˆX (m1 , m2 ) e−j2π(ν1 m1 +ν2 m2 ) (3.4)

m1 =−M1 m2 =−M2

where γˆX (m1 , m2 ) is an ACF estimation resulting from the outcome x(k1 , k2 ) of the X(k1 , k2 ) process. This estimation can be biased or non-biased. A biased estimation of ACF guarantees its semi-defined positivity and, for this reason, it is used more often than the non-biased estimation, as given in Chapter 1. If the 2-D signal x(k1 , k2 ) is available in a window (0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1), the biased estimation of ACF is given by: γˆX (m1 , m2 ) = ⎧ K 1 −1 K 2 −1 ⎪ 1 ⎪ ⎪ x (k1 , k2 ) x∗ (k1 − m1 , k2 − m2 ) ⎪ ⎪ ⎪ K K 1 2 ⎪ k1 =m1 k2 =m2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for 0 ≤ m1 ≤ K1 − 1 ; 0 ≤ m2 ≤ K2 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ K 1 −1 K2 −1+m (3.5) 2 1 x (k1 , k2 ) x∗ (k1 − m1 , k2 − m2 ) ⎪ K1 K2 ⎪ ⎪ k1 =m1 k2 =0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for 0 ≤ m1 ≤ K1 − 1; −K2 + 1 ≤ m2 < 0, ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ γˆX (−m1 , −m2 ) for − K1 + 1 ≤ m1 < 0; −K2 + 1 ≤ m2 ≤ K2 − 1, ⎪ ⎪ ⎪ ⎩ 0 elsewhere. The choice of window function w(m1 , m2 ) is not without constraints; in fact, it must also be positive semi-definitive to lead to a non-negative PSD across the spectral

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domain. Moreover, its form must be adapted to the quality required by the estimation, for example, by the frequency parameters in the framework of analysis of harmonic components. The Bartlett window is one of the most widely used. It is defined as follows:

⎧ |m2 | ⎪ ⎨ 1 − |m1 | 1− for |m1 | ≤ M1; |m2 | ≤ M2 , M1 M2 (3.6) w (m1 , m2 ) = ⎪ ⎩ 0 elsewhere. The estimator of Bartlett’s correlogram thus defined is equivalent to the periodogram estimator of relation (3.3) with K1 = M1 + 1, K2 = M2 + 1 and w(k1 , k2 ) = 1. In addition, the mean of the PSD estimated by the correlogram is the exact convolution of the PSD and Fourier transform of window function w(m1 , m2 ). This signifies that the correlogram is a biased estimator, whose bias strongly depends on the window chosen. For mean spatial frequencies the variance of the correlogram is given by the following relation: S 2 (ν , ν ) 1 2 C Var SˆX (ν1 , ν2 ) = X K1 K2

M1

M2

w2 (m1 , m2 )

(3.7)

m1 =−M1 m2 =−M2

The variance in the correlogram, as in the periodogram, is proportional to the square of the theoretical PSD of the process, but also depends on the form of the window function. In Chapter 4, we will see that the variance of traditional bispectral estimators is proportional to the triple product of the spectrum. Thus, there is a biasvariance compromise for the correlogram estimator. 3.3.3. Limits of traditional spectral analysis In the 1-D domain, the use of the Fourier transform imposes certain limits on the spectral resolution of estimators. The resolution is inversely proportional to the duration of the signal. In the same way, 2-D resolution is inversely proportional to the duration of the signal. This limit is a crucial point for both the methods described above. For example, for the correlogram technique, the size of the window function must be significant if we want to preserve good resolution, but, in this case, the precision in estimation of the ACF decreases for high indices m1 and m2 in absolute value. In the same way, for the Welch periodogram method, taking the mean of several estimates considerably reduces the resolution for a fixed duration of the signal. On the whole, we can say that the use of traditional methods is justified from the moment when the outcome of the studied WSS process is of sufficient size. Unfortunately, in many applications where spectral analysis is the proven basic tool, this condition is, materially or by definition, unrealizable. For this reason, traditional methods had only a moderate success in certain fields where the amount of


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data – assumed to be stationary – to be processed is limited (seismic, antenna processing, etc.) and so attention is given to the more powerful methods of spectral estimation, which we will examine now.

3.4. Parametric 2-D spectral estimation The parametric methods of 2-D spectral estimation are based on PSD modeling of random fields. These PSDs thus become generally reduced deterministic functions of a finite number of unknown parameters. Then, the problem of spectral estimation amounts to the equivalent problem of estimating these parameters from a finite resolution of the process. Parametering the PSD of a process is a difficult problem and a preliminary knowledge of its “form” is often desirable, before attributing any model to it. For example, we know that AR modeling is well-adapted to spectral analysis of narrow-band signals, whereas moving average (MA) modeling paves the way for a better description of broadband processes. However, in practice we often have little or no knowledge of the properties and/or PSD of the random field under study. The use of parametric methods of spectral estimation compels us to impose the hypothesis of a model a priori on the available data. This concept is fundamental for understanding parametric techniques. We will see at the end of this chapter how a physical model for obtaining images can influence the choice of a particular parametric method in a radar imagery application. One of the advantages of parametric methods is to produce PSD estimates which are less “fluctuating” than those obtained, for example, with the periodogram method. This implies a greater facility of extraction of frequency components for the analysis of periodic signals. However, the essential contribution of parametric methods is in their spectral resolution power, particularly in the case of short duration signals. It is precisely this property of these methods which facilitates their intensive use in fields of application like radar and radioastronomy where the number of sensors available is relatively reduced. Let us note that in a number of applications, stochastic AR(MA) models with regular PSDs are used for spectral estimation of non-regular PSD processes (with typical Dirac distributions). This apparent inadequacy of model and data should not interrupt anything; in these applications, the goal is not obtaining the PSD itself, but the extraction of frequencies and powers of harmonic components present in the signal from its estimation. Two families of methods are described here: first, 2-D spectral analysis methods by the linear AR(MA) model and by the GM model; second, techniques of maximum entropy and minimum variance or semi-parametric techniques which are based on specific principles.

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We will discuss the basic principles and significant results achieved for both methods. For explicit calculation of parameters, see Chapter 2 which deals with linear modeling, as well as descriptions of other techniques.

3.4.1. Spectral estimation by linear stochastic models 3.4.1.1. AR model AR modeling of a signal (see Chapter 2) assumes that outcome x of a 2-D X random process is obtained by inputting the outcome e of a centered 2-D E white noise, of variance σ 2 in a 2-D LIT system whose transfer function is (see Chapter 1): H (z1 , z2 ) = (m1 ,m2 )∈D

1 1 = A(z1 , z2 ) am1 ,m2 z1−m1 z2−m2

(3.8)

Therefore, the 2-D signal is obtained by the following differential equation: x (k1 , k2 ) = −

am1 ,m2 x (k1 − m1 , k2 − m2 ) + e (k1 , k2 )

(3.9)

(m1 ,m2 )∈D

Thus, we find ourselves in the particular case of linear ARMA modeling (see Chapter 2) where b0,0 = 1 and bm1 ,m2 = 0 for any (m1 , m2 ) = (0, 0). D is the prediction support chosen for the model and perhaps of the quarter plane (QP), nonsymmetric half plane (NSHP), symmetric half plane (SHP) or the non-causal (NC) type. The PSD of the random signal obtained in the output of the system constitutes the evaluation on the unit torus of the square of the module of the z transfer function of the system, multiplied by the PSD of the input noise (see Chapter 1). The PSD of an AR field is thus expressed by: σ2 AR (ν1 , ν2 ) = # SX #2 # # # # am1 ,m2 e−j2π(ν1 m1 +ν2 m2 ) # #1 + # # (m1 ,m2 )∈D

(3.10)

The PSD is a deterministic function of parameters {am1 ,m2 } and variance σ 2 of input white noise. As a result, the problem of spectral estimation of 2-D signals by AR modeling amounts to finding an estimate {ˆ am1 ,m2 } of parameters with D support defined by


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hypothesis (see Chapter 2) and an estimate σ ˆ 2 of the input noise variance, for which modeling is better adapted in terms of a defined criterion. These estimates are indeed obtained from a finite outcome x of a random WSS field. AR of Thus, from the estimation of these parameters we get a PSD estimation SˆX the random field.

Without going into the details of methods which enable this parameter estimation (see Chapter 2), we can say that, as in the 1-D case, choosing a causal AR system leads to a system of normal equations (or Yule-Walker equations) where the ACF intervenes, which itself must be estimated beforehand [CAD 81]. However, in contrast to 1-D processing, where the number of parameters of the model is equal to the number of ACF values (see [CAS 06]), there is no corresponding correlation matching property for 2-D AR methods: the calculation of parameters requires more values of ACF than the parameters themselves. The common techniques in 1-D processing, like the Levinson algorithm [JUS 77] and calculation of reflecting coefficients [MAR 80], have been the subject of many attempts in the extension to the 2-D field (see Chapter 2). On the other hand, choosing a semi-causal or non-causal AR model poses some problems where the application of traditional criteria of minimum mean-square error leads to inconsistent estimations of parameters of the model and thus of the corresponding AR PSD. However, there are some consistent estimation methods of these parameters, but they are highly non-linear and tricky to implement [SHA 86] (see Chapter 2). In other works, we can find many results, both theoretical and experimental, relating to the performances of some AR 2-D spectral estimators applied to complex or real sinusoidal signals. For example, a synthesis of these performances in complex value data is available in an article by Zou and Liu [ZOU 88]. For causal support AR methods (QP and NSHP), we know that the frequency estimation is not optimal. Even in a higher order, these models produce badly conditioned PSD: the peak position in a spectral domain is shifted with respect to the exact frequency of the sinusoid. Moreover, due to a particular form of spatial support, these PSDs exhibit disymmetry of energy around the peaks obtained. The resolution power can be altered, for example, in the case of two harmonic components, and according to the relative orientation of the two wave planes with respect to that of the support. Lastly, when the order of the model is over-estimated we see, as in 1-D processing, a phenomenon of spurious peaks due to an arbitrary positioning of the poles of the linear filter transport function.

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In order to solve all these problems, an estimator was proposed [JAC 79, KUM 81] called the spectral estimator by harmonic mean (HM), which is defined by the following relation:

1 = AR−HM ˆ 2 SX (ν1 , ν2 ) 1

1 AR−P Q SˆX (ν1 , ν2 )

+

1 AR−SQ SˆX (ν1 , ν2 )

(3.11)

AR−P Q AR−SQ (ν1 , ν2 ) and SˆX (ν1 , ν2 ) are two PSD estimations of the process where SˆX obtained by AR modeling with respective supports in the first and second QPs.

This spectral estimator cannot be interpreted in terms of the stochastic model. Indeed, a stochastic process cannot be AR-QP following two mutually exclusive quadrants. Thus, the AR-HM model should be regarded only as an empirical spectral estimator. However, as testified by other works, its properties are numerous, especially with regard to the analysis of 2-D sinusoids [ZOU 88]: – PSD anisotropy, which is noted in the estimate by the QP or NSHP model, is very considerably reduced by the HM estimator; – spurious lobes disappear, as the PSD HM will only have high values if the PSD PQ and SQ have high values at the same spatial frequencies, which is highly probable for the frequency of a sinusoid that we wish to find, but less probable for spurious lobes of each AR-QP estimates whose distribution in spectral domain ξ 2 is unspecified; – frequency estimation qualities of a sinusoid and the ability to distinguish between two sinusoid frequencies are found to be greatly improved compared to QP, NSHP and NC models. For a model with SHP support, we know that one of the major disadvantages is that it does not guarantee the obtaining of the real and positive PSD [JAI 81]. Its performances are nearly equivalent to those of AR-QP models, i.e., the estimated PSD reveals an asymmetry due to spatial support, poor frequency resolution as well as a high appearance likelihood of spurious peaks, especially in the case of a weak signalto-noise ratio. Finally, in the case of the NC model, which is more adapted to signal analysis of homogenous spatial dimensions (of imagery type), the PSD is generally wellconditioned according to the criterion adopted for parameter estimation. Choosing the criterion of the minimum mean-square error, for example, leads to similar performances of the HM method for resolution quality but, on the other hand, there is the problem of spurious peaks [ZOU 88]. The maximum likelihood criterion is alone capable of providing consistent parameter estimation [SHA 86].


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In Chapter 2, we showed parameter estimation of the 2-D AR-QP model with the help of the 2-D FLRLS algorithm. The choice of this support was partly motivated by the existence of the AR-HM PSD estimation method. Thus, by traversing the signal in two different ways (from top to bottom, from left to right and from top to bottom and from right to left), we have the possibility of estimating the parameters of the models resulting from the first quadrant (F Q) and second quadrant (SQ). In the course of such operation, the vectorial approach of QP support is not at all taken into account for the estimation of the PSD. In fact, the 2-D FLRLS algorithm not only allows the estimation of the parameters of the AR-QP model but also of all the parameters of the multichannel AR representation. Thus, we can define several 2-D linear prediction models for QP support of order (M1 , M2 ): x ˆμ (k1 , k2 ) = −

aμm1 ,m2 x(k1 − (m1 + 1), k2 + μ − m2 ) (3.12)

(m1 ,m2 )∈DP Q

with 1 ≤ μ ≤ M2 and εμ (k1 , k2 ) = x(k1 , k2 ) − x ˆμ (k1 , k2 ). Then, a PSD estimation can be calculated for each of these models: 2

σ μ P Qμ (ν1 , ν2 ) = # SˆX #2 (3.13) # # # # μ am1 ,m2 e−j2π(ν1 (m1 +1)+ν2 (m2 −μ)) # #1 + # # (m1 ,m2 )∈DP Q It is possible to carry out an empirical estimation of PSD by making a harmonic mean of all PSD associated with direct multichannel supports, which we will call PSD MHM (Multichannel Harmonic Mean): 1 = M HM (ν , ν ) M +1 2 SˆX 1 2 1

1 PQ SˆX (ν1 , ν2 )

+

M2

1

μ=1

P Qμ SˆX (ν1 , ν2 )

(3.14)

In order to solve a residual anisotropy problem of the PSD of an AR-MHM model, the usage of harmonic mean of the PSD MHM calculated for horizontal F Q and vertical SQ paths is advised [ALA 98]. This estimation, known as HMHV (Harmonic Mean Horizontal Vertical), allows us to obtain the isotropy of PSD. Then, the PSD HMHV is written as: 1 1 1 1 = + (3.15) 2 SˆM HM −P Qh (ν1 , ν2 ) SˆM HM −SQv (ν1 , ν2 ) SˆHM HV (ν1 , ν2 ) X

X

X

M HM −P Qh (ν1 , ν2 ) as the PSD MHM for a FQ support with a horizontal with SˆX M HM −SQv path and SˆX , for a SQ support with vertical path.

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3.4.1.2. ARMA model The general linear model, as described in Chapter 2, is an ARMA model. Therefore, the transfer function of the corresponding LIT filter contains poles and zeros, and the PSD of such a model is given by: # #2 # # # −j2π(ν1 m1 +ν2 m2 ) # bm1 ,m2 e # # #(m1 ,m2 )∈D # ARM A 2 SX (ν1 , ν2 ) = # #2 σ # # # # am1 ,m2 e−j2π(ν1 m1 +ν2 n2 ) # #1 + # # (m1 ,m2 )∈D

(3.16)

Thus, the PSD ARMA consists of a numerator (MA part) which is the function of spatial frequency. Compared to AR modeling, the development and the application of basic spectral estimation methods of 2-D ARMA models is very much underdeveloped; the majority of the methods commonly used are direct extensions of known 1-D methods. In 1-D processing, there are primarily two types of techniques of parameter estimation of an ARMA model: – techniques which are based on the principle of the maximum likelihood, which is often approximated [AKA 73], for which the set of AR and MA parameters are estimated simultaneously via iterative minimization of a criterion where the form of the estimated PSD directly intervenes. More precisely, we seek to minimize the following function: Q ({am }, {bm )}) =

1 2

− 12

! "−1 P ARM A SˆX (ν) SˆX dν (ν)

(3.17)

P where SˆX (ν) is the 1-D periodogram obtained from the initial data. These techniques make it necessary to implement optimization methods of non-linear parameter functions, which in many cases require very high calculating times to converge towards the solution; – modified Yule-Walker techniques for which we first estimate AR parameters of a model with the help of normal equations and then MA parameters, or directly the PSD numerator, from AR parameters and initial data. These techniques are obviously sub-optimal compared with the maximum likelihood techniques, but are considerably less expensive because of their algorithmic linearity [CAD 81].

The simulations on 1-D signals [KAY 88] seem to indicate that Yule-Walker methods are well-adapted to the analysis of signals involving marked periodicities. Thus, we can expect to obtain a similar result in 2-D processing. As a result, but also with the help of the algorithmic linearity of this technique, we will be able to proceed in the parameter estimation of the ARMA-QP model by successively identifying the AR


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and MA parts. The estimation of AR can be viewed as a least squares method applied to the Yule-Walker equations extended to 2-D field [CAD 80]. The method proposed in Cadzow and Ogino’s article [CAD 81] was treated in the same manner in Cariou’s thesis [CAR 91], only with the difference of creating a less expensive development for the estimation of the PSD numerator. This last technique is an extension to the 2-D domain of the work outlined in [KAV 79]. In addition, there are some results relating to the non-causal ARMA model, which, as a more general model, is much more difficult to comprehend [ARU 86] (see Chapter 2). When analyzing unspecified signals, ARMA modeling (in 1-D as in 2-D) is advantageous in the sense that it requires, in equal performances, fewer parameters than AR modeling. In addition, it allows a better analysis of the noise signals: the MA span of PSD is well-adapted to identify colored noise (i.e., whose ACF differs from the unit impulse). With very few results available in other works, it is difficult to measure the contribution of 2-D ARMA modeling from a practical point of view. In [CAD 81], the authors show that the application of the two AR and ARMA methods to noisy sinusoids mainly favors the ARMA method at the level of the final aspect of the resulting PSD, which decreases (but does not lose) certain defects of AR spectral estimation: spurious peaks and dissymmetry of PSD due to the choice of QP support. To avoid these problems, in conclusion, the authors suggest the usage of the quadratic combination of parametric estimation criteria from models with the support in four QPs. Based on this view, and inspired by the AR-HM technique described earlier, the ARMA-HM technique was developed [CAR 91] using the harmonic mean of two PSD ARMA-QP resulting from two QP models in adjacent quadrants. In this case, the PSD is written as: 1 = ARM A−HM ˆ 2 SX (ν1 , ν2 ) 1

1 ARM A−P Q SˆX (ν1 , ν2 )

+

1 ARM A−SQ SˆX (ν1 , ν2 )

(3.18)

3.4.1.3. Gauss-Markov model The growing interest for modeling random fields brought many researchers to develop algorithms (for example, segmentation of images) by relying on very general probabilistic models [BES 74, GUY 82, ORD 75, WHI 54]. Among them, the GM field model is one of the more well-known. The definitions relative to Markovian fields have already been seen in Chapter 2 and we will not come back to those again. However, we will recall that the PSD of an L-Markovian field is written as:

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GM SX (ν1 , ν2 ) =

2 σE

1+ (m1 ,m2 )∈D N C

cm1 ,m2 e−j2π(ν1 m1 +ν2 m2 )

(3.19)

2 represents the where {cm1 ,m2 } is the set of L-Markovian field parameters and σE variance of the innovation process, which, if the field studied is Gaussian, is a colored Gaussian noise.

The class of random GM fields is a more general class than that of Gaussian AR processes, in the sense that for any AR process, we can find a GM field of the same PSD. The inverse is not true, particularly if we consider the GM field, whose neighborhood is made up of four neighbors closest to the current site [SHA 86]. We observe that the analytical form of the PSD of a GM field is equivalent to that given by the ME method (see section 3.4.2). Consequently, to estimate the PSD ME of a Gaussian random field, a method will consist of obtaining {cm1 ,m2 } parameters 2 of a GM field verifying the property of correlation correspondence. We find and σE such methods in the work of Chellappa et al. [CHE 83] and Sharma et al. [SHA 85], where authors use the maximum likelihood criteria for the parameter estimation. The non-causality of the neighborhood employed in this method reveals the fact that maximization of the likelihood function can be operated only in an iterative way. Since the spectral estimation by the GM model is associated with spectral estimation by ME, their performances are very similar. In the analysis of broadband random PSD fields there are some good results for a relatively weak order of the model (typically, 12 parameters {cm1 ,m2 }) [SHA 85, WOO 76].

3.4.2. Maximum entropy method The ME method for the spectral estimation of signals, proposed by Burg [BUR 75], was widely circulated and applied successfully in many well-known fields such as speech processing and seismic signals. Preference for this method has two origins: its performances, considered excellent in other works, and the fact that it is no longer based on an explicit modeling of a random process, but on a very general principle. In the 1-D field, AR modeling and the ME method lead to an identical expression of the PSD of the process. However, if the extension of the AR methods to the 2-D field is relatively direct (at least for the models with causal support), it is not the same for the ME method. This is due to the impossibility of factorization of the multidimensional polynomials [BUR 75, MCC 82, WOO 76] (see Chapter 1).


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3.4.2.1. ME principle The ME principle is mainly inspired by physics and more particularly by statistical thermodynamics, and the concept of entropy itself is the basis of information theory [PAP 81, SHA 79]. Applied to signal processing, this principle consists of considering that the signal is as random as possible, while taking into account certain constraints linked with a priori knowledge of the signal. Let us assume that we know the ACF of a WSS process in a region A of network Z 2 . Then the principle is to maximize the following quantity H, known as the entropy of the process: H=

ξ2

ln SX (ν1 , ν2 )dν1 dν2

(3.20)

ME under the constraint that 2-D IFT of the particular PSD SX (ν1 , ν2 ) which maximizes H is equal to the ACF in area A. Let us note that it is possible, based on the context, to limit the frequency domain of integration for calculating H. In addition, the choice of a spatial (or spatio-temporal) distribution for the region A can be arbitrary. This factor makes the technique interesting, notably for multisensor signal processing.

In other words, the ME principle leads to extrapolate the ACF beyond the interval where it is known. However, it is not evident that a positive semi-definitive extrapolation exists in 2-D [DIC 80], even if the available ACF span is positive semi-definitive, whereas in 1-D, it was shown that there is an infinity of possible extrapolations [BUR 75]. It is difficult to evaluate if the given ACF segment is an extract of the positive semi-definitive ACF [LAN 82], but this hypothesis is generally retained in all implementations. Woods showed the existence and uniqueness of the ME solution under this hypothesis [WOO 76]. 3.4.2.2. Implementation Although there are methods of direct maximization of entropy [BUR 75] and digital optimization of PSD taking into account the error made in the measurement of ACF [WER 77], the problem of spectral estimation by ME is often formulated in a dual way. ME (ν1 , ν2 ) as the spectrum maximizing the entropy H and Thus, let us name SX ME (ν1 , ν2 ) according considering the constraints on the autocorrelation. By writing SX / A, then by to γX (m1 , m2 ) for (m1 , m2 ) ∈ A and γˆX (m1 , m2 ) for (m1 , m2 ) ∈ / A, the maximization of imposing constraints ∂H/∂ˆ γX (m1 , m2 ) = 0 for (m1 , m2 ) ∈ H, using the Lagrange multiplicator method, will be formulated differently.

130


ME Given γX (m1 , m2 ) for (m1 , m2 ) ∈ A, the PSD ME SX (ν1 , ν2 ) must be determined in the form:

1 λ(m1 , m2 ) e−j2π(ν1 m1 +ν2 m2 )

ME SX (ν1 , ν2 ) =

(3.21)

(m1 ,m2 )∈A ME where the parameters λ(m1 , m2 ) are such that the 2-D IFT of SX is equal to the ACF in the region A.

It is immediately seen that the PSD ME, which is dependent on a polynomial in two dimensions, is generally not factorizable. Since the problem of spectral estimation by ME amounts to that of the estimation of parameters λ, for which no canonical solution could be found, the majority of algorithms are iterative and proceed by non-linear optimization. The difficulty of the problem of 2-D ME inspired the research to a great extent and many algorithms were developed. Woods approaches random field ACFs by a truncated series of convolution products of a gradient operator [WOO 76]. For certain iterative methods, it is the parameters λ that are evaluated [JAI 78], while in other cases [ROU 80], ACF is calculated in a local way, i.e., point by point, according to a path specified in Z 2 network. Let us note that there are also some algorithms allowing a specific study of the isotropic random fields, i.e., whose ACF has the property of circular symmetry (see Chapter 1) [TEW 88]. 3.4.2.3. Example: alternate projection methods An iterative method of alternate projections [LIM 81] optimizes the PSD ME by alternatively acting on λ parameters and on the extrapolated ACF. This iterative method calls for algorithmic efficiency of the 2-D FFT. Let us assume that we know γX (m1 , m2 ) for (m1 , m2 ) ∈ A as segment of a positive defined ACF. To find the ME estimate, we express a PSD characterized by the two following relations:

SY (ν1 , ν2 ) =

+∞

+∞

γY (m1 , m2 )e−j2π(ν1 m1 +ν2 m2 )

(3.22)

m1 =−∞ m2 =−∞

and: 1 = λ(m1 , m2 )e−j2π(ν1 m1 +ν2 m2 ) SY (ν1 , ν2 ) m =−∞ m =−∞ +∞

1

+∞

2

(3.23)


131

From these two equations, we see that γY can be obtained from λ and vice versa with the help of the 2-D FT. Moreover, we know that SY is the single ME estimate / A and γY (m1 , m2 ) = γX (m1 , m2 ) for (m1 , m2 ) if λ(m1 , m2 ) = 0 for (m1 , m2 ) ∈ ∈ A. Thus, we are in the presence of two constraints in γY and λ. Then it is possible to construct an iterative algorithm; more precisely, we pass from γY to λ, then from λ to γY , each time trying to impose the respective constraints. By (0) starting the algorithm with an estimate λ(0) , we get an ACF estimate γY (m1 , m2 ), that we can correct, in region A, by γX (m1 , m2 ). This new ACF thus helps to determine a new function λ(1) that we truncate in adequate limits. The procedure is repeated until the convergence of γY (m1 , m2 ) towards γX (m1 , m2 ), for (m1 , m2 ) ∈ A. Generally, this algorithm cannot be used as such. In fact, due to multiple operated transformations, it is very probable that the PSD SY (ν1 , ν2 ) becomes negative, in which case the algorithm must be stopped. However, there can be a compromise by (i+1) and λ(i+1) . Denoting the 2-D IFT estimating, in stage (i + 1), the functions γY −1 and by showing: by FT γ (m1 , m2 ) = FT −1

λ (m1 , m2 ) = FT

−1

1 FT [λi (m1 , m2 )]

;

1 i+1 FT [γY (m1 , m2 )]

(3.24)

(3.25)

and: w (m1 , m2 ) =

1

for (m1 , m2 ) ∈ A

0

elsewhere

(3.26)

then, in each stage (i + 1), there is at least one couple (α(i+1) , β (i+1) ) ∈ 2 , allowing (i+1) and λ(i+1) are positive defined, such that: an update and assuring that γY " ! = γ + 1 − α(i+1) (γY − γ ) w

(3.27)

" ! λ(i+1) = β (i+1) λ(i) + 1 − β (i+1) λ w

(3.28)

(i+1)

γY and:

Thus, the problem is based on the choice of the sequence (α(i+1) , β (i+1) ). In [LIM 81], the solution of updating proposed by the authors seems to be a good response to the problem, though it is neither Newton’s method, nor a gradient method [MCC 82].

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3.4.2.4. Performances Even if several techniques exist, the uniqueness of the ME solution for a given ACF span reveals that the performances of the methods are nearly equivalent. The only important difference between the methods is actually in the implementation of algorithms. Performances of the 2-D ME spectral estimation, more deeply studied for harmonic signals [MAL 82], show that this method has a better quality than AR-QP modeling and the correlogram method concerning the estimation of frequencies and the advanced resolution power of several sinusoids, even in the case of a weak signal-to-noise ratio. On the contrary, there is little comparison between the ME approach and the more general ARMA modeling [CAR 91]. 3.4.3. Minimum variance method The minimum variance method, also known as spectral estimation by maximum likelihood (ML)2 [CAP 69], was originally developed for the analysis of seismic signals. This method, which is well-known in 1-D processing, can be directly extended to the 2-D domain. The definition of this estimator is closely connected to the concept of filtering: precisely, we construct a 2-D finite impulse response (FIR) filter through which we wish to pass a (generally complex) 2-D harmonic component without distortion, of the given frequency, and by minimizing the power of the signal at the output of the filter. Let X(k1 , k2 ) be the complex random 2-D signal: X(k1 , k2 ) = a ej(2π(F1 k1 +F2 k2 )+Φ) + B(k1 , k2 ) k1 = 0, 1, ..., K1 − 1 for k2 = 0, 1, ..., K2 − 1

(3.29)

where B is a Gaussian circular and centered complex white noise, Φ a random phase uniformly distributed in [0, 2π], a a complex constant and (F1 , F2 ) is a couple of normalized (constant) frequencies. After lexicographic rearrangement (concatenation of lines in 2-D table X(k1 , k2 )), we can write: x = a ejΦ e + b

(3.30)

2. In reality, this qualifier is incorrectly chosen, since this estimator is not of the ML type and does not have any of its properties.


133

with: x = [X(0, 0) · · · X(0, K2 − 1) X(1, 1) · · · X(1, K2 − 1) · · · X(K1 − 1, 0) · · · X(K1 − 1, K2 − 1)]T b = [B(0, 0) · · · B(0, K2 − 1) B(1, 1) · · · B(1, K2 − 1) · · · B(K1 − 1, 0) · · · B(K1 − 1, K2 − 1)]T e = e1 e2 where the symbol represents the Kronecker product and: e1 = [1 ej2πF1 ej4πF1 · · · ej2π(K1 −1)F1 ]T e2 = [1 ej2πF2 ej4πF2 · · · ej2π(K2 −1)F2 ]T

(3.31)

By using x in an FIR filter of coefficients w, the output is given by w† x. Then we impose that the coefficients satisfy the relation w† e = 1 in such a way that the sinusoid is not distorted at the output. In order to minimize the power of the filter output, we can show that it is necessary to minimize w† Γb w, where Γb = E[bb† ] (for example, see [KAY 88] for a detailed calculation in 1-D). The solution to this problem of quadratic minimization is given by optimal filter: w=

Γb −1 e e† Γb −1 e

(3.32)

and the minimum power at the output of the filter is: Pmin =

1

(3.33)

e† Γb −1 e

Thus, such a filter w has the property to let the spatial frequencies pass close to (F1 , F2 ) and to reject all the others. Power at the output of the filter is thus a good indication of the power of the signal in the neighborhood of this frequency. If we now assume that X(k1 , k2 ) is a WSS process more general than that of a simple harmonic component, the calculation of this power, for (ν1 , ν2 ) ∈ ξ 2 , gives access to the spectral contents of the process. The PSD MV (theory) is then written as: MV SX (ν1 , ν2 ) =

1 e† Γx −1 e

(3.34)

134


where now e is written as: e = e 1 e2 with e1 = [1 ej2πν1 ej4πν1 · · · ej2π(P1 −1)ν1 ]T e2 = [1 ej2πν2 ej4πν2 · · · ej2π(P2 −1)ν2 ]T

(3.35)

Γx is the theoretical autocorrelation matrix of dimension P1 P2 × P1 P2 , of the Toeplitz-blocks, Hermitian and positive definitive type, which we obtain with the help of relations (1.33) and (1.34) of Chapter 1. In practice, the autocorrelation matrix is estimated in the following way: ˆx = Γ

K 1 −P1 K 2 −P2 1 x(p, q)x† (p, q) (K1 − P1 + 1)(K2 − P2 + 1) p=0 q=0

(3.36)

where: x(p, q) = [x(p, q) x(p, q + 1) · · · x(p, q + P2 − 1) x(p + 1, q) x(p + 1, q + 1) · · · x(p + 1, q + P2 − 1) · · ·

(3.37)

x(p + P1 − 1, q) · · · x(p + P1 − 1, q + P2 − 1)]

T

Thus, we choose P1 K1 and P2 K2 , so as to get a reliable estimation of the ACF points from original data. Here, the couple (P1 , P2 ) plays a role similar to the order (M1 = P1 − 1, M2 = P2 − 1) of an AR-QP model. For the analysis of 2-D sinusoids, the performances of the MV estimator for spectral estimation are located mid-way between the traditional methods such as a correlogram and those of the AR and ME methods, irrespective of the signal-to-noise ratio, imposed on the data [MAL 82]. As in the 1-D case, it was shown that the MV method can be connected in 2-D to the AR spectral estimator [DOW 84]. 3.5. 2-D high resolution methods High resolution (HR) methods were suggested specifically to study complex harmonic signals showing close frequency components. Interest for these methods is justified for well-defined applications like telecommunications [VEE 98], SAR (Synthetic Aperture Radar) image processing [CAR 91, DEG 98, MAI 00], or even studies of measurements on the equivalent radar surface of a target [BER 98]. The common


135

point of these various applications lies in the modeling of the studied signals. Indeed, all of them admit a representation by a sum of complex exponentials in which certain parameters are to be determined. In the framework of spectral analysis, we seek to estimate signal frequencies, for example, from the calculation of a spectrum or a pseudo-spectrum of the signal. Methods based on the Fourier transform, and methods known as beamforming, present a resolution power which is insufficient to study signals having very close harmonic components which sometimes occur in the case of, for example, radar imagery. Thus, HR methods were developed by using decomposition into signal and noise sub-spaces associated with the data. This decomposition is the result of modeling the complex random signal X as the sum of a useful signal e and a noise B. Useful signal e is defined by:

e(k1 , k2 ) =

L

al ej(2π(F1l k1 +F2l k2 )+ϕl )

(3.38)

l=1

for

k1 = 0, 1, ..., K1 − 1 k2 = 0, 1, ..., K2 − 1

This signal is made up of L components corresponding to L complex exponentials characterized by normalized frequencies {F1l , F2l } and by terms of amplitude and phase al and ϕl . We also assume that B is a white Gaussian noise, of zero mean, 2 and circular. In addition, we consider that the useful signal and the noise variance σB are independent. 3.5.1. 2-D MUSIC The MUSIC (MUltiple SIgnal Classification) method [BIE 83, SCH 86] was introduced at the beginning of the 1980s [PIS 73]. It is in the latter article that for the first time the separation in signal and noise sub-spaces was used for calculating a pseudo-spectrum whose maxima corresponded to the studied frequencies. This method was developed for 1-D signals, but its extension to 2-D cases does not present any difficulties. Nevertheless, the variant of this method called root-MUSIC is not available for 2-D signals due to problems raised by factorization of 2-D polynomials. 3.5.2. Calculation of a pseudo-spectrum The MUSIC method makes use of orthogonality between the signal and noise sub-spaces which are defined in terms of decomposition in eigenvalues of the autocorrelation matrix.

136


Let Γx be an autocorrelation matrix (theory) of size P1 P2 × P1 P2 associated with the 2-D signal which is defined above. Since this matrix is Hermitian and positive definite, it accepts the following decomposition: 2 Γx = SΨS† + σB IP1 P2

(3.39)

Ψ is a diagonal matrix of size L × L containing squares of amplitudes: Ψ = diag{a2l }

(3.40)

and matrix S verifies: S = [e11 e21 , e12 e22 , · · · , e1L e2L ]

(3.41)

e1l and e2l respectively indicate vectors:

1 , · · · , ej2πF1l (P1 −1)

T

T and 1 , · · · , ej2πF2l (P2 −1)

Relation (3.39) exposes autocorrelation matrices of useful signal and noise, 2 IP1 P2 . Since matrix SΨS† is of rank L, the real eigenvalues respectively SΨS† and σB of autocorrelation matrix Γx classified by decreasing order verify the relation: 2 β1 ≥ β2 ≥ · · · ≥ βL > βL+1 = · · · = βP1 P2 = σB

(3.42)

Γx can also be diagonalized in an orthogonal base, which is written as: Γx =

P 1 P2

βl ul u†l = UDU†

(3.43)

l=1

ul and βl respectively being the lth column and (l, l) element of matrices U and D containing the eigenvectors and eigenvalues of Γx . The space generated by the columns of matrix S is called the sub-space signal. It is also generated by the eigenvectors of matrix Γx associated with L greatest eigenvalues. 2 is called The space generated by P1 P2 −L eigenvectors associated with eigenvalue σB the sub-space noise and it is orthogonal to the sub-space signal. Matrix Γx can therefore be written in the form: Γx = US DS U†S + UN DN U†N

(3.44)

Matrices US and DS contain eigenelements associated with the sub-space signal, and UN and DN are associated with the sub-space noise. The MUSIC method is based on the projection of vectors of sub-space signal on the sub-space noise. The projector associated with the latter sub-space is UN U†N .


137

The square of the projection norm of a vector e on the sub-space noise is therefore e† UN U†N e if the eigenbase is orthonormal. When e is a vector of the sub-space signal, i.e., when there is F1l and F2l , such that: T T (3.45) e = s(ν1 , ν2 ) = 1 , · · · , ej2πF1l (K1 −1) 1 , · · · , ej2πF2l (K2 −1) then we have: e† UN U†N e = 0

(3.46)

Estimating the signal frequencies X thus leads to looking for L couples (F1l , F2l ) for whom the pseudo-spectrum defined by PX M U SIC (ν1 , ν2 ) =

1 e† UN U†N e

(3.47)

tends towards infinity. 3.5.3. Pseudo-spectrum estimation In practice, we do not have the exact autocorrelation matrix of a noisy signal. Thus, it is estimated, for example, by using relation (3.36). This matrix is always Hermitian and positive definite, but this time its eigenvalues are given in a decreasing order: βˆ1 > βˆ2 > · · · > βˆL > βˆL+1 > · · · > βˆP1 P2

(3.48)

The order of the model, i.e., the number of signal components, can no longer be deduced from the multiplicity of the smallest eigenvalue. The statistical criteria obtained from the information theory [AKA 73, RIS 78] or the threshold criteria must thus be used to determine the number of L components of the signal. So, the estimated ˆ x associated sub-space signal corresponds to the space generated by L eigenvectors of Γ with L greatest eigenvalues and the estimated sub-space noise is defined as the space generated by P1 P2 − L, remaining eigenvectors. These spaces are orthogonal due to ˆ x , which can be broken up in the the Hermitian character of autocorrelation matrix Γ following manner: ˆ SU ˆ† +U ˆ NU ˆ† ˆ SD ˆ ND ˆx = U Γ S N

(3.49)

by using analog notations like those introduced by relation (3.44). Therefore, signal frequency estimation consists of finding L maxima of a pseudospectrum defined by: M U SIC PˆX (ν1 , ν2 ) =

1 ˆ NU ˆ† e e† U N

(3.50)

138


This method is the object of various studies, as much for 1-D as for 2-D signals, because of its good performances from a precise point of view and its high resolution power. In the 2-D case, the variants of this algorithm were developed to reduce the calculation time or to extend the use of this method to the study of deadened signals [LI 98]. Youlal et al. [YOU 94] presents a comparison of performances of some AR parametric estimators with the 2-D MUSIC method for the estimation of pure frequencies. 3.6. Other techniques This overview would not be complete without evoking two particular techniques and the characteristics of the diversity of multidimensional signals. These methods are mainly used when dimensions of the signal brought into play are not equivalent. A typical example is that of a signal coming from multisensors: one of the dimensions is of a spatial nature (spatial distribution of individual sensors) and the other of a temporal nature (measurement in time of a physical value on each sensor). These methods are as follows. Separable methods: with these spectral analysis methods, we use the fact that the kernel of Fourier transform m-D is separable. Thus, it is possible to decompose the Fourier transform m-D in m 1-D Fourier transforms, each reacting on a particular signal dimension. In addition, we will see that this is what is carried out in the calculation of the FFT of a multidimensional signal. Nevertheless, it is possible to use a different technique for each dimension. For example, in the case of the multisensor signal mentioned earlier, since the amount of temporal data for each sensor is high, a non-parametric technique (the FT of the signal emerging from each sensor) can be used. Conversely, a lesser number of sensors leads instead to the use of a parametric or semi-parametric ME technique according to this dimension [JAC 78, JOY 79]. Data extension methods: here, the technique consists of extending the data in an artificial way to apply a non-parametric spectral estimator. The extension can be carried out separately according to each dimension or simultaneously based on two dimensions. In the first case, the ME principle enables us to construct, using the Burg algorithm [BUR 75], a linear predictor of 1-D data (different or well-connected by phase information) for each line and each column of data [FRO 79]. In the second case, the use of four 2-D linear predictors in QP structure leads to a direct 2-D extension of data [FRO 80]. In both cases, conventional techniques of 2-D spectral analysis (periodogram or correlogram) can be used. 3.7. Comparative study of some techniques In the preceding sections, we described a set of 2-D spectral estimation techniques, by insisting more particularly on the modern methods of spectral analysis because of


139

good performances which we can expect, and also their popularity and their relative simplicity of implementation. In this section, by comparing them, we examine the performances of some 2-D spectral estimation methods presented earlier, through the analysis of synthetic signals. Based on an experimental approach, this stage is often made necessary owing to the difficulty of theoretical analysis. Two types of 2-D signals allow us to evaluate the performances of the selected estimators: – 2-D data of the component type or sum of harmonic components, whose frequency and amplitude parameters are specified. An outcome of a noise of known characteristics is added to this data. Thus, we can particularly study the parameters of frequency estimation variance and the separation power of two harmonic components; – carrying out a 2-D stochastic process, for which the theoretical PSD are known and possibly parametrizable. In this case, it is possible to evaluate the performances of spectral estimators proposed by the analysis of outcomes of random simulated fields in an experimental way. 3.7.1. Analysis of 2-D harmonic components Spectral analysis is often used with the goal of highlighting the presence of one or more harmonic components in a signal and estimating their frequency and amplitude parameters. In fact, for such signals, the PSD contains one or more maxima, whose amplitudes are related to the amplitudes of these components, and the positions in the spectral domain provide spatial frequencies of these signals. In this case, the general pace of PSD estimated by any method (parametric or not) has little importance and cannot influence the estimation of the sought parameters [KAY 84]. In other works, this type of 2-D signal is often employed to evaluate the relative performances of spectral estimators (for example, [CAD 81, LIM 81]). In general, we fix the frequency and amplitude parameters of one or several harmonic components (real or complex) and the variance of an additive white noise. The theoretical ACF of the signal is then known, on a spatial support that we fix, and this information is manipulated to obtain the PSD which can be compared with an estimator in another PSD. This evaluation technique has the advantage of being simple, but it does not replace, for example, the access to statistics of frequency estimation of the sinusoid. This last problem, although important, is relatively difficult to solve in an analytical way, more so as the theoretical ACF is generally not available and must be estimated. In this section, after a short description of the behavior of the 2-D PSD obtained for each method, we compare the performances of some estimators on the variance criteria of the frequency estimation and separability of close frequency components.

140


3.7.1.1. Synthesis of 2-D sinusoidal signals In order to determine the performances of the 2-D spectral estimators presented in terms of frequency estimation, we synthesize the real 2-D signal and the finite duration x(k1 , k2 ), 0 ≤ k1 ≤ K1 − 1, 0 ≤ k2 ≤ K2 − 1 in the following way: x(k1 , k2 ) =

L √

2al sin(2π(F1l k1 + F2l k2 ) + ϕl ) + b(k1 , k2 )

(3.51)

l=1

where: – L is the number of real 2-D sinusoids; – (F1l , F2l ) ∈ ξ 2 is the normalized couple of spatial frequencies corresponding to the lth sinusoid; – ϕl represents the phase of the lth sinusoid, randomly selected with a uniform distribution in an interval [0, 2π]; √ – 2al is a (real) constant representing the amplitude of the lth . Since the signal is real, the quantity a2l defines the power of the lth sinusoid; 2 – b(k1 , k2 ) is the outcome of a centered white noise B, of variance σB . We will choose a Gaussian distribution for this noise. Since the phase of each sinusoid is random and their amplitude constant, the random field is WSS. The theoretical ACF of the signal is given by the relation: γX (m1 , m2 ) =

L

2 a2l cos(2π(F1l m1 + F2l m2 )) + σB δ(m1 , m2 )

(3.52)

l=1

for (m1 , m2 ) ∈ Z 2 , and the theoretical PSD of the signal has the value: SX (ν1 , ν2 ) =

L a2 l

l=1

2

2 (δ(ν1 − F1l , ν2 − F2l ) + δ(ν1 + F1l , ν2 + F2l )) + σB (3.53)

for (ν1 , ν2 ) ∈ ξ 2 . In this last relation, δ(ν1 , ν2 ) represents the Dirac distribution. Let us recall that spectral analysis of a random field observed on a finite part of the Z 2 network cannot produce such a PSD. We define the signal-to-noise ratio (SNR), for real signals, by the following quantity expressed in decibels (dB) [LIM 81]: ⎞ ⎛ L

a2l ⎟ ⎜ ⎜ l=1 ⎟ SNR = 10 log ⎜ 2 ⎟ ⎝ σB ⎠

(3.54)


141

In the case of complex exponential harmonic components, the data are modeled by relation (3.38), but the definition of SNR remains unchanged. 3.7.1.2. General behavior of PSD estimates Before quantitatively evaluating the performances of spectral estimators, it is interesting to compare the behavior of the PSD obtained for each method with some simple examples. Let us first consider a synthetic real 2-D signal made up of a sinusoid of normalized frequencies (F1 = 0, 2; F2 = 0, 3), of zero phase at origin, of SNR 0 dB and of size K1 ×K2 = 8×8 points. To simplify, later on we will choose to analyze data on a square support, by assuming that K1 = K2 = K. We will observe that these conditions of study are relatively strict when compared with those that we find in other works. An analysis of this numerical data is operated by each spectral estimator previously described (except the MUSIC method, presented in section 3.5.1 solely for complex signals and of which we will see an example of spectral estimation later on). For each method requiring a preliminary estimation of ACF γX from 2-D data, we chose to use a support of size 3 × 3 for the latter. Taking into account the property of even symmetry in the real case (see Chapter 1), only five points of the ACF are to be estimated, in fact: γˆX (0, 0), γˆX (0, 1), γˆX (1, −1), γˆX (1, 0), and γˆX (1, 1) This constraint, applied to the use of a parametric model for spectral analysis, conditions the order of the model to be used. For example, given an AR-QP model we will be limited to the choice of a model of order (M, N ) = (1, 1) (see Chapter 2). The resulting PSD are given in Figures 3.1 to 3.13, under two modes of representation: the first mode is a 3-D view of the PSD in linear scale, while the second is a contour graph of 0 (×), −3 (–), −6 (—), −9 (—), and −12 (· · · ) dB below the absolute maximum of the PSD. The results of this first application of different methods are as follows: 1) Periodogram (Figure 3.1): the PSD estimated by this method reveals a pair of localized main peaks around the exact normalized frequency. These peaks have a rather large width at −3 dB. Moreover, this estimate comprises a series of sidelobes of relatively high power but which do not correspond to any harmonic phenomenon in the starting image. These spurious lobes come from the window function of data by the rectangular window, which causes a convolution of the exact PSD by the square of a 2-D cardinal sine function in the frequency field. 2) Correlogram (Figure 3.2): for this method, we obtain a PSD estimate of rather coarse behavior. Contours in −3 dB are very broad and the maximum of the PSD is

142


rather far away from the exact frequency. This lack of precision in the estimation is due to the reduced size of the support available for the ACF. 3) AR-QP (Figure 3.3): for this method, we observe a PSD comprising relatively fine peaks, but incorrectly located compared to the exact frequency, and which also contains a great dissymmetry consisting of lengthening contours in a direction − π4 compared with trigonometrically directed space (ν1 , ν2 ). This dissymmetry is a direct consequence of the choice of linear prediction support in the first quadrant for this method. No sidelobe is observed here. 4) AR-NSHP (Figure 3.4): here also, the anisotropy of NSHP prediction support generates a dissymmetry of PSD with a lengthening of contours in a preferred direction, although much less marked than with the AR-QP estimator. 5) AR-HM (Figure 3.5): here, we notice a clear improvement in the behavior of the PSD estimate compared to the AR-QP method; the main peaks, even relatively broad, are well localized. The inherent dissymmetry in AR-QP estimator has disappeared and the contour with −3 dB approaches circularity. Here, the sidelobe no longer exists. 6) AR-MHM (Figure 3.6): this estimator leads to a rather poor PSD; the main lobe is very broad. We can also show that the contour in −3 dB in this case is an ellipse whose main axis presents a slight inclination when compared to frequency axis ν2 [ALA 98]. 7) AR-HMHV (Figure 3.7): the harmonic combination of the PSD MHM in the horizontal and vertical path improves the estimate, although the result remains worse than that obtained by the AR-HM method. 8) AR-NC (Figure 3.8): this estimator produces a PSD estimate of virtually identical behavior to that obtained by AR-HM estimation, with broader contours in −3 dB and a worse frequency localization. 9) ARMA-QP (Figure 3.9): with this method, we find the problem of dissymmetry of main peaks due to the choice of QP spatial support. The localization of peaks is incorrect, but we can notice the almost zero level of PSD on a good part of the spectral field. 10) ARMA-HM (Figure 3.10): in this example, the quality of spectral estimation by the ARMA-HM method is clearly established; the peaks are well marked, have a circular symmetry and are rather correctly located as compared to exact frequency. No sidelobe is present. 11) Gauss-Markov (GM) (Figure 3.11): here we get a PSD which is slightly better than that obtained by the AR-HM estimator in terms of contours with −3 dB, but the sidelobes are present. 12) ME (Figure 3.12): this method gives a satisfactory result: the peaks are fine, the frequency localization is correct, contours are relatively circular and there is no secondary lobe. In this example, in comparison with the correlogram method, we observe the importance of the extension of the ACF beyond the field where it is supposed to be known. 13) MV (Figure 3.13): among the parametric techniques, the MV estimator produces the least well conditioned PSD with very broad lobes.


Figure 3.1. 2-D periodogram

Figure 3.2. Bartlett correlogram

143

144


Figure 3.3. PSD AR-QP

Figure 3.4. PSD AR-NSHP


Figure 3.5. PSD AR-HM

Figure 3.6. PSD AR-MHM

145

146


Figure 3.7. PSD AR-HMHV

Figure 3.8. PSD AR-NC


Figure 3.9. PSD ARMA-QP

Figure 3.10. PSD ARMA-HM

147

148


Figure 3.11. Gauss-Markov PSD

Figure 3.12. PSD by maximum entropy


149

Figure 3.13. PSD in minimum variance

In order to learn more about the qualities of a 2-D MUSIC estimator, we also present in Figure 3.14 the result of spectral (pseudo-spectrum) estimation of a harmonic complex signal (see relation (3.38)), of size 8×8 points, of normalized spatial frequency (0.2; 0.3) and SNR 0 dB. Let us note that the pseudo-spectrum, like the PSD, does not have the property of parity in the case of signals with complex values, unlike the real case: only a single peak appears here. The analysis of eigenvalues of the estimated autocorrelation matrix Γx , calculated in this example for P1 = P2 = 5, makes it possible to conclude in the presence of only one harmonic component. The peak obtained in the pseudo-spectrum is very pronounced, which indicates a good behavior of this spectral estimator. Of course, this result is the consequence of an ad hoc modeling of 2-D data, which is not the case for other estimators presented. These implementations, which are very specific and only qualitative, are not enough to draw final conclusions on the performances of different spectral estimators. Even if we note good behavior of the ME, AR-HM and ARMA-HM estimators in the example chosen, it is necessary to evaluate the performances of all these estimators on several planes; what is the influence of parameters linked to data itself on the result (localization of frequency (or frequencies), SNR, size of the signal) and the influence of the model for parametric methods (type of method, order of the model)?

150


Figure 3.14. 2-D MUSIC pseudo-spectrum

3.7.1.3. Statistics of frequency estimation What is the behavior of spectral estimators presented above for the estimation of frequency parameters (F1 , F2 ) of a harmonic 2-D component buried in a Gaussian white noise? Such a study was realized in [CAR 91]3 and considered again in [ALA 98] by a Monte Carlo method: several simulations of the same sinusoid, comprising various outcomes of the same noise, if their number is sufficiently large, allow us to evaluate the statistics of the estimation. Here we recall its essential points. In this study, once the 2-D frequency of the harmonic component (F1 = 0.25; F2 = 0.25) is chosen, the analysis takes into account the following parameters: – SNR(−3, 0, +3 dB); – size K × K of the signal, K varying from 8 to 64 by steps of 8; – if necessary, the order of parametric model corresponding to the size of support of the ACF estimate (3 × 3, 5 × 5, 7 × 7).

3. Use of the AR-NSHP, MP and 2-D MUSIC estimators do not appear in these works.


151

For a given method, and for each of these three parameters, 100 signals were simulated for various outcomes of the noise process and for a random phase of the sinusoid uniformly distributed on [0, 2π[. A simple statistical analysis consisted of a calculation of means and empirical variances of estimates (Fˆ1 , Fˆ2 ), deduced from the maximum position of each PSD. Part of the results obtained, given in Figure 3.15, shows, for each method suggested, the evolution of the logarithm of empirical variance of the estimation of F1 , according to K, for various values of SNR and for a size 5 × 5 of the support of the ACF used. Here, we also find the evolution of the approximate Cramér-Rao bound giving the minimum variance of estimation of F1 . The calculation detail of this bound is given in [CAR 91]. The results lead to the following conclusions: 1) Periodogram: irrespective of the noise parameters, only the results for K = 8 and K = 16 are presented. Beyond that, the empirical variance of F1 could not be estimated, since the construction of the estimator does not allow it. However, we can observe that for given K values, the empirical variances obtained approximate the Cramér-Rao bound, which indicates the good behavior of this estimator, even in the presence of a small SNR. 2) Correlogram: irrespective of the size parameters of the ACF and SNR, this estimator gives relatively satisfactory and stable results. For example, for the ACF of size 7 × 7 (not shown in Figure 3.15), the variances obtained are very close to those given by the ME method. This estimate stability must be attributed, on the one hand, to relative insensitivity of the method to noise conditions [ZOU 88] and, on the other hand, to the reduction of variance of the consecutive spectral estimation in the imposition of an observation of the estimated ACF. 3) AR-QP: this method is not very suitable for the frequency parameter estimations of a sinusoid. Indeed, the variance of the estimate is high: for example, if we choose 3 × 3 points of the ACF (order (1; 1) of the model), the position of the peak is found in an almost equiprobable way in (0; 0), (0; 0.5), (0.5; 0) and (0.5; 0.5). More than the SNR, it is therefore necessary to increase the order of the model to observe a reduction in the dispersion of the detected peaks. 4) AR-HM: the performances of this method for frequency estimation are relatively good. However, it seems that it is less well adapted than the ME method in the case of a restricted order of the model. On the other hand, the increase in the order of the model supports the reduction of estimation variance, especially in the case of a weak SNR. 5) AR-NC: this method produces estimations of variance greater than those of the ME and GM methods. The difference between the AR-NC and GM methods, however, appears to lessen for an average size of ACF (5 × 5) and an SNR of 3 dB. Under these conditions, for K = 8, the resulting variance is much weaker than for the GM method.


0

8

16

24

32

40

48

56

64

40

48

56

64

0 -1 -2 Log (var(F1))

-3 -4 -5 -6 -7 -8 -9 K

0

8

16

24

32

0 Periodogram

-1

Correlogram

Log (var(F1))

-2

AR-QP

-3

AR-HM

-4

ME

-5

GM

-6

ARMA-QP ARMA-HM

-7

AR-NC

-8

CR bound

-9 K

0

8

16

24

32

40

48

56

64

0 -1 -2 -3 Log (var(F1))

152

-4 -5 -6 -7 -8 -9 K

Figure 3.15. Variance of frequency estimation in −3 (top), 0 (middle) and +3 dB (bottom)


153

6) ARMA-QP: for a given ACF of size 3 × 3 (model of order AR-(1, 1), MA-(1, 1), the ARMA-QP estimator is much worse than all the others. This is explained in the same manner as for the AR-QP estimator (distribution of peaks on extreme frequencies), but also when K, the SNR and the order of the model increase, by an excessive distortion of peaks in a preferred direction and the distortion is much more marked than for the AR-QP method. Therefore, this induces a great variability of frequency estimation according to this direction. 7) ARMA-HM: asymptotically, when K increases, this method is the best in all the scenarios. On the other hand, we note that if the dispersion of the peaks is very strong for small K and K grows, the method approaches the Cramér-Rao bound. When the SNR grows, the variance of the frequency estimation tends towards that of the correlogram methods, i.e. ME and AR-HM. The difference between the ARMA-HM and AR-HM methods for a weak SNR is due to the fact that ARMA modeling is better adapted to the analysis of signals disturbed by an additive white noise [KAY 88]. 8) GM: for an ACF of size 3 × 3 and an SNR of value 0 dB or 3 dB, the results obtained are comparable to those given by the ME method. This is not astonishing if it is considered that these two methods lead to the same analytical form of PSD. On the other hand, the GM method is less powerful than the ME method when the SNR increases. This can be explained by the fact that the technique employed for the estimation of parameters (ML) is mainly adapted to broadband signal analysis [KAY 88] which is not the case here. 9) ME: if the difference between the ME methods and the correlogram is weak for an average size of ACF (5 × 5) or high (7 × 7), it is more obvious when this size decreases (3 × 3). The contribution of maximum entropy principle, which consists of extending ACF beyond the spatial domain where it is defined, is important here. When SNR increases, the ME method seems to be even more effective than the correlogram method. It is interesting to note how much the harmonic combination of two PSDs obtained from causal QP models improves the quality of the frequency estimation of a sinusoid. Dispersions resulting from the application of the ARMA-QP and ARMAHM methods confirm this fact, since they constitute, in the majority of cases, the extremes among all the results presented. 3.7.1.4. Separability of two frequency components One of the important qualities that we expect in a spectral estimator is to be able to distinguish in a signal the presence of two or several harmonic components of close frequencies. The separability of two frequency components is generally defined in −3 dB. This means that two peaks are regarded as separate if the level of the PSD in median frequency between two peaks is lower than or equal to half of

154


the mean level of the PSD at two peaks. This definition of separability was originally selected to evaluate the performances of the estimators based on the Fourier transform (periodogram and variants) and to provide greater confidence in significant detection of the periodic signals. Indeed, the PSD obtained by these methods often comprise parasitic sidelobes that could be taken for the spectral signature of periodic phenomena. This definition is too strict and we will not adopt it here; the application of parametric methods leads to PSDs which generally do not contain sidelobes of great power, as is the case with traditional methods. Moreover, the risk of seeing splitting peaks is practically zero in normal conditions of use of these techniques. Lastly, based on the example of some authors [KAY 84, ZOU 88], we will consider that separability is assured when the estimated PSD contains two distinct peaks corresponding to frequency components that are really present, and this without any condition on the level of PSD between the two peaks. An experimental approach to the study of the separation power of two close harmonic components is introduced in [CAR 91] and reconsidered in [ALA 98]. This general approach makes it possible to supplement the results which we can find in other works. Usually some simple examples are presented: two couples of normalized frequencies are arbitrarily fixed and we observe the separation of peaks, possibly according to the SNR and the order of the model for parametric methods (see, for example, [CAD 81, LIM 81, MAL 82]). These results are very specific and do not highlight the separation power of an estimator according to the relative position of frequencies in the spectral field. However, if some theoretical results exist in this subject [ZOU 88], no experimental result comes in support of these works. Moreover, for certain methods (for example, ME, GM, AR-NC methods), we do not know any analytical result on the separation power of two harmonic components because of the complexity of the problem. The only solution to this lack of information thus lies in the development of an experimental approach, an example of which we give now. Let us consider the 2-D signal shown in Figure 3.16 in the form of a digital image and representing a radial sinusoid whose spatial frequency is linearly modulated from (0, 0) (in the center) to (±F, ±F ) (in four corners). In this example, F = 0.25 and the SNR is of 0 dB. By cutting this image into blocks of reduced size, and by assuming that the signal in each block is sinusoidal within the context of relation (3.51), it is possible to locally obtain an estimation of the spatial frequency and the direction of this harmonic component. For each block, the estimated spatial frequency can be compared with the real spatial frequency, by calculating the Euclidean distance which separates them. When the analysis blocks approach the center of the image, the local spatial frequency tends towards (0, 0) and the spectral estimator will produce a PSD whose symmetric peaks (the signal being real, the PSD is an even function) will approach each other until they merge into a single peak at zero frequency. Thus, we will be in a zone of non-separability (ZNS) (Single Peak Area in [ZOU 88]). The more extended


155

the ZNS, the less powerful the estimator concerned for the separation of harmonic components. Figure 3.17 shows the performances of correlogram estimators, ME, AR-QP, AR-HM, AR-NSHP and AR-HMHV. For the periodogram method, the ZNS is reduced in the majority of cases to one single point; this is to relate to the performances of this estimator, already well-known in 1-D processing, for the detection of signals containing pure harmonics. For the adopted methods, we fixed the size of the available ACF to 5 × 5 points (order (2, 2) for AR methods). 2-D data was uniformly cut into 32 × 32 windows of 32 × 32 pixels (approximately 50% overlapping of one window in another). For each analyzed window, the position of the main peak was found in the estimated PSD. Two cases can occur: – if the peak is localized in (0, 0), then the spectrum zone corresponding to the local frequency of the block belongs to the ZNS. Then we attribute to the analysis block a quantity 0; – if the peak is not localized in (0, 0), we calculate the quantity δF which determines the radial error of localization of the peak. To better highlight its variations outside the ZNS, we attribute to the analysis block a quantity proportional to log(δF ). In order to display the results, it is then enough to give the attributed value to the analysis block, according to its position in the original image. “Cards” thus obtained are superimposable in the spectral field, in the interval of normalized frequencies [−0.25; 0.25] × [−0.25; 0.25]. The central zones (in black) in Figure 3.17 are the estimated ZNS. The analysis and interpretation of the results must be carried out in various levels. With regard to the form of the ZNS, the results differ based on the methods: the ZNS of the correlogram and AR-HMHV methods are of circular form, those of the AR-QP and AR-DNSHP methods have a directionality strongly marked in the direction of distortion of the PSD, while those of the ME and AR-HM methods have a rhomboidal form. We note that ZNS forms for the AR-QP and AR-HM estimators, at different noise levels, were predicted in [ZOU 88]. For example, in weak SNR, we know that the ZNS of the AR-QP method has a non-convex form: in the interior of the zone defined by the external contour of the ZNS, there are two symmetric zones for which the separation of peaks occurs. When the SNR increases, these two zones increase to join the zone of external separability. The results presented here thus confirm the theory. The size of the ZNS is an important parameter because it is primarily that which conditions the performance of a method. For the correlogram method this size is consistent and it has been shown that it almost does not vary with the SNR [CAR 91, ZOU 88].

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Figure 3.16. Image analyzed for the estimation of ZNS

The ZNS size for the ME method is much lower than for the correlogram. Thus, the ME method is much more powerful than the correlogram for the discrimination of close frequencies. The best result leads to AR-HMHV technique for which the ZNS is of more reduced size than the others. The ZNS sizes are difficult to determine, even approximately, for the estimators of ARMA type (not represented here). We, however, note that for the ARMA-HM estimator, the ZNS is much smaller than for all the other estimators and that it seems to be further reduced with the increase of SNR [CAR 91]. Variations of the radial localization error of peaks, in separability zones, are also important and can indirectly make it possible to judge the precision of spectral estimation when two harmonic components are present. AR-QP and ARMA-QP methods, for example, lead to these rather significant errors in estimation when compared to other methods and irrespective of the SNR. For the AR-QP method, the zones where the error is most significant are pairs of lobes located on both sides of the ZNS, which are clearly visible in Figure 3.17. For ME and AR-HM correlogram estimators, closer to the ZNS, we observe a sudden increase in the error. This is attributed to the fact that the proximity of the true frequency components causes a bringing together of the estimated peaks, which are “attracted” towards one another and thus moved from


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their ideal position. For the AR-HMHV technique, we notice a consistent increase in the localization error in the areas located far from the ZNS. 3.7.1.5. Estimation of the power of a sinusoid If, for the analysis of harmonic signals, the performances in frequency localization are important, it is also of great use to know the properties of the spectral estimators for the evaluation of the power of such signals. Certainly, this can enable us to go up to parameters associated with energy of a periodic phenomenon, but also to classify, according to their importance, the spectral signatures of the phenomena which we try to detect. In [CAR 91], a short study enabled us to compare the power estimation of a buried sinusoid in a noise for periodogram, correlogram, ME, ARQP and AR-HM estimators. With the help of simulations, the performance of these estimators are examined with respect to the stability of PSD maxima according to the SNR and the size of the data analyzed. This study confirms the linearity of the maximum measurement of the PSD estimated (in dB) based on the SNR, at least for traditional methods (periodogram and correlogram) and for AR-QP and AR-HM methods. Such a conclusion is less clear for the ME estimator, for which the power of a sinusoid is estimated proportionally to the volume in the corresponding peak of the PSD [MAL 82]. 3.7.1.6. Summary As a conclusion of this study, we can establish a qualitative comparison of different spectral estimators proposed according to various criteria of performances traditionally retained for the analysis of harmonic signals. Table 3.1 summarizes the results obtained from which we can draw the following conclusions: – although the periodogram method produces a PSD which can, if the size of the window is reduced, contain secondary peaks, it is observed that, in total, its performances are as good as those of parametric methods; – the Bartlett correlogram method does not have the qualities of an estimator in high resolution; – the GM and AR-NC methods, on the studied criteria, are slightly less powerful than the ME method for the analysis of sinusoids. Moreover, they lead to prohibitive calculative costs; – the ARMA-HM method is the most powerful of all the parametric methods for frequency estimation despite its inability to estimate the power of a sinusoid; – among the parametric methods compared here, ME, AR-HM and AR-HMHV methods are those which show the overall best and closest performances; – although not compared with other techniques presented in [CAR 91] or [ALA 98], the case of MP and MUSIC methods is quite symptomatic of the paradigm of “a priori added”. In fact, according to other works [KAY 88], the MP estimator is considered to

158


Figure 3.17. ZNS of various estimators: correlogram (hg); ME (hd); AR-QP (mg); AR-HM (md); AR-NSHP (bg); AR-HMHV (bd). For the AR-HMHV technique, the image was modified in a manner so as to better emphasize the ZNS

2-D Spectral Analysis Behavior

of

DSP

Dep. θ

-3dB contour

Sec. peaks

Estim. freq. Bias

Var.

Separability

Proport.

Estim.

P

no

average

yes

yes

zero

very good

∝P

∀K

BC

no

large

yes

no

weak

weak

∝P

K→∞

159

Power Stabil.

AR-QP

yes

small

yes

no

strong

average

∝ P2

K→∞

AR-HM

no

small

no

no

weak

good

∝ P2

∀K

AR-NC

no

average

yes

no

weak

-

-

-

ARMA-QP

yes

very small

yes

no

strong

good

unsuitable

unsuitable

ARMA-HM

no

very small

no

no

very weak very good unsuitable

unsuitable

GM

no

average

no

no

weak

-

ME

no

small

no

no

weak

good

-

-

∝ volume K → ∞

Table 3.1. Summary of the characteristics of 2-D spectral estimators for the analysis of harmonic components

obtain average performances, which are often lower than those of the ME technique [MCC 82]. So, for the measurement of autocorrelation matrix, it is sufficient to add a piece of additional information on the nature (and possibly the number) of signals present – specifically the harmonic signals – to see the performances in frequency estimation of the MUSIC method becoming one of the best reported in other works [YOU 94]. 3.7.2. Analysis of random fields It is quite obvious that the study of performances of the 2-D spectral estimators does not limit itself to the evaluation of the frequency analysis of 2-D harmonic components. It is also necessary to examine the behavior of methods in the presence of more general 2-D random fields. In this case, it is no longer a question of detection of pure frequencies, but of the adequacy of the PSD estimated with the theoretical PSD of the studied field. We immediately understand the difficulty of such a task: even if restrictive but obligatory assumptions of a real (or complex), stationary and ergodic field are made, there is an infinity of fields of this type. This is why, to simplify such a study, we often restrict ourselves to the analysis of random fields obtained from a few parametric models such as causal AR(MA) or GM models. It can be noted that these models were mainly studied in other works as tools of synthesis and analysis of textures in the field of image processing (see Chapter 2 and, for example, [ALA 98, CHE 81, CHE 82, KAS 83]). Here, we recall the techniques which make it possible to generate random field outcomes with these models, and then we present an example of synthesis and spectral analysis of signal with regular PSD.

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3.7.2.1. Synthesis of parametric random fields Fields with causal AR(MA) structure: the simplest technique to synthesize a causal field is based on the definition itself of the AR or ARMA field (see Chapter 2). Since the LTI filter is causal, results of the ARMA QP or NSHP field can be obtained from the outcome of a Gaussian white noise by the relation of definition (3.9). In order for the response in the output of the filter to be stationary, the first values of the signal obtained must be eliminated. Fields with GM structure: here, the synthesizing technique is different because of implicit non-causality of the GM model (see Chapter 2), as defined earlier. The following method [CHE 81] enables us to obtain an outcome of a GM field of parameters {cm1 ,m2 } and involves five stages: 1) generation of an outcome u(k1 , k2 ), of size K × K, of a centered Gaussian white noise of variance σ 2;

2) application of the 2-D DFT in this table and safeguards in table u(l1 , l2 ); 3) application of the 2-D DFT in the table of parameters {cm1 ,m2 } and safeguards

in table c (l1 , l2 );

7

4) calculation of table x(l1 , l2 ) = u(l1 , l2 )/

c (l1 , l2 );

5) application of the 2-D IFT in table x(l1 , l2 ) and safeguards in x(k1 , k2 ). In this case, x(k1 , k2 ) is an outcome of a random GM field of parameters {cm1 ,m2 }. 3.7.2.2. Spectral estimation of a GM field Here we give an example of PSD behaviors obtained by spectral estimation of an outcome of a random GM field. An outcome of a GM field of size 64 × 64 points is synthesized by the method described earlier, by choosing a model of order 2 and parameters: ⎧ c0,1 = c0,−1 = 0.1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ c1,0 = c−1,0 = −0.1 ⎪ c1,1 = c−1,−1 = −0.19 ⎪ ⎪ ⎪ ⎪ ⎩ c−1,1 = c1,−1 = 0.19 Knowing these parameters, the theoretical PSD can be calculated by relation (3.19). Figure 3.18 shows the 2-D signal obtained and its theoretical PSD, both represented in the form of digital images. The parameters of the field were selected in such a way that the latter exhibits a quasi-periodic spatial structure. The theoretical PSD indeed presents a pair of fairly strongly pronounced peaks. The position of these peaks in the spectral field corresponds to a spatial frequency normalized from


161

(0.3; 0.2). In the same figure, we find the PSDs estimated by the methods presented earlier. For parametric and correlogram methods, the ACF is estimated on 3 × 3 points, which corresponds to the order of the original GM model. The results are as follows: 1) Periodogram: the values of the PSD in the neighborhood of theoretical peaks are fluctuating and bring out a strong noisy aspect. Thus, we realize the kind of disadvantages which this method can induce during the analysis of small outcomes of random fields. 2) Correlogram: the PSD contains very broad peaks and its mean level is much higher than that of the theoretical PSD. Nevertheless, the estimate of the main frequency is more satisfactory than what is found with the periodogram method. 3) AR-QP: this method completely fails in the analysis of the GM field. 4) AR-NSHP: the presence of peaks is noted, but those are slightly lengthened in a preferred frequency direction corresponding to the least represented spatial dimension in the support of NSHP prediction. 5) AR-HM: there again, we see how the HM principle makes it possible to improve the PSD compared to the AR-QP estimator. Although a pair of peaks is found in a spatial frequency quite close to the real value, the behavior of the PSD is relatively far from that of the theoretical PSD. 6) AR-MMH and HMHV: estimations produced by these two methods are quite different from the theoretical PSD. Curiously, this result seems to contradict those presented in [ALA 98] which show a certain advantage, particularly for the HMHV technique, and comparatively to the HM method, for the Wold field analysis with a very complex structure. This must be explained by the fact that the HMHV technique requires more data than the HM technique for obtaining correct results. 7) AR-NC: for this estimator, we know that modeling is not suited to the structure of the analyzed field. However, nothing prevents us from examining the possible adequacy of the estimated PSD in theoretical PSD. In this case we notice a pair of broader peaks, but the spatial frequency is fairly correctly estimated. 8) ARMA-QP: as for the AR-QP estimator, we observe that this estimator is not suitable. 9) ARMA-HM: the PSD obtained here is closer to the original PSD than that of the AR-HM estimator. The peaks are indeed much finer, but on the other hand, they are not well localized. 10) GM: this estimator is optimal since it is based on the adequate stochastic model and it uses the ML technique for parameter estimation. The resulting PSD is that whose total form better approaches that of the theoretical PSD. 11) ME: here, the estimated PSD is very close to the theoretical PSD. This result is coherent because we know that the PSD ME and GM have the same analytical form.

162


Figure 3.18. PSD estimated from Gauss-Markov field (line by line and from left to right): analyzed image; theoretical PSD; periodogram; correlogram; ME; GM; AR-NC; AR-QP; AR-HM; AR-NSHP; AR-MHM; AR-HMHV; ARMA-QP; ARMA-HM; MV


163

12) MV: the result obtained here is a little better than that of the correlogram technique, but not as good as the AR-HM method, which corroborates a behavior that we have already stated earlier concerning this approach. 3.7.2.3. Summary Conclusions regarding the spectral analysis of random WSS fields with regular PSD are relatively difficult to draw. Here, we have presented only one example of 2-D data, based on a general random field model. The experiment undertaken on a random field with GM structure shows that the choice of ad hoc model for the PSD leads to an optimal estimation (and it is good!), even if the calculation technique differs from one method to another (ME estimator). Let us note that there are other diagrams, like AR-NC modeling, allowing us to generate varied random fields [CHE 85]. Some of these modelings were the subject of research on the synthesis and identification of “natural” stationary textures, such as those which can be found in Brodatz [BRO 66], or those that are observed on the images coming from spatial remote sensing systems [FRA 87]. The problem of identifying a stochastic model in unspecified 2-D data is not yet solved. However, in certain cases, a knowledge a priori of the context of analysis is desirable and very important; for example, in the case of digital images, because of intrinsic non-causal nature of 2-D data, an anisotropic modeling (of QP or NSHP type) appears unrealistic and to be prohibited. However, to what extent should we deduce the inadequacy of a PSD estimated by an AR-QP model (AR-HM estimator) with theoretical PSD of the process? This type of question is still outstanding and is likely to find a response only by means of experimental comparisons between the methods carried out on real data.

3.7.3. Conclusion The set of results presented here makes it possible to draw some conclusions relating to the performances of 2-D spectral estimators. Regarding traditional methods, we can first of all say that the periodogram method is better adapted to the analysis of sinusoidal signals than to the analysis of random fields with regular PSD, especially when the size of observation is reduced. Then, the correlogram method shows overall weak performances, except perhaps with regard to the variance of frequency estimation of harmonic components. This estimator requires data with a large number of ACF points of the field in order to give acceptable results in the case of analysis of random broadband fields. Here, we find a conclusion which is already well-known in 1-D signal processing: – periodogram techniques are more adapted to detection aspects of harmonic components;

164


– correlogram techniques are more adapted to frequency estimation aspects of harmonic components. Among the parametric methods, those methods that call for a non-trivial modeling (AR-QP and ARMA-QP) introduce in the considered PSD a dissymmetry capable of obstructing the simultaneous detection of two periodic or quasi-periodic phenomena, and close spatial frequencies. Moreover, they are not reliable for the precise localization of frequency components. It is thus desirable to avoid these techniques for an application with 2-D data, except of course if the process of formation of the data corresponds precisely to this model, or even with an aim of using the HM principle. Although it does not explicitly correspond to a stochastic modeling of the data and it is only empirical, the HM principle leads to relatively robust spectral estimators, particularly with regard to frequency estimation of harmonic components. Models bringing into play a non-causal structure of the dependence between samples of 2-D signal, appear in practice, apart from slight differences, better adapted to the problem of spectral estimation of broadband processes than to that of analysis of 2-D sinusoids [CAR 91]. Only the ME method, which is built on a more general principle, gives simultaneous satisfactory results for the analysis of 2-D sinusoids and the analysis of signals with regular PSD. It is quite obvious that these conclusions do not have anything either exhaustive or definite. However, the number of parameters to be taken into account, to evaluate and compare the performances of methods is so important that a complete experimental study would soon become prohibitive. An important problem that has not been mentioned until now is that of calculation cost. For algorithmically linear methods (periodogram, correlogram, AR/MA-QP/HM, MV, MUSIC), the calculating time of a PSD can be evaluated with a high degree of accuracy. On the other hand, for iterative methods (ME, GM, AR-NC), this cost cannot be predicted. The experiment shows, for example, that the speed of convergence in a solution for the ME algorithm [LIM 81] is inversely proportional to the SNR and to the size chosen for the ACF. To determine ideas on this aspect, we give the typical values of calculation time of a PSD for some methods in Table 3.2, the reference time is the one used for calculating the periodogram on the same data. It is immediately seen that the calculating periods are reduced for linear methods as compared with iterative methods. In this case, the ratio of calculating time between ME and AR-HM methods is higher than 5 but, under certain conditions, can reach several tens. P BC AR-QP AR-HM AR-NC ARMA-QP ARMA-HM GM ME 1.00 1.00 1.05 2.13 17.6 4.00 7.92 20.7 11.1 Table 3.2. Calculation time of a PSD reported in a periodogram


165

3.8. Application: spectral analysis of remote sensing images 3.8.1. Position of the problem The phenomena of the ocean surface such as the swell and waves have a quasiperiodic structure, spatially and temporally at the same time. The spatial periodicity is translated, on the images of remote sensing, in the form of alternatively clear and dark fringes, having widths related to wavelengths of variable phenomena and directions according to the acquired scene and the type of remote sensing system. The temporal periodicity of the swell or the waves is found in spatial periodicity of the intensity levels of pixels of remote sensing images. It is thus possible, by spectral analysis, to consider the frequency parameters of the phenomenon studied and to go up thereafter to the physical parameters of wavelength, direction and amplitude (trough). The 2-D spectral estimation methods were employed for obtaining the first satellite images of the ocean surface with the objective of determining these physical parameters. The spatial (and temporal) stationarity of the swell in high tide indeed validates the application, on images of huge size, of traditional 2-D periodogram or correlogram spectral estimation methods, which are always largely used by oceanographers. However, near the coasts, the swell changes in a considerable way: the wavelength is shortened, propagation direction changes, the trough decreases. Then, the long range stationarity is no longer verified, which involves the impossibility of using such methods. In this case it is necessary to study these physical parameters locally, i.e., on small zones of the image. By admitting that on this small zone scale the stationarity is preserved, it is then possible to consider the use of 2-D spectral estimation methods presented before [GAR 91, LAR 92]. The works presented here, taken from [CAR 91], are related to the application of spectral estimation methods in the analysis of swell, from digital images coming from remote sensing radar systems. The specificity of the radar images can indeed lead us to better determine the spectral estimation methods to be used. The process of formation of an ocean image by a radar system is much more complicated than in the case of optical imaging. The radar system is of active type: it emits a hyperfrequency signal in the form of impulses, which interact with the illuminated terrestrial medium and are retrodiffused towards the (real or synthetic) antenna of the system. We can exactly know the process of acquisition of a radar image only if the radar-ocean interaction is perfectly known. This type of interaction was the object of several modelings (see, for example, [ELA 88, MAI 00]). To simplify, we will deal only with noise inherent in radar images. In passive optical imaging, we can admit, to an extent, that the original 2-D signal is disturbed by an additive noise which is the sum of different noises of atmospheric origin or coming from the material itself. This is not the case with radar images.

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Indeed, they contain a speckle noise, which is due to the constructive or destructive interferences of radar signals reflected by land scatterers and integrated in a resolution cell by the system. This noise is identical to the one that we observe by illumination of a rough surface with a laser beam. In order to reduce the incidence of this noise in images, the majority of remote sensing radar systems are known to carry out an average of independent N looks of the same scene: this is the principle of Synthetic Aperture Radar (SAR). We often regard speckle noise as multiplicative and statistically independent of the useful signal [BUS 75, ULA 86, MAI 00]. Consequently, if I(k1 , k2 ) represents the observed radar image process, S(k1 , k2 ) the “useful image” and Z(k1 , k2 ) the speckle noise, we can write: I (k1 , k2 ) = S (k1 , k2 ) Z (k1 , k2 )

(3.55)

It should be noted here that, generally, each one of these random fields is strictly positive. In addition, we assume that these fields are WSS. It was experimentally shown in [CAR 91, GAR 91] that speckle noise could be modeled to a good extent by a non-centered white noise, which corroborates the theory [GOO 75, ULA 86]. Thus, its second ACF is written as: 2 rZ (m1 , m2 ) = μ2Z + σZ δ(m1 , m2 )

(3.56)

3.8.2. Stochastic modeling of a radar image Statistical independence of noisy and non-noisy image leads to the following relations expressing the mean and the second ACF of the image in square root of intensity: E[I] = E[S] E[Z] = μS μZ = μI

(3.57)

E[I(k1 , k2 ) I(k1 − m1 , k2 − m2 )] = rS (m1 , m2 ) rZ (m1 , m2 ) = rI (m1 , m2 )

(3.58)

By using relation (3.56), we can show that the first ACF of the image is written as: 2 γI (k1 , k2 ) = μ2Z γS (k1 , k2 ) + σZ (σS2 + 2μ2S ) δ(k1 , k2 )

(3.59)


167

and the application of the Fourier transform gives: 2 SI (ν1 , ν2 ) = μ2Z SS (ν1 , ν2 ) + σZ (σS2 + 2μ2S )

(3.60)

The superimposition of a multiplicative white noise in the original image modifies only the original spectrum in a linear way. Thus, in a close multiplicative constant, we are brought back to the case of additive noise. Let us now assume that we model the original image S by an AR process, of parametric PSD: 2 σE SS (ν1 , ν2 ) = # #2 # # # a (ν1 , ν2 )#

(3.61)

with a (ν1 , ν2 ) as the transfer function of the inverse filter evaluated on the unit torus. The PSD of the noisy image (now centered) has the value: # #2 # # 2 2 + σZ (σS2 + 2μ2S ) # a (ν1 , ν2 )# σE (3.62) SI (ν1 , ν2 ) = # #2 # # # a (ν1 , ν2 )# This PSD is of ARMA type. The AR and MA parameters are also proportional except those corresponding to the development of the PSD in (ν1 , ν2 ) with order 0. ARMA modeling for spectral estimation is thus, in theory, better adapted to the analysis of radar images than AR modeling. 3.8.3. Example of application Figure 3.19 shows an application result of ARMA-HM technique on a SEASAT4 image of the island region of Glénan in South Brittany. One of the possible applications of the local wavelength series of swell is the bathymetry estimation in shallow water (see [CAR 91, GAR 91] for an example in optical imagery). In order to detect the local variations of wavelength and direction of the swell, this image, of size 512 × 512, was cut into 32 × 32 blocks of 16 × 16 pixels without overlapping. The ARMA-HM technique was then applied to each block, by choosing AR (5, 5)-MA (5, 5) models for the first and second quadrants, and by raising the first two maxima in each PSD. The latter were then converted into vectors representing the wavelength and the direction of the swell, given in Figure 3.19b. This estimated cartography of surface phenomena can be superimposed on the original image.

4. This image was kindly provided to us by Ph. Forget, of LSEET of Toulon.

168


(a)

(b)

Figure 3.19. Analysis of a field of a locally stationary swell: a) SEASAT radar image; b) local direction and wavelength vector field


169

Experimentally, it was shown that this cartography, obtained with ARMA spectral estimation, is, among those obtained with different estimators presented in this chapter, the most coherent with respect to initial data [GAR 91].

3.9. Conclusion In this chapter, we have presented various approaches enabling us to carry out the spectral analysis of 2-D signals. A first family of traditional estimators does not require any knowledge a priori of the 2-D data to be analyzed and simply uses the original definition of the PSD of a signal. These methods have the advantage of simplifying the implementation, at the cost of certain disadvantages in terms of detection ability or estimation of harmonic components, but mainly adequacy to theorectical PSD in the case of a WSS process with regular PSD. The second family corresponds to parametric approaches. For some of these methods, a model a priori of the PSD must be selected according to the nature of the data. The set of approaches by stochastic linear models fall within this framework. For others, like the methods of maximum entropy or the minimum variance, obtaining an estimation of the PSD is subjected to the adoption of a general principle applied to 2-D data. A third approach is reserved for processing 2-D signals which can be modeled as sums of harmonic components. For corresponding methods, like 2-D MUSIC, good performances obtained are characteristic of considering this information a priori on the nature of the signal. All the methods presented in this chapter are not specific to the analysis of 2-D signals; actually, only methods based on linear models seen in Chapter 2, the ME method and the separable approaches are not direct extensions of the 1-D case. This comes, once again, from the non-existence of an implicit causality on a Z 2 network. Contrary to the 1-D case, the performances, which are compared to traditional spectral, parametric estimators in high resolution in the 2-D cases, are relatively unknown. Here, we tried to gather some important elements about them. Lastly, we saw that 2-D spectral analysis could find applicability in imagery, i.e. the analysis of quasi-periodic WSS signals with regular PSD. This framework differs, to a great extent, from other traditional 2-D applications and also by using spectral analysis such as beamforming methods.

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Chapter 4

Bispectral Analysis of 2-D Signals

4.1. Introduction In this chapter, we tackle higher order statistics (HOS) for 2-D signals. In the first part, we recall the definitions of moments and spectra of higher orders as well as their generalization to 2-D signals. We will see that the third-order moments as well as the bispectra are 4-D quantities and that the exploitation of the information contained in this structure is found to be difficult. We have to define the structures (projections following given axes) to interpret the bispectral information – which is done in section 4.5.2. In contrast to the first three chapters, the variables with indices 1 or 2 no longer represent the variables following a particular axis. Indeed, for the HOS, we use a sequence of 2-D variables with indices from 1 to p, each of these variables ki thus having two components noted by νxi and νyi . For reasons of readibility, we will detail the explicit 2-D writing only when necessary.

4.1.1. Higher order moments and cumulants Here, we regroup the definition of moments and cumulants. First, we have to take a collection of distinct or non-distinct random variables, X = (X1 , . . . Xi , . . . , Xp ). This collection can, in particular, be arranged according to a 2-D structure Xi = X(k, l). Any analysis from moments and spectra passes, beforehand, through the first

Chapter written by Jean-Marc LE CAILLEC and René GARELLO.

176


joint characteristic function defined by: φ(ω1 , . . . , ωi , . . . , ωp ) = E[ej(ω1 X1 + · · · + ωi Xi . . . + ωp Xp ) ] = E[ej ω X ] (4.1) where E[.] denotes the expected value, ω the vector ω = (ω1 , . . . , ωp ) and X = (X1 , . . . , Xp )T . The joint moment of order p is thus defined (first definition) as the derivative of the joint characteristic function of order p as compared with each one of the variables ωi , since the value of the derivative is taken at the origin: # # p p ∂ φ(ω1 , . . . , ωi , . . . , ωp ) # (4.2) EX1 ,...,Xi ,...,Xp = (−j) # ∂ω1 , . . . , ∂ωi , . . . , ∂ωp # ω1 =...=ωi =...=ωp =0

By using a development in Taylor series, φ(ω1 , . . . , ωi , . . . , ωp ) is explained by: φ(ω1 , . . . , ωi , . . . , ωp ) =

X μ1 +...+μp ≤n

j μ1 +...+μp μ μ E[X1μ1 . . . Xp p ] ω1μ1 . . . ωp p + o(ωin ) μ1 ! . . . μp !

Only the final factor of ω1 . . . ωp is non-zero during the derivation with respect to each of the variables and hence we obtain a second definition of the moment of order p which can be written as the expected value of the product of all the variables of the collection X: EX1 ,...,Xi ,...,Xp = E[X1 . . . Xi . . . Xp ]

(4.3)

The properties of the moments are simple enough and we will explain them in section 4.1.2. The second tools for the statistical analysis are the cumulants, which are defined from the second joint characteristic function, which is the Napier logarithm of the first joint characteristic function: ψ(ω1 , . . . , ωi , . . . , ωp ) = ln(φ(ω1 , . . . , ωi , . . . , ωp )) = ln(E[ej(ω1 X1 +...+ωi Xi +...ωp Xp ) ])

(4.4)

The cumulant of order p of the collection X is defined, as previously, as the derivative of the original second characteristic function at the origin as compared to each of the variables ωi: # # p p ∂ ψ(ω1 , . . . , ωi , . . . , ωp ) # (4.5) CX1 ,...,Xi ,...,Xp = (−j) # ∂ω1 , . . . , ∂ωi , . . . , ∂ωp ) # ω1 =...=ωi =...=ωp =0


177

Just as the characteristic functions are linked to each other, the cumulants and moments are not independent quantities. They are connected by the relation [BRI 65, LEO 59, MEN 91, ROS 85]: (4.6) (−1)q−1 (q − 1)! E Xi E Xi . . . E Xi CX1 ,...,Xi ,...,Xp = i∈s1

i∈s2

i∈sq

where denotes the sum of all the partitions of Np = (1, 2, . . . , p), since these sets are denoted by si and q is the number of non-empty sets forming this partition1. In order to better understand this relation, we will illustrate it with an example, i.e. the third order cumulant. The N3 partitions have been given in Table 4.1. q 1 2 3

s1 1, 2, 3 1, 2 1, 3 2, 3 1

s2 ? 3 2 1 2

s3 ? ? ? ? 3

Table 4.1. Partition of N3

These partitions give the following relation between the third order cumulant and the moments of collection X: CX1 ,X2 ,X3 = EX1 ,X2 ,X3 − EX1 EX2 ,X3 − EX2 EX1 ,X3 −EX3 EX1 ,X2 + 2EX1 EX2 EX3 If the variables are of zero mean, the third order cumulant and moment are confused, which is not the case for the fourth order cumulant and moment (but the expression of this cumulant is simplified in a significant manner). The question that naturally comes to mind is: why is there such a duality in the two statistic tools? Moments are quantities that we can qualify as “physical” since they are linked to the values obtained by the process, which is not so in the case of cumulants. Cumulants verify the “intrinsic” distribution property of the law. For example, the second order moment gives the mean interval, which can take the values of a random process. This mean interval cannot be compared for two statistical laws with different means, while for the variance, which is the second order cumulant, the two laws can be easily compared.

1. A partition of example Np is a collection of sets si , i ∈ Nq such that: q si = Np and sj ∩ si = ?(i = j) i=1

178


In the same way, the moments can be expressed according to cumulants of the lower order by the formula: EX1 ,...,Xi ,...,Xp =

C Xi C Xi . . . C Xi i∈s1

i∈s2

(4.7)

i∈sq

where denotes the sum of all the partitions of Np = (1, 2, . . . , p) and q the number of non-empty sets forming this partition. For a 1-D or 2-D signal which is stationary up to order p, the moment of order p will be noted: EX (n1 , n2 , . . . , np−1 ) = E[X(i) X(i + n1 ) . . . X(i + np−1 )]

(4.8)

The symmetric relations for 1-D signals are given by: EX (n1 , n2 ) = EX (n2 , n1 ) = EX (−n2 , n1 − n2 ) = EX (−n1 , n2 − n1 ) = EX (n1 − n2 , −n2 ) = EX (n2 − n1 , −n1 )

(4.9)

which gives for 2-D signals: EX (n1x , n1y , n2x , n2y ) = EX (n2x , n2y , n1x , n1y ) = EX (−n2x , −n2y , n1x − n2x , n1y − n2y ) = EX (−n1x , −n1y , n2x − n1x , n2y − n1y ) = EX (n1x − n2x , n1y − n2y , −n2x , −n2y ) = EX (n2x − n1x , n2y − n1y , −n1x , −n1y ) (4.10) The support of the third order moment (which will be noted by SM (N )) is defined by: |n1x | ≤ N, |n2x | ≤ N, |n2x − n1x | ≤ N and |n1y | ≤ N, |n2y | ≤ N, |n2y − n1y | ≤ N

(4.11)

The non-redundant support that will be noted by NRSM(N ) is given by: 0 ≤ n2x ≤ n1x ≤ N

and

|n1y | ≤ N,

|n2y | ≤ N,

|n2y − n1y | ≤ N

(4.12)

The complete and the non-redundant support are represented in Figure 4.1. The non-redundant support can still be slightly reduced [ERD 92].


N

179

n2 n 2= n 1

n1 -N

N

-N

Support of the third order moment for SM(N) 1-D signals Non-redundant support of the third order moment for NRSM(N) 1-D signals

Figure 4.1. Complete and non-redundant support of the third order moment for 1-D signals

4.1.2. Properties of moments and cumulants There are three properties for moments and seven for cumulants. The demonstrations which are not described here can be found in an article by Mendel [MEN 91]. These properties do not assume any distribution hypotheses of the random variables. We will begin with the simplest properties, i.e. the symmetry properties. 1) Permutability of the arguments of moments and cumulants: EX1 ,...,Xi ,...,Xj ,...,Xp = EX1 ...,Xj ,...,Xi ,...Xp CX1 ,...,Xi ,...,Xj ,...,Xp = CX1 ...,Xj ,...,Xi ,...Xp

(4.13)

180


2) The scale change of the arguments implies the same scale change for moments and cumulants: Eλ1 X1 ,...,λi Xi ,...,λp Xp =

p

λk EX1 ...,Xi ,...Xp

(4.14)

λk CX1 ...,Xi ,...Xp

(4.15)

k=1

Cλ1 X1 ,...,λi Xi ,...,λp Xp =

p k=1

3) Distributivity of the moment and cumulant as compared with the arguments: EX1 ,...,Yi +Zi ,...,Xp = EX1 ...,Yi ,...Xp + EX1 ...,Zi ,...Xp

(4.16)

CX1 ,...,Yi +Zi ,...,Xp = CX1 ...,Yi ,...Xp + CX1 ...,Zi ,...Xp

(4.17)

4) The cumulants are insensitive to the additive constants: CX1 +α,...,Xi ,...,Xp = CX1 ...,Xi ,...Xp

(4.18)

5) The cumulant of a random variable collection, which can be expressed as the sum of two independent random variable collections, is equal to the sum of the cumulants of each of the two families: CX1 +Y1 ,...,Xi +Yi ,...,Xp +Yp = CX1 ...,Xi ,...Xp + CY1 ...,Yi ,...Yp

(4.19)

6) If there is a sub-collection of independent random variables X1 , . . . , Xn of the rest of the collection Xn+1 , . . . , Xp then the cumulant of the collection is zero. 7) Cumulants of orders higher than two of a collection of jointly Gaussian random variables are zero. From this latter property, two other results should be noted: – the moment of order p of a collection of Gaussian random variables can be expressed according to the moments of orders strictly less than p of sub-collections of X, which, by applying relation (4.6) for order 3, gives: EX1 ,X2 ,X3 = EX1 EX2 ,X3 +EX2 EX1 ,X3 +EX3 EX1 ,X2 −2EX1 EX2 EX3 (4.20) Using descending recurrence on the order of the moment, we find that a Gaussian law is perfectly defined thanks to its first two moments; – in addition, if the mean vector μ is identically zero then, only for third and fourth orders the definition of the cumulant of a collection of any random variables can be expressed by: CX = MX − MI

(4.21)

where I is a collection of Gaussian random variables having the same covariance matrix as collection X.


181

However, let us note that the hypotheses, for order 3, of a zero mean vector is not necessary for relation (4.21) to be true. If this definition seems limited due to the fact that it is valid only for two orders, it is extremely useful as in practice we often restrict ourselves to the study of these two orders. For this reason, many authors express the cumulant as a measure of the deviation of a law as compared to the equivalent Gaussian law, which is not strictly correct. More generally, it is to be remembered that any symmetric distribution has a third order cumulant, which is identically zero [NIK 93]. 4.1.3. Polyspectra of stationary signals The polyspectra or spectrum of higher orders is defined as the Fourier transform (FT) of the (cumulant) moment of order p for stationary signals:

SX (k1 ,. . ., kp−1 ) =

+∞ n1 =−∞

...

+∞

EX (n1 ,. . ., np−1 )e−j(n1 k1 +...+np−1 kp−1 )

np−1 =−∞

(4.22) The existence of SX (k1 , . . . , kp−1 ) has been supposed by assuming, for example, that the moment of order p is absolutely summable. However, the definition is dual and we can define the polyspectra of order p as the mean of the Fourier coefficients, which gives: ˜ 1 ) . . . X(k ˜ p−1 ) X ˜ ∗( SX (k1 , . . . , kp−1 ) = E[X(k

p−1

ki )]

(4.23)

i=1 N −1

˜ with X(k) = i=0 X(i)e−jki and the conditions of addition which induce the existence of the FT of the stationary process. The two definitions are of course equivalent. The demonstration of equivalence of the definitions is made by considering the specns : trum of non-stationary signals noted by SX ns ˜ 1 ) . . . X(k ˜ p−1 ) X(k ˜ p )] SX (k1 , . . . , kp−1 , kp ) = E[X(k

˜ then by replacing X(k) with its expression according to X(n). The stationarity of order p of the signal implies that the spectrum of order p is zero outside the hyperp p−1 planes ki ≡ 0[2π] and thus kp = − ki is called the hidden variable of stationi=1

i=1

arity. The definitions of the polyspectrum, and more importantly of the bispectrum, imply certain properties, which we present below. The higher order spectra are, contrary to the second order spectrum, complex quantities (the moment of order p does not verify the Hermitian symmetry

182


EX (n1 , . . . , np−1 ) = EX (−n1 , . . . , −np−1 )). The second order spectrum is real which implies that all the phase information is lost by the second order statistics, as will be shown again in section 4.2.1 for non-minimum phase filters. The second order statistics and the spectrum are said to be phase blind. The polyspectra are periodic: SX (k1 , . . . , ki + 2π/Te , . . . , kp−1 ) = SX (k1 , . . . , ki , . . . , kp−1 )

(4.24)

Te being the sampling period which will be considered equal to 1 in what follows.

␻3 (0 ,⫺␲ , ␲)

(⫺␲ ,0, ␲)

(␲ , ⫺␲ , 0)

␻1

(⫺␲ ,␲ ,0) (␲ , 0 , ⫺␲)

␻2 (0 , ␲ , ⫺␲) Support of the bispectrum

Figure 4.2. Intersection of the plane ω3 = −ω1 − ω2 and of the hypercube (−π, +π)3


183

The support of the bispectrum (third order polyspectrum) for stationary signals is ns (k1 , k2 , k3 ) thus the intersection of the hypercube [−π, +π]3 , due to the periodicity of SX and the planes: k1 + k2 + k3 ≡ 0[2π] The support is constituted from a main hexagon provided by the intersection of the plane k1 + k2 + k3 = 0 and of two triangles [ROS 85] due to the intersection of: k1 + k2 + k3 = ±2π In fact, a non-zero bispectrum on these two triangles indicates a problem of aliasing the data (see [HIN 68]). By considering that there is no cover of the spectrum for the considered signals, the bispectrum will be taken to be zero on these two triangles. Thus, the definition support for 2-D signals is more precisely defined by: |kx1 | < π

and

|kx2 | < π

and

|kx1 + kx2 | < π

and:

(4.25) |ky1 | < π

and

|ky2 | < π

and

|ky1 + ky2 | < π

Certain authors include this condition in the definition of the bispectrum, while it refers only to a reduction of the support to the data without cover of the spectrum. The following are the properties of symmetry of the bispectrum for real signals: B(k1 , k2 ) = B(k2 , k1 ) = B ∗ (−k2 , −k1 ) = B(k1 , −k1 − k2 ) = B(−k1 − k2 , k1 ) = B(k2 , −k1 − k2 ) = B(−k1 − k2 , k2 )

(4.26)

or in developed writing: B(kx1 , ky1 , kx2 , ky2 ) = B(kx2 , ky2 , kx1 , ky1 ) = B ∗ (−kx1 , −ky1 , −kx2 , −ky2 ) = B(kx1 , ky1 , −kx1 − kx2 , −ky1 − ky2 ) = B(−kx1 − kx2 ,−ky1 − ky2 , kx1 , ky1 ) =

(4.27)

B(kx2 , ky2 ,−kx1 − kx2 ,−ky1 − ky2 ) = B(−kx1 − kx2 ,−ky1 − ky2 , kx2 , ky2 ) For a 1-D signal, the non-redundant support is thus the triangle AOB of Figure 4.3. It is necessary to note that the symmetries are non-orthogonal axial symmetries. As an example, the node A(N, 0) of the plane is symmetric to A (N, −N ) by the symmetry: x→x

y → −x − y

184


and the node B (N/2, −N ) is the symmetric of B(N/2, N/2) by the same symmetry. The axial symmetry is the axis ω1 = −2ω2 and the direction of the symmetry is the axis 0x. The axial symmetry for the real signals, as well as the support of the bispectrum for 1-D signals, are represented in Figure 4.3. The functions having bispectral symmetry (see equation (4.26)) form a vectorial space. Moreover [TEK 90], for any sequence having a symmetry of the third order moment (and hence by bijection for any sequence having bispectral symmetry), a stochastic process corresponds whose third order moment is equal to this sequence (the demonstration for 2-D signals is given in [LEC 97]). The definitions and properties of the cumulants and polyspectra of complex signals [JOU 92] can be multiple because of the positioning of one or several conjugate complex quantities in the moment of order p and in the spectrum of order p. The definition of moments and spectra of higher orders of complex signals implies that there exists less symmetry for these signals.

␻1 ⫽ 0

␻1 ⫽⫺␻2

␻1 ⫽ ␻2 B

0

A

␻2⫽ 0

␻1 ⫽ ⫺2␻2 B'

A'

⫺2␻1 ⫽ ␻2

Region with conjugate symmetry with respect to the triangle AOB

Figure 4.3. Bispectrum support for 1-D signals


185

4.1.4. Polyspectra The polyspectrum of order p is defined as the inverse Fourier transform of the logarithm of the polyspectrum of order p: CeX (n1 , . . . , np−1 ) = +∞

+∞

... k1 =−∞

ln SX (k1 , . . . , kp−1 ) ej(n1 k1 +...+np−1 kp−1 )

(4.28)

kp−1 =−∞

When developing the expression for the bispectrum, this is equivalent to: CeX (n1x , n1y , n2x , n2y ) =

+∞ −∞

+∞

−∞

+∞

−∞

+∞ −∞

1 1 1 1 2 2 2 2 ln B(kx1 , ky1 , kx2 , ky2 ) ej(nx kx +ny ky +nx kx +ny ky ) dkx1 dky1 dkx2 dky2

(4.29)

This definition is calculated on diverse definitions of the spectrum of the existing second order statistics (complex spectrum, of power, differential, etc.). The properties of the symmetry of the polyspectra are equivalent to those of the moments of the same order. The other properties, especially with regard to linearity, are detailed in section 4.2.4.

4.1.5. Definition of the coherence of order p The coherence of order p is defined in the following manner: SX (k1 . . . , kp−1 )

P (k1 , . . . , kp−1 ) =

SX (k1 ) SX (k2 ) SX (k1 + k2 )

(4.30)

Introduced in a study by Brillinger [BRI 65], the bicoherence can be assimilated with a normalized spectrum of order p and possesses the properties of linear signals (see section 4.2.5).

4.2. Moments and spectra of order p for linear signals This section introduces the properties of the spectra of order p for linear signals which can be produced by a white noise using a linear filter. It tackles the calculation of a spectrum of order p and certain properties deduced for this type of signals.

186


4.2.1. Moments and cumulants of order p for linear signals In the case of linear filters, the moment of order p of the output signal is easily expressed by: EX (n1 , n2 . . . , np−1 ) +∞

h(m1 ) W (i − m1 ). . .

= E[ m1 =−∞ +∞

mp =−∞

+∞

...

= m1 =−∞

+∞

mp =−∞

h(mp ) W (i + np−1 − mp )]

(4.31)

h(m1 ) . . . h(mp )EW (n2 + m1 − mp ,. . ., np−1 + m1 − mp )

In particular, if the input signal is a white noise, which possesses a moment of order p non-zero only at node 0, EW (0, . . . , 0) = γp , then: EX (n1 , n2 . . . , np−1 ) = γp

+∞

h(m1 ) h(m1 +n1 ) . . . h(m1 +np ) (4.32)

m1 =−∞

This calculation allows us to show that the second order statistics cannot distinguish certain filters. The example initially developed in the works of Nikias et al. [NIK 87, NIK 93] is adapted here in the 2-D case. Let us consider the following three moving average filters: H1: Y (n) = X(n) − (a + b)X(n − 1) + abX(n − 2) H2: Y (n) = −aX(n + 1) + (1 + ab)X(n) − bX(n + 1)

(4.33)

H3: Y (n) = abX(n + 2) − (a + b)X(n + 1) + X(n) H1 , H2 , H3 EX (0) 1 − (a + b)2 + a2 b2 EX (1) −(a + b)(1 + ab) EX (2) ab Table 4.2. Second order moment for three filters

If we consider that these three filters have an input of the same noise, we can verify that the output signals have the same second order statistics, which are given in Table 4.2. On the other hand, a calculation of the third order moment allows us to differentiate them, as is described in Table 4.2. The identification of a non-minimum phase filter with the help of a sequence of the second order moment is thus not possible, as this sequence always leads to the identification of the spectrally equivalent minimum phase filter.


187

H1 H2 H3 EX (0,0,0,0) 1 −(a +b)3 +a3 +b3 1 −(a +b)3 +a3 +b3 (1 +ab)3 −a3 −b3 EX (1,0,1,0) (a +b)(−1 +a2 b +ab2 ) (a +b)(a +b −a2 b2 ) (1 +ab)(a2 −b +ab2 ) EX (1,1,1,1) (a +b)(a +b −a2 b2 ) (a +b)(−1 +a2 b +ab2 ) (1 +ab)(−a −a2 b +b2 ) EX (2,0,2,0) ab a2 + b 2 −a2 b EX (2,1,2,1) −(a + b)ab −(a + b)ab ab(1 + ab) EX (2,2,2,2) a2 b 2 ab −ab2 Table 4.3. Third order moment for three filters

4.2.2. Spectrum of order p for a linear signal This calculation, initially developed by Brillinger and Rosenblat [BRI 67] for the 1-D case, is easily generalized for the 2-D case. By once again taking expression (4.31), we see that it refers to a convolution of p dimensions and by the Fourier transform, we get:

SX (k1 , . . . , kp−1 ) = H(k1 ) . . . H(kp−1 ) H(−

p−1

ki ) SW (k1 , . . . , kp−1 )

i=1

(4.34) If the input signal is a white noise, in other words SW (k1,. . ., kp−1 ) = γp , the spectrum of order p of the output signal is thus: SX (k1 , . . . , kp−1 ) = γp H(k1 ) . . . H(kp−1 ) . . . H(−

p−1

ki )

(4.35)

i=1

4.2.3. General properties of the bispectra of linear signals From the properties of linear signals in the previous section, we can introduce the following definition. A signal X(n) will be called linear (or linearizable) in order p if there is a transfer function H(k) such that its spectrum of order p can be written as: SX (k1 , . . . , kp−1 ) = γp H(k1 ) . . . H(kp−1 ) H(−

p−1

ki )

with

γp = 0

(4.36)

i=1

In an equivalent manner, the spectrum of order p will be called factorable. For any spectrum of order p factorable by H(k), there exists a filter with finite impulse response whose transfer function is equal to H(k) [TEK 89]. Hence, three properties are deduced from this:

188


– the second order spectrum is always linearizable. Indeed, we can always write S(k) = S(k) S(k) (here we do not consider the modeling conditions of the spectrum by causal filters). Hence, all the transfer functions whose module is equal to S(k) allow us to verify the linearization of the spectrum. Under this property, which seems simple, it is possible to feel the intrinsic notion of non-linearity as it refers, in fact, to a relation existing between the amplitude and phase information of the Fourier transform. In the spectrum, the phase information being absent, all the signals are hence linear for the second order spectrum; – the factorization in the form of a linear bispectrum is not always possible [TEK 89]; – the set of factorable bispectra is of zero measure and the set of factorable bispectra is not dense in the set of bispectral functions [TEK 90]. In conclusion, it should be remembered that a bispectrum could almost never be equal to a bispectrum of the output of a linear filter. Moreover, in this case, it cannot be approximated (supposing that we have normalized the set of bispectral functions) by the bispectrum of an output signal of a linear filter. This implies that a parametric estimation of the bispectrum cannot converge towards this bispectrum. 4.2.4. Polyspectrum of a linear signal The most important property of polycepstra applies only to linear signals. Indeed, the polyspectrum of order p of a linear signal is zero outside the planes: = 0; np−1 P b1 = (n1x = 0; n1y = 0), . . . , P bp−1 = (np−1 = 0) x y

(4.37)

and: 1 p−1 P bp = (n1x = . . . = np−1 ) x , ny = . . . = ny

for 2-D signals. By introducing (4.35) in (4.29), we get: CeX (n1 , . . . , np−1 ) = !

+π −π

...

+π

−π

∗

ln(H(k1 )) +. . .+ ln(H(kp−1 )) + ln(H (

p

" ki )) e

p j i=1

ki n i dk1 . . . dkp−1

i=1

(4.38) Hence, the first terms are distributed according to ni , the last term being nonzero for n1 = . . . = np−1 , which in the 2-D case is explained by the planes in (4.37).


189

In the case of linear signals, the bispectrum is linked to the complex spectrum [PAN 88], the latter being defined [OPP 89] by:

+π

ln(H(k)) ej n k dk

c(n) =

(4.39)

π

In the 1-D case, for the complex spectrum, the components situated on n ≥ 0 arise from the minimum phase side of the filter, while the components found on the side of the negative axis contribute to the maximum phase transfer function.

4.2.5. Coherence of order p for linear signals By applying relation (4.30), for a linear signal whose spectrum of order p is expressed by (4.35), we get: |P (k1 , k2 )|2 =

γp p/2

γ2

(4.40)

The important property of the coherence of order p for linear signals is that it is flat, irrespective of p on the set of the definition of the spectrum of order p. In the case of (4.40), bicoherence (p = 3) is equal to the normalized Skewness of the input signal. In case of tricoherence (p = 4), we would have obtained the normalized Kurtosis. Contrary to a frequently held idea, it should be noted that bicoherence is not necessarily less than 1 [RAG 90]. This can make the use of bicoherence tricky and the constant of bicoherence is not an easy phenomenon to note, as there can be some variations due to errors of estimation which can be taken for non-linearity (we will see in section 4.5.1 the hypotheses tests for the detection of non-linearity based on bicoherence). Moreover, in the case of 2-D signals, bicoherence being a 4-D structure, it is necessary to project the information in 2-D structures to understand better the information contained in bicoherence.

4.3. Signals in quadratic phase coupling, non-linearity and the Volterra system When we apply the polyspectra and polycepstra for the detection of non-linearity, the results are difficult to interpret from a physical point of view (as we will see in section 4.3.2.2). Another possibility, to introduce the detection of non-linearity, whose meaning is more immediate, rests in the phase coupling. In this section, we are going to describe the mechanisms for detection of phase coupling with the help of higher order statictics; in the second part, we will see that the quadratic coupling of the phase induces a non-zero bispectrum for a non-linear system having Gaussian data as input.

190


4.3.1. Bispectrum of a signal in quadratic phase coupling In order to introduce the notion of phase coupling, we have to consider a signal made up of three waves whose vector of one of the sinusoids is the sum of the wave vectors of the other two: X(n) = A1 cos(k1 n + ϕ1 ) + A2 cos(k2 n + ϕ2 ) + A3 cos((k1 + k2 ) n + ϕ3 ) (4.41) where Ai is a positive random variable and ϕi is a random phase which is uniformly distributed on [0, 2π[. Moreover, we suppose that ϕ1 and ϕ2 are independent. ϕ3 , as we are going to see, may or may not be independent of the other two phases. If we calculate the product X(n) X(n + n1 ) X(n + n2 ), we get a sum product of the three cosines (with a multiplicative coefficient dependent on Ai ). This product can then be written in the form of a sum of four cosines. For example: A1 cos(k1 n +ϕ1 ) A2 cos(k2 (n +n1 ) + ϕ2 ) A3 cos((k1 +k2 ) (n +n2 ) + ϕ3 ) = A1 A2 A3 cos(k1 n + ϕ1 + k2 (n + n1 ) + ϕ2 + (k1 + k2 ) (n + n2 ) + ϕ3 )+ 4 cos(k1 n + ϕ1 + k2 (n + n1 ) + ϕ2 − (k1 + k2 ) (n + n2 ) − ϕ3 )+ cos(k1 n + ϕ1 − k2 (n + n1 ) − ϕ2 + (k1 + k2 ) (n + n2 ) + ϕ3 )+ cos(k1 n + ϕ1 − k2 (n + n1 ) − ϕ2 − (k1 + k2 ) (n + n2 ) − ϕ3 ) (4.42) The phase of each of these terms can be expressed as ϕ1 ± ϕ2 ± ϕ3 . Thus, there are two separate cases: – either ϕ1 , ϕ2 , ϕ3 are the independent random variables uniformly distributed on [0, 2π] and their sum is also uniformly distributed on [0, 2π], hence the expected value of the cosines is zero; – or the last variable is linearly linked to the other two variables, for example, ϕ1 + ϕ2 = ϕ3 and thus the phase in the second term of (4.42) is no longer random, its expected value is no longer zero and it is equal to: E[A1 A2 A3 ] cos(k2 n1 − (k1 + k2 ) n2 ) 4

(4.43)

In this last case, by again taking all the possibilities, we obtain: E[A1 A2 A3 ] [cos (k2 n1 − (k1 + k2 ) n2 ) + 4 cos (k2 n2 − (k1 + k2 ) n1 )) + cos (k1 n1 − (k1 + k2 ) n2 ) +

EX (n1 , n2 ) =

cos (k1 n2 − (k1 + k2 ) n1 ) + cos (k1 n1 + k2 n2 ) + cos (k1 n2 + k2 n1 )] (4.44)


191

Hence, we again find the fact that the third order moment is the sum of six cosines, all of them symmetric by the symmetry of the third order moment (see equation (4.9)), since the non-redundant information is contained in a single cosine. The sinusoids in phase coupling of the signal X(n) constitute a perfectly symmetric triplet (that we find in the structure of the third order moment and thus in the bispectrum). For example, the second wave of the wave vector k2 can be seen as being in coupling with the waves of the wave vector k1 , [−k1 − k2 ] (we have −ϕ2 = ϕ1 − ϕ1 − ϕ2 ). Moreover, we note that in the traditional case of phase coupling of (4.41), the moment is stationary and does not depend on n. In the case where we have the relation, ϕ3 = ϕ1 + ϕ2 , we will say that the signal is in quadratic phase coupling. In fact, it would be more correct to say that the signal is in phase coherence, as a relation of the type ϕ3 = ϕ1 + ϕ2 + c, where c is a constant, induces similar results to those of (4.44) (modulo a de-phasing of c). This approach can clearly be generalized to higher orders and we will talk of the cubic phase coupling between four waves if their phases verify ϕ4 = ϕ1 + ϕ2 + ϕ3 + c. The bispectrum of a signal in phase coupling can be obtained by the Fourier transform of (4.44), which gives: E[A1 A2 A3 ] B(k1 , k2 ) = 8 δ(k − k2 , k + k1 + k2 ) + δ(k + k2 , k − k1 − k2 ) +δ(k − k1 , k + k1 + k2 ) + δ(k + k1 , k − k1 − k2 ) +δ(k + k1 + k2 , k − k2 ) + δ(k − k1 − k2 , k + k2 )

(4.45)

+δ(k + k1 + k2 , k − k1 ) + δ(k − k1 − k2 , k + k1 ) +δ(k + k2 , k + k1 ) + δ(k − k2 , k − k1 )

+δ(k + k1 , k + k2 ) + δ(k − k1 , k − k2 )

This result is more easily found in the frequency domain by directly applying the second definition of the bispectrum. Indeed, the Fourier coefficients of the signal of equation (4.41) are: 1 ˜ X(k) = A1 δ(k − k1 )ej ϕ1 + A1 δ(k + k1 )e−j ϕ1 + A2 δ(k − k2 )ej ϕ2 + 2 A2 δ(k + k2 )e−j ϕ2 + A3 δ(k − k1 − k2 )ej ϕ3 + A3 δ(k + k1 + k2 )ej ϕ3 (4.46) ˜ )X(k ˜ )X(k ˜ + k ) is non-zero, for example, if k = k1 and The product X(k k = k2 , and in this case we get:

˜ 1 )X(k ˜ 2 )X ˜ ∗ (k1 + k2 )] = E[A1 A2 A3 ] E[ej (ϕ1 +ϕ2 −ϕ3 ) ] E[X(k 8 and thus we again find the previous results.

(4.47)

192


The expected value E[ej (ϕ1 +ϕ2 −ϕ3 ) ] is zero if the three phases are independent and non-zero if we have a relation of type ϕ3 = ϕ1 + ϕ2 . These simple results are easily applicable to non-linearity. If we take a signal constituted of two waves of wave vectors k1 and k2 , respectively, and we square it, we get a signal which is also constituted of two wave vectors k1 − k2 and k1 + k2 . k1 + k2 is thus one of the two wave vectors of natural interactions, but it is also the hidden variable of the stationarity. It is this ambivalence of the wave vector, as well as the position of the interaction and hidden variable of the stationarity, which explains the simple results obtained for the bispectrum of a signal in phase coupling and which explains that the third order moment is stationary. If the quadratic place of interaction is no longer the sum of the wave vectors, the moment is not stationary anymore [LEC 96]. Finally, a last interesting point is that if we now suppose that for the wave vector k1 + k2 , there is a wave of amplitude A3 which is in phase coupling and a wave of amplitude B which is not in phase coupling and if, in addition, we suppose that A3 = αA1 A2 (as is the case for Volterra modules, as we will see in section 4.3.2), then the square module of bicoherence allows us to quantify the energy in phase coupling by: |P (k1 , k2 )|2 = α2

E[A23 ] (E[A21 ] E[A22 ])2 = E[A21 ] E[A22 ] E[A23 + B 2 ] E[A23 + B 2 ]

(4.48)

Unfortunately, in the more general cases, this result cannot be applied, as often there simultaneously exists some linear and quadratic components to wave vectors k1 and k2 .

4.3.2. Volterra models and decomposition of non-linear systems This section is especially devoted to non-linear systems and the approximation of these systems by Volterra models, as well as the results obtained for the output bispectrum signal of these models. Therefore, it is divided into two sub-sections, one for the calculation of the bispectrum and bicoherence of the output of these models and another for the presentation of the non-linearity of Volterra models (see section 4.3.2.2). In this part, we will assume the data of the Gaussian input to be that which corresponds to the hypotheses of section 4.3 since the Fourier transform of the Gaussian data is complex Gaussian; in other words, the amplitude follows a Rayleigh law and the phase is uniformly distributed on [0, 2π[. 4.3.2.1. General principles In order to arrive at these general principles, we will assume a system whose output depends on the number of finite input samples (hypotheses that seems realistic in the case of real systems). That implies that output Y (n) is given by: Y (n) = F (X(n − p− ), . . . , X(n), . . . , X(n + p+ ))


193

where X(n) are input samples and p− , p+ are two integers (or vectors of integers in the case of 2-D signals). In addition, if we assume that F can be developed in power series, we then get the following development. Y (n) = F (0, . . . , 0, . . . , 0)+ +

p ∂F (X(n − p− ), . . . , X(n), . . . , X(n + p+ )) X(n + k) ∂X(n + k) −

k=p

+

+

k=p

l=p

p p ∂ 2 F (X(n − p− ), . . . , X(n), . . . , X(n + p+ )) X(n + k) + ∂X(n + k) ∂X(n + l) − −

X(n + l) + . . . (4.49) All the derivatives are taken at the origin. If we limit ourselves to order 2 of equation (4.4) and if we ignore the constant term, we get a second order Volterra model for 2-D finite impulse response signals: p+

p+

Y (n) = k=p−

ak X(n + k) +

p+

k=p− l=p−

ckl X(n + k) X(n + l)

with: ak =

∂F (X(n − p− ), . . . , X(n), . . . , X(n + p+ )) ∂X(n + k)

ckl =

∂ 2 F (X(n − p− ), . . . , X(n), . . . , X(n + p+ )) ∂X(n + k) ∂X(n + l)

and

(4.50)

The quadratic term generates the waves by interactions between those of the input signal. The relation between the wave amplitude of the input signal and the output signal is not always taken into account properly; it is an unknown notion in the processing of linear signals. Thus, for the study of the processing of the non-linear signal, it is necessary to fix all the input magnitudes of the system. The stability conditions in the case of infinite impulse response models are badly defined, especially for the quadratic part. A sufficient condition in order for the impulse response of the quadratic part with a bound excitation to also be bound is [SCH 80]: +∞

+∞

k=−∞ l=−∞

|ckl | < +∞

194


Up to now it has not been shown that this condition was necessary. For the entire model including the linear part, we can only state one sufficient stability condition, which is the stability of each of the two parts, and therefore a sufficient condition is: +∞

|ak | < +∞

+∞

and

k=−∞

+∞

|ckl | < +∞

k=−∞ l=−∞

If the frequency response of a linear filter (attenuation and phase of the transfer function) is identifiable by the collection of the responses of the filter to a sinusoidal input describing the set of wave vectors, this is not the case in the quadratic part, as Hq (k1 , k2 ) is identifiable only by the collection of frequency responses obtained by making k1 and k2 vary. All the physical notions of the second order Volterra model and of the non-linear systems are found in this result, as a quadratic kernel is identifiable from the interactions that it produces between two sinusoids. However, the scale factor also plays an important role in the identification of this kernel (which is not the case for linear filters). In the same way, a cubic kernel can be identified only with the help of interactions that are produced between three sinusoids. The identification of the complete (linear and quadratic) model is even less evident. It is possible to identify the linear part and the quadratic part at the same time, by supposing that the input data (if it is accessible) can be orthogonalized [KIM 88]. A canonical decomposition of the second order kernels into the form of linear and quadratic filters (which is useful for the synthesis of these models) is obtained [SCH 80] by showing that any quadratic kernel can be written in the form Hq (k1 , k2 ) = Ha (k1 + k2 )Hb (k1 )Hc (k2 ). 4.3.2.2. Bispectrum of a Volterra model with Gaussian input data In the frequency domain, the Fourier coefficients for output Y˜ (k) can be expressed by: ˜ Y˜ (k) = H1 (k) X(k) +

+∞

−∞

˜ i ) X(k ˜ − ki )dki Hq (ki , k − ki ) X(k

(4.51)

Without the knowledge of what we have already done in section 4.2.2 for the linear models, we can develop the bispectrum calculation for such a system. If we take the case of the white Gaussian input data of zero mean and of variance equal to γ 2 , the output signal Y (n) has a spectrum: S(k) = γ22 |H(k)|2 + γ24

∞

−∞

(1 + 3 δ(k1 − k/2))|Hq (k1 , k − k1 )|2 (4.52)


195

and a bispectrum: . +∞ B(k1 , k2 ) = δ(k2 )γ 4 |H(k1 )|2 −∞ Hq (−ki , ki )dki . +∞ +δ(k1 )γ 4 |H(k2 )|2 −∞ Hq (−ki , ki )dki . +∞ +δ(k1 + k2 )γ 4 |H(k1 )|2 −∞ Hq (−ki , ki )dki +2γ 4 H(k2 ) H(−k1 − k2 ) Hq (k1 + k2 , −k2 ) +2γ 4 H(k1 )H(k2 ) Hq (−k1 , −k2 )+2γ 4 H(k1 )H(−k1 − k2 ) Hq (−k1 , k2 + k1 ) 3 . +∞ +γ 6 δ(k1 )δ(k2 ) −∞ Hq (ki , −ki ) . +∞ . +∞ +2 δ(k2 ) γ 6 −∞ Hq (ki , −ki )dki −∞ |Hq (ki , k1 − ki )|2 dki . +∞ . +∞ +2 δ(k1 ) γ 6 −∞ Hq (ki , −ki )dki −∞ |Hq (ki , k2 − ki )|2 dki . +∞ . +∞ +2 δ(k1 + k2 ) γ 6 −∞ Hq (ki , −ki ) −∞ |Hq (ki , k1 − ki )|2 dki . +∞ +8 γ 6 −∞ Hq (ki , k1 − ki )Hq (−ki , k2 + ki )Hq (−k1 + ki , −k2 − ki )dki (4.53) From this calculation, we can draw two conclusions. We can limit ourselves to the study of Volterra systems having white input data, since for any input signal admitting the decomposition of its higher order spectra as described in (4.36), it is sufficient to replace: H1 (k) ↔ H1 (k) H(k)

and

Hq (ki , k−ki ) ↔ Hq (ki , k−ki )H(ki )H(k−ki )

We note that, in expression (4.53), there are two types of terms: those that make the linear kernel occur twice and the quadratic kernel once and those that only make quadratic terms occur. The first type of term translates the coherence of phases that exists between two waves (linear kernel) and their interaction (quadratic kernel) as we have seen in section 4.3. This type of coupling is called the primary phase coupling phenomenon (PPCP). The terms which consist only of quadratic terms translate a coherence of phases between three generated waves; this phenomenon is called the secondary phase coupling phenomenon (SPCP). 4.4. Bispectral estimators for 2-D signals In this section we present the four bispectral estimators (these 2-D bispectral estimation methods have been presented in [LEC 95a]). The first two sections represent the conventional bispectral estimators and the last two present two parametric estimators. The class of conventional bispectral estimators is based on the two equivalent definitions of the bispectrum. The direct method uses the mean of the Fourier coefficients and the indirect method goes by the transform of the third order moment (see sections 4.4.1 and 4.4.2). The class of parametric estimators is

196


much richer than the class of conventional methods, as each method of identification of a filter with a non-minimum phase constitutes a parametric estimator of the bispectrum. However, as we have already mentioned in section 4.2, one of the main motivations of the use of higher order moments and spectra is to find the transfer functions with non-minimum phase. A number of methods are presented in other works on higher order moments and spectra (for an exhaustive review see [GIA 90, NIK 93]). The identification method for the autoregressive filter is based on Yule-Walker equations, which are well-known for second order statistics (see section 2.4.2) and distributed to the higher orders in [RAG 87]. The ARMA method is based on the same method as the previous one for the autoregressive part and on the fortified Giannakis formula for the identification of the MA parameters. The reader, however, should not forget the theoretical limits of parametric estimation that we have seen in section 4.2.3, especially the fact that parametric estimation cannot converge towards a bispectrum when it is not factorable. The first motivation for the use of these parametric models rests in their great resolution power on the small windows of signal analysis, as has been shown for the second order moments (spectral analysis) in Chapter 3.

4.4.1. Indirect method The indirect method is thus the application of the first definition of the bispectrum. It consists of estimating the bispectrum by the Fourier transform of the third order ˆX (n1 , n2 ), but weighted by an estimation window with moment estimated to be E a finite support SM (L) (see section 4.1.1) in order to reduce the variance of the estimator: ˆ ind (k1 , k2 ) = B

ˆX (n1 , n2 ) e−j(k1 n1 +k2 n2 ) (4.54) w(n1 , n2 ) E

(n1 ,n2 )∈SM (L)

This window function allows us to truncate the infinite addition by a finite addition for the Fourier transform. The biased estimation of the third order moment is generally defined by the authors [BRI 67, NIK 87, NIK 93, RAG 87] by: s2 ˆX (n1 , n2 ) = 1 X(n) X(n + n1 )X(n + n2 ) E N 2 n=s 1

where s1 = max(0, −n1 , −n2 ) and s2 = min(N, N − n1 , N − n2 ). Since: lim

N →+∞

s2 1 X(n) X(n + n1 )X(n + n2 ) = M3x (n1 , n2 ) s2 − s1 n=s 1

(4.55)


197

if N is sufficiently large, equation (4.55) can be approximated by: ˆX (n1 , n2 ) = 1 − p EX (n1 , n2 ) E N with p = max(|n1 |, |n2 |, |n1 − n2 |). The estimation of the third order moment is biased, but asymptotically (N big) [BRI 67, ROS 65, VAN 66] this estimation becomes non-biased. For the indirect method, the bispectrum is estimated by the Fourier transform of the third order moment estimated and weighted by a weighting (or smoothing) window, which induces a bias. The absence of such a weighting window implies that this mode of estimation is inconsistent [LEC 97]. This weighting (or smoothing) window has to follow several conditions, as in the case of bispectral estimation of 1-D signals: – the window w(n1 , n2 ) = w(n1x , n1y , n2x , n2y ) has to respect the symmetries of the third order moment in order to conserve those of the bispectrum; – the window has to be zero beyond a certain value L: w(n1 , n2 ) = 0

if

(n1 , n2 ) ∈ / SM (L)

SM (L) being the support of the third order moment of length L; – the energy of the window has to be equal to 1, which is translated by introducing W (k1 , k2 ) = F T (w(n1 , n2 )) by the following relation: +∞ +∞ W (k1 , k2 )dk1 dk2 = 1 ⇔ w(0, 0) = 1 −∞

−∞

This condition is called the normalization condition and implies that the estimation is asymptotically non-biased. Moreover, we will suppose that W (k1 , k2 ) takes only real positive values. This condition makes it possible to avoid modifying the phase that is implicitly contained in the third order moment. For 2-D signals, the simplest way is to define the axially separable window functions (it is the immediate idea for the extension of the 1-D window function to 2-D window functions), i.e.: w2d (n1x , n1y , n2x , n2y ) = w1d (n1x , n2x ) w1d (n1y , n2y )

(4.56)

These windows, which are axially separable in the time field, are also separable in the frequency field since: W2d (kx1 , ky1 , kx2 , ky2 ) = W1d (kx1 , kx2 ) W1d (ky1 , ky2 ) The normalized maximum bias – introduced as the real maximum bias introduces the maximum of the second derivative – for such windows is given for 2-D signals by:

198


Jmax =

. +∞ . +∞ . +∞ . +∞ ! ∞

−∞

−∞

−∞

(kx1 − kx2 )2 + (ky1 − ky2 )2

"

W2d (kx1 , ky1 , kx2 , ky2 )dkx1 dky1 dkx2 dky2

(4.57)

and the variance is given by: ˆ in (k1 , k2 )) V ar(B

with

V2d =

+∞

+∞

V2d N2

S(k1 ) S(k2 ) S(k1 + k2 ) 1 + δ(k1 − k2 ) + δ(2k1 + k2 ) + δ(2k1 + k2 ) (4.58)

2 W2d (k1 , k2 ).

k1 =−∞ k2 =−∞ 2 , since By admitting an axial decomposition of the window, we obtain V2d = V1d V1d is the variance parameter for 1-D estimation [BRI 65, NIK 93]. In the same way, we find that J2d = 2 J1d . We also note that the variance is double on the symmetric axes of the bispectrum. The windows usually used for the estimation of the bispectrum of the 1-D signals can be decomposed into the form [NIK 87, NIK 93]:

w1d (i, j) = d(i) d(j) d(i − j) In order to respect the parity of w1d (i, j), let us note that d(i) has to be even. Several windows of this kind are given in Table 4.4. The non-separable window functions for 1-D signals exist [NIK 93]; however, their study has not been extended, for the moment, to the case of 2-D signals. If w2d (ix , iy , jx , jy ) is axially separable and if w1d (i, j) is separable, window w2d (ix , iy , jx , jy ) will be called totally separable. Table 4.4 recapitulates the values of the indices Jmax and V for the different windows.

Daniell Hamming Parzen

Priestley Sasaki

Analytical expression sin(π m) if |m| ≤ 1 π m d(m) = 0 otherwise 0.54 + 0.46 cos(π m) if |m| ≤ 1 d(m) = 0 otherwise 8 < 1 − 6u2 + 6|m|3 if |m| ≤ 0.5 d(m) = 2(1 − |m|)3 if 0.5 ≤ |m| ≤ 1 : 0 otherwise ( i sin(π m) 3 Big[ − cos(π m) if |m| ≤ 1 π u (π m)2 d(m) = 0 otherwise 1 | sin(π m)|+(1 −|m|) cos(π m) if |m| ≤ 1 π d(m) = 0 otherwise

Table 4.4. Different weighting windows for 2-D signals 2 and J2d = 2 J1d with V2d = V1d

V2−D

J2−D

0.0143

198937

0.8221

121329.6

0.0016

16784.9

0.0413

576004.2

0.0024

2630.4


199

4.4.2. Direct method The idea of a higher order periodogram, or to be more precise of a mean periodogram, is based on the second definition of the bispectrum and leads to an estimation given by: ˜ 1 )X(k ˜ 2 )X ˜ ∗ (k1 + k2 ) ˆ pe (k1 , k2 ) = 1 X(k B N2

with

˜ X(k) =

N −1

X(i)e−j k i

i=0

The calculation of the variance of the periodogram for 2-D signals (given in [LEC 97]) shows that it does not tend towards 0 when the number of samples tends towards infinity (inconsistent estimator). With the knowledge of what we have already seen for the indirect method, the idea here is to define a window in a frequency field (a window whose weighting factor is uniform on the support) and to average the estimation on this window. If we call Δ0 , in conformity with the other works on the direct method [HIN 68, HUB 70, LII 81], the necessary space between the two frequency samples, along a bispectrum axis, and we estimate the bispectrum between the normalized frequencies to be [−0.5, 0.5], since the number of samples along an axis is 1 (see Figure 4.4). No = 2 Nb , we have the relation Δ0 = No Ν0

Δ0 li k

i+1

k

i

k

k i-1

Figure 4.4. Averaging method for the direct method

In order to estimate the sample of the bispectrum it is possible to estimate the bispectrum by an averaging on a hypercube by: ˆ dir (k1 , k2 ) = B

1 Δ40 N 6

J

J

˜ 1− X(k

l1 =−J l2 =−J

2π l1 ˜ N )X(k2

−

2π l2 N )

2π l1 N

−

2π l2 N )

2π l1 N , k2

−

2π l2 N )

˜ ∗ (k1 + k2 − X =

1 (2J+1)4

J

J

l1 =−J l2 =−J

ˆ pe (k1 − B

(4.59)

200


It would be possible to average on a hexagonal window having the form of the bispectral support [HUB 70]. However, in other traditional works this method uses only one square window [HIN 68, LII 81]. If we assume that M = 2 J + 1, the number of samples of the signal on which the Fourier transform operates is N = N0 M . One of the most tricky points of this method consists of finding a compromise between N , which is generally fixed, and N0 , which is the number of samples of the expected bispectrum. Obviously, this estimator is biased as it is also necessary to take into account the multiplication by the maximum of the second derivative of the bispectrum. The normalized bias is given by: Jmax =

16π 2 J (J + 1) 3 N2

(4.60)

and for variance has: ˆ dir (k1 , k2 )) Var(B

!

"2 N S(k1 ) S(k2 ) S(k1 + k2 ) 2 (2J + 1)

(4.61)

In fact, and we see it in the expression of the bias and of the variance, J has to be an increasing function of N , but the increase of J should not be very fast so that the estimator is asymptotically non-biased. The function J(N ) particularly has to verify: lim

N →+∞

J(N ) =0 N

and

lim

N →+∞

J 2 (N ) = +∞ N

(4.62)

Other methods, which have not been extended to the 2-D case, especially the complex demodulation method [GOD 65], are reported in the works of Nikias et al. [NIK 87, NIK 93]. After presenting the two conventional methods, we are now going to present the two parametric methods, one based on AR modeling and the other on ARMA modeling.

4.4.3. Autoregressive model The first estimator presented for 2-D signals is one of the most traditional and has been, without a doubt, the first parametric bispectral estimator for the 1-D case. This estimator is based on the well-known autoregressive model: p+ i=p−

ai X(n − i) = W (n) ⇔

p x+

py +

aix ,iy X(nx − ix , ny − iy ) = W (nx , ny )

i=px − i=py −

(4.63) where p− and p+ constitute the orders of the filter. Given a delay in the nearby phase, we can assume that p− = 0 and p = p − +p+. We will assume as before that W (n)


201

is centered, hence implying that X(n) is necessarily centered. By multiplying the two members by X(n − k)X(n − l) and by taking the expected value, we get equations of the type: p

ai EX (i − k, i − l) = 0

(4.64)

i=0

Under a developed 2-D form, the system is thus written as: py px

aix ,iy EX (ix − kx , iy − ky , ix − lx , iy − ly ) = 0

(4.65)

i=0 i=0

The choice of equation remains the crucial point for this type of method. In order to avoid redundant equations, we have to first limit the choice of (k, l) to NRSM(L) (by not taking the point 0) with L chosen in such a way that we obtain a number of equations at least equal to the number of unknowns. As for the 1-D case, we will make the traditional hypothesis that a0,0 = 1, which implies that the unknown vector is: a = (a1,0 , . . . , apx ,0 , a0,1 , . . . , apx ,1 , . . . , apx ,py )T where aT is the transpose of a and by noting: p = −(EX (−1, 0, 0, 0),. . ., EX (−kx , −ky , −lx , −ly ),. . ., EX (−L, −L, −L, −L))T matrix A is: 0

EX (0, 0, 0, 0) B . B . . B B E (−k ,−k B X x y ,−lx,−ly ) B B . . @ . EX (1−L,−L,−L,−L)

··· EX (ix −1, iy , ix , iy ) . .. . . . · · · EX (ix−kx ,iy−ky ,ix−lx ,iy−ly ) . .. . . . · · · EX (ix−L, iy−L, ix−L, iy−L)

1 ··· EX (px −1, py , px , py ) C . .. C . . . C C · · · EX (px−kx , py−ky ,px−lx ,py−ly ) C C C . .. . A . . · · · EX (px−L, py−L, px−L, py−L)

the equation system is easily noted by: Aa=p

(4.66)

In order to resolve the system of relation (4.66), it is necessary that the matrix is of sufficient rank, i.e. equal to (px + 1) (py + 1) − 1. It is not an easy task to test all the equations in order to find (px + 1) (py + 1) − 1 equations which are nonlinearly dependent. For this reason, we often choose a least squares (or pseudo-inverse) algorithm [HAY 86], in other words, we are led to resolve: a = (AT A)−1 AT p

(4.67)

202


However, even in this case, the inversion of (AT A) is not assured. The third order moment (γ3 ) of the supposed noise in the input of the autoregressive model, necessary for the bispectral estimation, is made with the help of pair (0, 0) by: γˆ3 =

p

ˆ X (i, i) a î E

i=0

Finally, the bispectrum is thus estimated by: ˆ 1 ) H(k ˆ 2) H ˆ ∗ (k1 + k2 ) ˆ ar (k1 , k2 ) = γˆ3 H(k B

(4.68)

This estimator is non-biased because, asymptotically, the estimation of the third order moment is non-biased, as well as the estimation of the parameters of the autoregressive filter by the least squares method [HAY 86]. It is not an easy task to give an expression of the variance of the parametric estimator for the second order statistics just as for the third order statistics. For different estimators, no expression (or even consideration) of variance has been proposed by the authors [RAG 87]. The most complicated point remains the estimation of the noise variance, which introduces at the same time the variance of the estimated parameters a î and that of the third order moment (the parameters and the moment being, of course, non-independent). 4.4.4. ARMA modeling The ARMA model is given by: p+

ai X(n − i) =

i=p−

q+

bi W (n − j)

j=q−

By assuming that p = p+ + p− , p− = 0, q = q + + q − and q − = 0, and by introducing a pure phase delay we return, as in the previous case, to: p i=0

ai X(n − i) =

q

bi W (n − j)

(4.69)

j=0

The initial idea to use an ARMA model is to introduce supplementary phase information with the help of a modified mean part, which cannot make up a single AR model. This method [GIA 90] consists of identifying the coefficients of the ARMA model by using the transfer function given in the fractional form, and by identifying the infinite impulse response with the help of the bispectrum (and hence of the third order moment): B(k1 , k2 ) = γ3 H(k1 )H(k2 )H ∗ (k1 + k2 ) = γ3

C(k1 , k2 ) A(k1 , k2 )

(4.70)


203

C(k1 , k2 ) is the transfer function associated with the modified mean part and A(k1 , k2 ) with the autoregressive part. Moreover, we get:

C(k1 , k2 ) =

β(n1 , n2 )e−jk1 n1 e−jk2 n2

(n1 ,n2 )∈SM (q)

with: β(n1 , n2 ) =

q

bk bk+n1 bk+n2

k=0

By multiplying by the autoregressive part (4.70) and by identifying the coefficients of the polynomial in z1 = ej k1 and z2 = ej k2 of each side, we deduce the following equations:

α(n1 , n2 )EX (k − n1 , l − n2 ) =

(n1 ,n2 )∈SM (p)

0

if (k, l) ∈ / SM (q)

γ3 β(k, l) otherwise (4.71)

with α(n1 , n2 ) =

p

ak ak+n1 ak+n2 . The identification of the coefficients of the

k=0

ARMA model is made by a fortified Giannakis formula [GIA 87, GIA 90] extended to the 2-D case by: b(ix ,iy ) = 4 qx qy (qx + 1) (qy + 1)

qx X

qy X

my mx X X

mx =0 my =0 nx =0 ny =0

β 1/3 (mx , my , nx , ny ) β(qx , qy , jx , jy ) −my −mx qyX qxX kx =0

1/3

β(qx ,qy ,kx ,ky ) β(qx ,qy ,kx +mx ,ky +my ) β(qx ,qy ,kx +nx ,ky +ny )

ky =0

(4.72) The tests conducted in 2-D on this method show in general the repeated difficulties of inversion of the matrix. It is preferable to work by identification of AR parameters with the help of Yule-Walker equations as shown in section 4.4.3, since the process of identification of the MA parameters remains the same as the original method, in other words, equation (4.72). The choice of the support of the equations has to be made in such a manner as to obtain an oversized system resolved by pseudo-inverse (least squares). As before, the estimator is non-biased (since the least squares solution as well as the fortified Giannakis formula is not biased [GIA 90]). As for the other estimator, no explicit formula of the variance is given in [GIA 90].

204


4.4.5. Measure of bias and variance of estimators A comparison of the values of bias and variance measured on real signals, with the theoretical values given in the previous sections, for the conventional methods shows, in general, a stronger value of variance and bias. The reason for this is simple: all the values obtained in the previous sections are asymptotic values that can agree only with large values of the number of samples of the image. Finally, the parametric estimators are found to be bad quantifiers but remain high resolution methods, as was seen in Chapter 3.

4.5. Hypothesis test for non-linearity and bicoherence tables This section presents the tools which enable a localization of the non-linearity, i.e., the bicoherence tables, that are projections in a 2-D structure of the information contained in the 4-D bicoherence. The hypothesis tests for detection of non-linearity are presented here.

4.5.1. Hypothesis tests Hypothesis tests consist of testing in a statistical way a hypothesis H0 , said to be “zero hypothesis”, against another hypothesis, said to be “alternative a hypotheses” noted by H1 . The first stage of this kind of a statistical test consists of fixing the probability of committing a type I error (in other words rejecting H0 even though true) to a value α said to be the significance level of the test. In the case of detection of non-linearity via higher order moments, it refers to testing if the spectrum of order p can be written in the factorized form (4.35) (in fact, here we discuss only order 3). Several techniques are possible, but we only present two methods, which are explicitly formulated in the form of hypothesis tests [HIN 80, RAO 82]. These two tests are based on the fact that the bicoherence is constant in the frequency domain and is immediately extendable to the 2-D case. Indeed, they are only based on the bicoherence value and the fact that these values are organized according to a 2-D or 4-D structure makes little difference in the detection process. The test described in [RAO 82] first of all consists of choosing L points in a frequency domain (k1i , k2i ), i = 1, . . . , L then taking the outcomes of the bicoherence in (k1i = k1i + pd, k2i = k2i ) and (k1i = k1i , k2i = k2i + pd) with p = 1, . . . r and d as a distance chosen in such a way that B(k1i , k2i ) = B(k1i , k2i ). This is possible under the conditions already expressed in (4.62) for the direct method. Hence, it is necessary to form n outcomes of the vector: Yi = (|P (k11 , k21 )|2 , . . . , |P (k1L , k2L )|2 )T

(4.73)


205

with i = 1, . . . , L. Finally, vector βi = B Yi is calculated by: ⎛

1 ⎜ 0 B=⎜ ⎝··· 0

−1 1 ··· 0

0 −1 ··· ···

··· ··· ··· 1

⎞ 0 0 ⎟ ⎟ ···⎠ −1

(4.74)

β is thus a vector with L − 1 Gaussian zero mean components. The test to determine if a Gaussian vector has zero mean is the T 2 Hostelling test. For this test, we estimate the mean and the correlation matrix of β: 1 βi β¯ = n i=1 n

S¯ =

n

¯ i − β) ¯T (βi − β)(β

(4.75)

i=1

and the test is accepted if: T 2 = nβ¯T S¯β¯ < λ0

(4.76)

the value λ0 being fixed via α as T 2 follows a law F with (L − 1, n − L + 1) degrees of freedom. The second approach [HIN 80] consists of estimating the interquartile of the histogram of the bicoherence (by supposing that if the bicoherence is flat, this interquartile will be weak2). Thanks to order statistics, we show that the interquartile follows a Gaussian law whose mean and variance are given in [HIN 80]. In this case, also, the test is accepted if the value of the interquartile is smaller than a threshold deduced from α via the quartiles of the Gaussian law. Constructive spectrum

Destructive spectrum

Original spectrum

Figure 4.5. Output spectrum of the second order Volterra model having as input an ocean surface

2. The interquartile is defined by 3 − 1 with P (|P 2 (k11 , k21 )| < 1 ) = P (|P 2 (k11 , k21 )| > 3 ) = 1/4.

206


Primary phenomenon of phase coupling

Secondary phenomenon of phase coupling Primary phenomenon of phase coupling

Secondary phenomeon of phase coupling

Figure 4.6. Output bicoherence tables of a second order Volterra model having as input an ocean surface (at the top cross-table, in the middle and at the bottom homogenous tables)


207

4.5.2. Bicoherence tables One of the problems of using the bicoherence information is its 4-D structure which is not properly visualized, since its global nature is difficult to access for the purpose of comprehension and analysis of non-linearity. The initial idea of the bicoherence tables is to calculate the deviation of the bicoherence as compared to a supposed constant value and to summarize the information thus obtained in a 2-D structure. We present below the results for the Gaussian input data of a non-linear system, but we can adapt them to other types of data, under the condition that we have a preliminary idea of the result. The immediate idea, in order to measure a deviation as compared to a constant, is to use [LEC 95b]: T (k1 , k2 ) =

+π −π

+π −π

(|P (k1 , k2 , k3 , k4 )| − C)2 dk3 dk4

(k1 , k2 ) can be one of the following non-redundant couples: (kx1 , ky1 )

(kx1 , kx2 )

(ky1 , ky2 )

Thus, we define a cross-table T (kx1 , ky1 ) and two homogenous tables T (kx1 , kx2 ) and T (ky1 , ky2 ). T (k1 , k2 ) effectively measures the bicoherence variations as compared to a certain constant value C. The choice or the estimation of this value is far from evident. It is preferable to have a preliminary idea of the result, in other words, of the value of the output bicoherence in the absence of non-linearity or to take C as equal to the measured mean of the bicoherence, which minimizes the integral of (|P (k1 , k2 , k3 , k4 )| − C)2 . Quadratic spectrum

Linear spectrum

Independent spectrum

Figure 4.7. Output spectrum of a second order Volterra model having as input an ocean surface to which an independent system has been added

208


Figure 4.8. Output tables of bicoherence of a second order Volterra model having as input an ocean surface to which an independent system has been added (at the top cross-table, in the middle and at the bottom homogenous tables)


209

Thanks to these tables, it is possible to distinguish the two phenomena for phase coupling. In the example that we are going to develop, we consider an ocean surface using a second order Volterra system, made of an identity filter for the linear kernel and a simple quadratic filter for the quadratic part. This type of modeling is very useful for the study of the creation of an image of the ocean surface with the help of a synthetic aperture radar (SAR), as is observed in [LEC 00]. On the spectrum of the output signal (see Figure 4.5), we observe the original spectrum and two quadratic spectra of which one is situated around the continuous component and the other at twice the wavelength of the original spectrum. The first of these two spectra will be called destructive as it is produced by the interaction of two waves having wave vectors of opposite direction. The constructive spectrum is produced by the interaction of two waves having wave vectors of the same direction. If we note by o a wave of original spectrum, by d a wave of destructive spectrum and by c a wave of constructive spectrum, the primary phase coupling takes place for the couples: (o1 , o2 )

(o, d)

(o, c)

indices 1 and 2 denoting two distinct or non-distinct waves of the original spectrum. The secondary coupling of phase takes place for: (c1 , c2 )

(c, d)

(d1 , d2 )

Indeed, if we again take expression (4.53), the resultant bispectral is significant, in the case of an original signal with a limited spectral band, if: k 2 − kj ∈ o

k1 + k2 − kj ∈ o

kj ∈ o

This implies that k2 belongs either to the destructive spectrum (the adherance of k2 − kj being assured by the adherance of kj to the original spectrum as k2 0), or k2 is in the constructive spectrum: – if k2 is in the destructive spectrum, k1 also has to be in the destructive spectrum so that k1 + k2 − kj is situated in the original spectrum; – if k2 is in the constructive spectrum, two cases are distinguished: either k1 is also in the constructive spectrum, but in the opposed quadrant and thus k1 + k2 − kj belongs to the original spectrum (because kj belongs to this spectrum), or k1 belongs to the destructive spectrum and thus k1 + k2 − kj belongs to the original spectrum as k2 − kj is in this spectrum. In fact, the two latter cases are the symmetric cases by bispectral symmetry while the first case is different. Indeed, the two latter cases introduce two waves of the constructive spectrum and one wave of the destructive spectrum, and the first case, three waves of destructive spectrum. Hence, we can conclude that the detection of the non-linearity remains theoretically possible when the constructive spectrum and the

210


original spectrum are eliminated, but not when the destructive and original spectra are eliminated. In homogenous tables (T (ky1 , ky2 ) and T (kx1 , kx2 )), the primary and secondary couplings appear in different places in the frequency plane defined by these tables. Thus, for a primary phase coupling found situated on (kx1 , kx2 ), since these two components are found on the projection of the original spectrum on the axis Ox , a secondary phase coupling will be found situated on (2 kx1 , 2 kx2 ) or on (2 kx1 , 0). In these homogenous tables, the original spectrum is easily localized since it is necessarily found on the axis kx1 = kx2 and ky1 = ky2 (see Figure 4.6), which enables us to isolate the spectra generated by quadratic interactions. In the cross-table, the primary and secondary phase coupling have the quadratic spectra T (kx1 , ky1 ) for localization and cannot be distinguished in these tables.

4.6. Applications 4.6.1. Image restoration Since one of the most common application of higher order moments and spectra is the identification of the model with a non-minimum phase, primordial for 2-D signals, it was natural that the higher order moments (and more specifically the bispectrum) help the deblurring. However, the results in this field are still relatively limited. We will note the results by Chang et al. [CHA 91] for the identification of the non-minimum phase system, Jin et al. and Newman et al. [JIN 92, NEW 92] for optics, as well as Lawrence et al. [LAW 92] for the restoration of astronomical images, which are corrupted by a multiplicative noise (speckle). 4.6.2. Artifact removal Another application, which we have mentioned previously, is the removal of artifacts due to an acquisition process. This problem often arises in the observation of the ocean surface by an SAR sensor. In order to describe this proposition, we are again going to take the example of the ocean surface seen in section 4.5.2, which passes “through” a second order Volterra system and to this system we add a completely independent swell system, as shown in the spectrum on Figure 4.7. On the bicoherence cross-table (represented at the top of Figure 4.8), we observe that the independent system is much weaker than on the spectrum. Thus, we have isolated, in this spectrum the components having non-linear interactions. The linear part (original spectrum) may be isolated, as is explained in section 4.5.2. The following stage consists of defining a filter (which eliminates the non-linear components), from localized non-linearity in the frequency domain, in order to obtain an image without artifact. However, this method fails if the linear and quadratic spectra overlap.


211

4.7. Bibliography [BRI 65] B RILLINGER D. R., “An introduction to polyspectra”, Ann. Math. Statist., vol. 36, p. 1351-1374, 1965. [BRI 67] B RILLINGER D., ROSENBLATT M., “Asymptotic Theory of k-th Order Spectra”, in Spectral Analysis of Time Series, B. Harris (ed.), Wiley, New York, p. 153-188, 1967. [CHA 91] C HANG M., T EKALP A. M., E RDEM A. T., “Blur identification using the bispectrum”, IEEE Transactions on Signal Processing, vol. 39, p. 2323-2325, 1991. [ERD 92] E RDEM A. T., T EKALP A. M., “New Theoritical Results on the Bistatistics of 2D Signals”, in Higher Order Statistics, J.L. Lacoume (ed.), Elsevier, p. 99-102, 1992. [GIA 87] G IANNAKIS G. B., “Cumulants: a powerful tool in signal processing”, Proceedings of the IEEE, vol. 75, p. 1333-1334, 1987. [GIA 90] G IANNAKIS G. B., S WAMI A., “On Estimating Non-causal Non-minimum Phase ARMA Models of Non-Gaussian Processes”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-38, p. 478-495, 1990. [GOD 65] G ODFREY M. D., “An Exploratory study of the bispectrum of Economic Time Series”, Applied Statistics, p. 48-69, 1965. [HAY 86] H AYKIN S., Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, NJ, 1986. [HIN 68] H INICH M. J., C LAY C. S., “The application of the discrete Fourier Transform in the estimation of Power Spectra, Coherence, and Bispectra of Geophysical Data”, Reviews of Geophysics, vol. 6, p. 347-363, 1968. [HIN 80] H INICH M. J., “Testing For Gaussianity and linearity of a stationnary time series”, Journal of Time Series Analysis, p. 169-176, 1980. [HUB 70] H UBER P. J., K LEINER B., G ASSER T., D UMERMUTH D., “Statistical Method for phase investigating Relations in stationary Stochastic Processes”, IEEE Transactions on Audio and Electroacoustics, vol. 19, p. 78-86, 1970. [JIN 92] J IN S., W EAR S., R AGHUVEER M. R., “Reconstruction of speckled images using Bispectra”, Journal of Optical Society of America A, p. 371-376, 1992. [JOU 92] J OUNY I. I., M OSES R. L., “The Bispectrum of complex signals definitions and properties”, IEEE Transactions on Signal Processing, vol. 40, p. 2833-2836, 1992. [KIM 88] K IM K. I., P OWERS E. J., “A Digital Method of Modeling Quadratically Nonlinear Systems with a General Random Input”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-36, p. 1758-1769, 1988. [LAW 92] L AWRENCE T. W., F ITCH J. M., G OODMAN D. M., .M ASSIE N. A., S HERWOOD R. J., J OHANSSON E., “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry”, Optical-Engineering, p. 627-636, 1992. [LEC 95a] L E C AILLEC J.-M., G ARELLO R., C HAPRON B., “Comparison between four 2D signal Bispectral estimators”, Proceedings of the IEEE Workshop on Nonlinear Signal and Image Processing, p. 547-550, 1995.

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[LEC 95b] L E C AILLEC J.-M., G ARELLO R., C HAPRON B., “Detection of Nonlinearities in Sea Surface Imaging Process Using Bispectrum Method Estimation”, Proc. of ICASSP, vol. 3, p. 1585-1588, 1995. [LEC 96] L E C AILLEC J.-M., G ARELLO R., “Detection and Quantification of Nonlinearities in a Modified Volterra Model by Using Wigner-Ville Higher Order Transform”, Proceedings of International Symposium on Time-Frequency and Time-Scale Analysis, p. 181-184, 1996. [LEC 97] L E C AILLEC J.-M., Etude et traitement des images SAR grâce aux moments et spectres d’ordres supérieur, PhD Thesis, University of Rennes I, 1997. [LEC 00] L E C AILLEC J.-M., G ARELLO R., “RadarOcéanographie”, Le Traitement des images radars a` synthèse d’ouverture, Henri Maˆıtre (ed.), Chapter 14, p. 307-328, Hermes, IC2 series, 2000. [LEO 59] L EONOV V. P., S HIRYAEV A. N., “On a method of Calculation of Semi-Invariants (translated by J.R. Brows)”, Theory of Probability and its Applications, vol. 4, p. 319-328, 1959. [LII 81] L II K. S., H ELLAND K. N., “Cross Bispectrum Computation and Variance Estimation”, ACM Transaction on Mathematical Software, p. 284-294, 1981. [MEN 91] M ENDEL J. R., “Tutorial on Higher-Order Statistics (Spectra) in Signal Processing and System Theory: Theoretical Results and Some Applications”, Proceedings of the IEEE, vol. 79, p. 278-304, 1991. [NEW 92] N EWMAN D., V RANKEN R. C. V., “Image-reconstruction algorithm using Kalaman filter weighting of the bispectrum”, Journal of Optical Society of America A, p. 1724-1739, 1992. [NIK 87] N IKIAS C. L., R AGHUVEER M. R., “Bispectrum Estimation: A digital Signal Processing Framework”, Proceedings of the IEEE, p. 869-891, 1987. [NIK 93] N IKIAS C. L., P ETROPULU A. P., Higher-Order Spectra Analysis -a nonlinear signal processing framework, Prentice Hall, Englewood Cliffs, NJ, 1993. [OPP 89] O PPENHEIM A. V., S CHAFER R., Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1989. [PAN 88] PAN R., N IKIAS C. L., “The Complex Spectrum of Higher Order Cumulants and Nonminimum Phase System Identification”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 36, p. 186-205, 1988. [RAG 87] R AGHUVEER M. R., N IKIAS C. L., “Bispectrum Estimation: A Digital Signal Processing Framework”, Proceedings of the IEEE, vol. 75, p. 869-891, 1987. [RAG 90] R AGHUVEER M. R., “Time-Domain Approaches to Quadratic Phase Coupling estimation”, IEEE Transactions on Automatic Control, vol. 35, p. 48-56, 1990. [RAO 82] R AO T. S., G ABR M. M., “A test for linearity of time series analysis”, Journal of Time Series Analysis, p. 145-158, 1982. [ROS 65] ROSENBLATT M., VAN N ESS J. W., “Estimation of the Bispectrum”, Ann. Math. Statistics, p. 1120-1136, 1965.


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[ROS 85] ROSENBLATT M., Stationary Sequences and Random Field, Birkhauser, Boston MA, 1985. [SCH 80] S CHETZEN J. S., The Volterra and Wiener Theories of Nonlinear Systems, John Wiley, New York, 1980. [TEK 89] T EKALP A., E RDEM A., “Higher-Order Spectrum Factorization in One and Two Dimensions with Applications in Signal Modeling and Nonminmum Phase System Identification”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 37, p. 1537-1549, 1989. [TEK 90] T EKALP A., E RDEM A., “On the Measure of the set of Factorizable Polynomial Bispectra”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 38, p. 1637-1539, 1990. [VAN 66] VAN N ESS J. W., “Asymptotic Normality of Bispectral Estimate”, Ann. Math. Statistics, p. 1257-1272, 1966.


Chapter 5

Time-frequency Representation of 2-D Signals

5.1. Introduction Bilinear time-frequency representations (TFR) constitute a spectral analysis process of great benefit for signals whose modeling is not known. The absence of modeling of the signal brings TFR closer to traditional Fourier analysis (periodogram and correlogram methods) and moves them away from the parametric spectral analysis methods presented in Chapter 3. This chapter describes various aspects of spectral analysis of ocean surface images carried out with TFR adapted to 2-D signals. In particular, problems due to the absence of perfect causality of an image will be tackled. This is a recurring difficulty in image processing; an example in Chapter 3 identifies failure to define a causal support for prediction parameters. We will see that the problem of TFR is of another type: the failure to define an analytic image in a unique manner (see section 5.4). 5.1.1. Bilinear time-frequency representation TFR present signal analysis in a space double the size of the natural representation of the signal: the time, which is in fact the most common description or recording of signals, and frequency. For images, time is replaced by two coordinates noted by (x, y), while the frequency naturally becomes the couple (fx , fy ). Let us also specify

Chapter written by Stéphane GRASSIN and René GARELLO.

216


that images are static and no additional time dimension is necessary. Whether bilinear or not, when the kernel is correctly selected, the benefit of TFR is to localize the spectral content of the signal and allow a description of the evolution of the spectral content during the existence of the signal (this can be done, for example, by instantaneous frequency). For images, the local spectral content is naturally attached to two coordinates. These bilinear representations are defined for the signals with real or complex scalar values on a multidimensional space. These representations, like those of gray level, can be applied to monospectral images. The expressions in those particular images will be detailed specifically. Several results presented below return to or extend those of the work devoted to time-frequency analysis in [HLA 01] and can also be seen in more detail in the thesis by Grassin [GRA 97] from where they are extracted.

5.1.2. Four spaces of representation Three bilinear representations are closely associated with the bilinear TFR of a signal. These representations form a family of four bilinear representations and all their dimensions are double that of the signal. The bilinearity property of a transform C is expressed by: Caf +bg = |a|2 Cf + |b|2 Cg + ab∗ Cf g + ba∗ Cgf where a and b are two complex scalar values. For a signal known by its temporal description, the most direct bilinear representation to obtain is simply the autocorrelation (localized) function of the signal. This bilinear representation is a function of two time variables: date and delay (difference between two dates). Generally, all four types of bilinear representations are functions of two variables, which are two vectors with two components in case of images. For each one of these variables, representation can be done in terms of a variable or its dual variable obtained through the Fourier transform. These variables are date, delay, Doppler frequency (dual variable of date) and frequency (dual variable of delay). For this family of bilinear representations, the four spaces of signal representation are: – time-delay space: this is the domain of autocorrelation functions of nonstationary signals, where the utility of additional time variable (date) is to localize the autocorrelation; – Doppler-delay space: this is the ambiguous field, which represents the correlation between a signal and its versions translated into time and frequency;


217

– time-frequency space: this domain is generally interpreted as representing the evolution of spectral content of a signal according to time; – Doppler-frequency space: this is the equivalent field of the time-delay domain for the Fourier spectrum of the signal. Interpretation of this representation is less intuitive and is not useful when the signal is represented in the temporal field. Strictly speaking, the names of the representation spaces mentioned above are normally valid for 1-D signals (for example, a function of time). In this case, the term “space” always refers to a plane. For signals defined for a space with several dimensions, such as images, it would be more accurate to speak of an equivalent space. If the signal is 2-D, the representation in the time-frequency space equivalent will take place in a 4-D space, which could be strictly called “spatial coordinates – spatial frequency space”. These four representations present information differently. However, the information content is always the same since Fourier transforms make it possible to pass from one representation space to another. Put simply, the term “time-frequency space” indicates the equivalent space for a multidimensional signal. Time thus indicates a coordinate vector. This principle also extends to other spaces, although the individual denomination of other variables remains the same for each of them. Thus, for a signal defined in an N dimensional space, the Doppler frequency, delay and frequency are all real N dimensional vectors (same dimensions as the coordinate vector).

5.1.3. Restriction to bilinear representation The study of the representations of two variables, including time-frequency, is restricted to bilinearity, for the sake of simplicity. The benefit of bilinearity is also theoretical: it guarantees independence of representation in the time-frequency field with respect to the choice of origin of dates or frequencies. In other words, the set of bilinear time-frequency representations identifies with the set of covariant time-frequency representations through temporal and frequency translation which is intuitively satisfactory. This set is often called Cohen’s class in scientific works. In the two fields of time (date) and frequency variable, these covariances take the form of a phase angle. The four types of representation are characterized entirely by a kernel. For bilinear representations, this kernel is a function of two variables only and it can be represented in the same four spaces as the bilinear representations. It is the independence of the kernel with respect to the signal which ensures the bilinearity of a representation. There are, however, methods to build kernels which adapt to specific characteristics of a signal [BAR 93a, BAR 93b].

218


5.1.4. Spectral description using bilinear representations For stationary signals of the second order, the correlogram method provides an estimate of PSD as the Fourier transform of the autocorrelation function (ACF) (see Chapter 1). For second order non-stationary signals, the generalized ACF is the equivalent of the ACF. The TFR is the FT following the variable delay of this generalized ACF. The “generalized ACF” is a representation of the signal in time-delay space which also has descriptive characteristics of correlation. 5.1.4.1. Bilinear representations in time-delay space For an image f (θ) of complex values with θ = (x, y) ∈ R2 , the representation in equivalent time-delay space results from the convolution product: TDRf (θ, τ ) = ΦTR (−θ, τ ) ∗θ Rf (θ, τ )

(5.1)

where 2-D delay on the image τ = (τx , τy ) ∈ R2 is a difference between two variables that are homogenous to θ, Rf is the instantaneous deterministic autocorrelation function (ACF-I) of the image: Rf (θ, τ ) = f (θ +

τ τ ) × f ∗ (θ − ) 2 2

(5.2)

while ΦTR is the kernel of the bilinear representation expressed in the equivalent time-delay space. If a signal is described by distributions, autocorrelation functions (5.1) and (5.2) should be taken in the sense of the functions defined by distributions (discrete signals fall into this category). Similarly to ACF-I, these representations in time-delay space are homogenous to an autocorrelation function. However, in order for them to have the characteristics of an autocorrelation function localized at the point of coordinates (x, y), it is necessary that the kernel ΦTR is mainly concentrated around the origin and tends to zero while moving away. The rate of decrease of the kernel must agree with the characteristic length for which any particular signal can be regarded as stationary, so that the representation of this signal accounts for internal correlations of the signal. That could seem to contradict the bilinearity character of the representation as it is related to the independence of the kernel with respect to the signal. This is in fact not true, as such a dependence would amount to retaining the kernel on the date of analysis of the signal which does not have any relationship with the choice of a size of the kernel adapted to an analysis scale of the signal. The instantaneous deterministic autocorrelation function (ACF-I) admits a Hermitian symmetry compared with the delay (τx , τy ). The TDR of an image preserves this symmetry if kernel ΦTR itself is also Hermitian with respect to the delay.


219

The benefit of such a Hermitian symmetry is to lead to time-frequency representations with real values which can intuitively be better interpreted from an energy point of view. The main benefit of the TDR is to make it possible to write the TFR of kernel ΦTR as a Fourier transform of this generalized deterministic ACF. Here we find the same step as in the method of spectral estimation of the correlogram. 5.1.4.2. Bilinear representations in equivalent time-frequency space The expression of the TFR, based on the kernel in time-delay space, is very often used, particularly for numerical implementations when the signal is known by a sequence of samples. On the contrary, its expression, based on the kernel in Dopplerdelay space, is preferred for the study of theoretical properties. The TFR of an image is the Fourier transform of the TDR according to the variable delay: % & (5.3) TFRf (θ, ω) = TFτ TDRf (θ, τ ) (θ, ω) ω being a spatial frequency on the image: ω = (ωx , ωy ) ∈ R2 . The basic expression for a practical implementation is obtained by introducing the development of the TDR in the preceding expression: ΦTR (α − x, β − y, τx , τy ) Rf (α, β, τx , τy ) TFRf (x, y, ωx , ωy ) = × exp −j(ωx τx + ωy τy ) dα dβ dτx dτy

(5.4)

Expression (5.4) often serves as the starting point in the calculation of the TFR of discrete signals. However, it is not obligatory and certain authors have already proposed discretizations in the Doppler-delay plane. If this formulation of the TFR is the most adapted to practically calculate the TFR of signals described as an evolution of a magnitude over time, it remains perfectly usable in the state to calculate a TFR of a known signal described by a sequence of samples. 5.2. TFR application to sampled images The practical implementation requires having a calculation formula adapted to sampled images and whose sums are finite. These two conditions cover distinctly separate aspects and for that are handled individually. 5.2.1. TFR expression of discrete images The extension of the calculation formula of the TFR of a 1-D signal to that of images is not immediate. As in 1-D, discrete images are modeled with a Dirac

220


impulse, thus allowing the use of a certain number of extremely useful calculation rules. In what follows we consider that discrete images result from a sampling on a rectangular grid with a period of sampling which can possibly be different on the two dimensions. The expressions of bilinear representations which follow are thus valid only within this framework. In particular, it would be necessary to repeat all of the study below if the sampling grid is hexagonal. 5.2.1.1. Autocorrelation function of a discrete image The expression of the autocorrelation function of 1-D discrete signals cannot be generalized directly to images, as the method of sampling an image is not unique. Let f be an image with continuous coordinates x and y. Let us consider the image fˆ obtained by sampling on a rectangular grid with Xe and Ye as periods. The sampling equation is: fˆ(x, y) =

kx

f (x, y) δ(x − kx Xe , y − ky Ye )

(5.5)

ky

Given the calculations, which must be followed, it is interesting to consider two stages for sampling (row and column). This makes it possible to replace the Dirac impulse with 2-D support by a product of Dirac impulse with separate variables. In other words this is like imposing having a square support rather than any other support, particularly a circular support, to a 2-D Dirac impulse. In fact, while proceeding in this manner, nothing is lost in most cases. The sampling equation of the image then becomes: fˆ(x, y) =

kx

ky

=f (x, y)

f (x, y) δ(x − kx Xe ) δ(y − ky Ye ) kx

δ(x − kx Xe ) δ(y − ky Ye )

(5.6)

ky

The ACF-I of the sampled image in terms of the ACF-I of image (5.2) is therefore:

Rfˆ(α, β, τx , τy ) = Rf (α, β, τx , τy )× ×

lx

kx

δ(α −

δ(α+

τy τx −kx Xe ) δ(β+ −ky Ye ) 2 2

τx − lx Xe ) 2

ly

ky

δ(β −

τy − ly Ye ) 2

(5.7)

Thus, similarly to sampling equation (5.6), the ACF-I of the sampled image is defined as a sum of Dirac impulses which takes the form of a 4-D Dirac comb.


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The search for the support of this distribution shows that for each sample of the ACF-I Rfˆ corresponds a single quadruple (kx , ky , lx , ly ) [GRA 97] for which the four Dirac impulses have the same support (the support is the real 0): ⎧ τx ⎪ − kx Xe ⎪α + ⎪ ⎪ 2 ⎪ ⎪ τx ⎪ ⎪ − lx Xe ⎨α − 2 τ y ⎪ − ky Ye β+ ⎪ ⎪ 2 ⎪ ⎪ ⎪ τ y ⎪ ⎪ − ly Ye ⎩β − 2

=0 =0 =0

⇐⇒

=0

⎧ Xe ⎪ ⎪α = (kx + lx ) 2 ⎪ ⎪ ⎪ ⎨τx = (kx − lx )Xe ⎪ β = (ky + ly ) Y2e ⎪ ⎪ ⎪ ⎪ ⎩τy = (ky − ly )Ye

(5.8)

The ACF-I of sampled image (5.7) can thus be written finally as:

Rfˆ(α, β, τx , τy ) =

kx

ky

lx

ly

Ye Xe Rf (kx + lx ) , (ky + ly ) , (kx − lx )Xe , (ky − ly )Ye 2 2 Xe δ τx − (kx − lx )Xe × δ α − (kx + lx ) 2 Ye δ τy − (ky − ly )Ye × δ β − (ky + ly ) 2

(5.9)

Taking into account the definition of TDR, which is homogenous to a generalized ACF, and the Dirac distribution properties, the generalized ACF expression of the sampled image becomes:

γfˆ(x, y, τx , τy ) =

kx

ky

lx

ly

Ye Xe − x, (ky + ly ) − y, (kx − lx )Xe , (ky − ly )Ye ΦTR (kx + lx ) 2 2 Xe Ye × Rf (kx + lx ) , (ky + ly ) , (kx − lx )Xe , (ky − ly )Ye 2 2 (5.10) × δ τx − (kx − lx )Xe × δ τy − (ky − ly )Ye

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Thus, the generalized ACF is also defined by a Dirac comb distribution, this time 2-D, and can be expressed directly from the ACF of the image before sampling. 5.2.1.2. Time-frequency representation of a discrete image The TFR kernel ΦTR for image fˆ resulting from a sampling on a rectangular grid can be obtained using the Fourier transform of the generalized ACF (5.10): Cfˆ(x, y, ωx , ωy ) =

Cfˆ(x, y, ωx , ωy ) =

γfˆ(x, y, τx , τy ) e−j(ωx τx +ωy τy ) dτx dτy

(5.11)

kx

ky

lx

ly

Ye Xe − x, (ky + ly ) − y, (kx − lx )Xe , (ky − ly )Ye ΦTR (kx + lx ) 2 2 Xe Ye × Rf (kx + lx ) , (ky + ly ) , (kx − lx )Xe , (ky − ly )Ye 2 2 (5.12) × exp −j ωx (kx − lx )Xe + ωy (ky − ly )Ye Variation intervals of indices (kx , ky , lx , ly ) depend on the support of the discrete image and kernel, in the case where this has been truncated. Taking into account all values of the generalized ACF will provide a non-overlapping TFR if the kernel of the TFR lends itself to it.

For an image spectrum, two repetitions can take place: one in X and the other in Y . For example, if in the distribution of autocorrelation, even terms such as kx + lx are removed, then the TFR will have a spectral overlapping in X. The TFR can, in fact, be expressed directly from the samples of the discrete image: Cfˆ(x, y, ωx , ωy ) =

kx

ky

lx

ly

Ye Xe − x, (ky + ly ) − y, (kx − lx )Xe , (ky − ly )Ye ΦTR (kx + lx ) 2 2 " ! × f (kx , ky ) × f ∗ (lx , l× y ) exp −j ωx (kx − lx )Xe + ωy (ky − ly )Ye

(5.13)

which constitutes a formulation that lends itself well to a numerical implementation.


223

The evaluation of the samples of the image, i.e.: (x, y) = (nx Xe , ny Ye ) with (nx , ny ) ∈ Z2 , leads to: Cfˆ(nx Xe , ny Ye , ωx , ωy ) =

kx

ky

lx

ly

Ye Xe ΦTR (kx + 2lx ) , (ky + 2ly ) , kx Xe , ky Ye 2 2 × f (nx + kx + lx , ny + ky + ly ) × f ∗ (nx + lx , ny + ly ) × exp −j ωx kx Xe + ωy ky Ye

(5.14)

For calculations at these positions only, the values the 4-D kernel ΦTR occur discretely. This operation resembles a kernel sampling, although it is inaccurate and the sampling equation is then: ˆ TR (x, y, τx , τy ) = Φ

kx

ky

lx

ly

" Xe ) × δ τx − (2kx + 1)Xe + δ(x − lx Xe ) × δ(τx − 2kx Xe ) 2 ! " Ye × δ(y − ly Ye − ) × δ τy − (2ky + 1)Ye + δ(y − ly Ye ) × δ(τy − 2ky Ye ) 2 !

δ(x − lx Xe −

× ΦTR (x, y, τx , τy )

(5.15)

5.2.2. Support of the sums All the sums of the above-mentioned formulae are performed on the integer set. It will be appropriate for ensuring sums on finite supports are obtained for any numerical implementation. This is not difficult. Indeed, all the images that are treated result from an image sampling and are of limited duration. In this case, the sums will be always finite, whether the kernel is of finite duration or not. The problem persists in the case where we seek to also evaluate the TFR of synthetic images described by analytic formulae (for example, a sum of sinusoids of infinite range) with a kernel of unlimited support. If an analytic expression of the TFR cannot be found, we cannot hope to get anything better than an approximate numerical evaluation. 5.3. Minimum properties and constraints on the kernel It would be preferable if spectral analysis, carried out with bilinear TFR on images, did not depend on the orientation of the image, as every direction of the image is then

224


handled equally. In the same way, it would be better if the TFR of an image enabled an energy interpretation of several majority spectral components. Many other properties could still be desired. They depend on the form of the kernel and each one is characterized by the constraints that they impose on it. There are many references on the subject; the constraints are mostly well-known and are unfortunately often contradictory. In practice, it is important to locate essential properties for an application and deduce the acceptable kernel. In this case, for an application of spectral analysis, it will be important to have correct spectral estimation properties and, if possible, freedom for the choice of image orientation. 5.3.1. Compatibility with reversible linear transformations It would be desirable that the spectrum follows the signal properties in elementary transformations which are rotations and symmetries. They are reversible linear transformation, with the characteristic of being isometric, as opposed to homotheties and affinities. More generally, for the signals defined on N dimensional space, matrices associated with reversible linear transformation which apply to these signals are square matrices with real coefficients noted by A, of dimension N × N and invertible. Let us note by fA the transformed signal: fA (θ) = f (Aθ)

(5.16)

The expression of compatibility of a TFR with a linear transformation associated with matrix A is written: 1 Cf (Aθ, t A−1 ω) (5.17) |det A| This formula expresses the hope that the spectrum follows the signal properties in its transformations. In the case of images, the presence of the inverse of the matrix in factor of the frequency vector would provide compatibility with changes in scale, while transposition of the matrix would restore the correct direction of rotation. Finally, the occurrence of the determinant at the denominator ensures the conservation of total energy. CfA (θ, ω) =

This compatibility is verified if and only if [GRA 97]: 1 ΦTR (A−1 θ, A−1 τ ) |det A|

∀ (θ, τ ),

ΦTR (θ, τ ) =

∀ (ξ, τ ),

ΦDR (ξ, τ ) = ΦDR (t Aξ, A−1 τ )

(5.18)

or: (5.19)


225

These constraints are flexible to ensure exact compatibility with rotation and symmetry, since in this case, the determinant is 1 or −1. 5.3.2. Positivity It would be interesting that the TFR, in order to achieve SAR ocean surface images, accounts for the energy of swell phenomena or internal waves. Even if the imaging process does not respect the conservation of spectral energy, the negative values of the TFR prevent any energy interpretation; for most of them, the TFR does not represent energy densities. The spectrograms are the only bilinear TFR to take only positive values. Their kernel in Doppler-frequency space is a weighted sum of combined ambiguity functions of windows by positive weights ([FLA 93], p. 120): ΦDR (ξ, τ ) =

ck A∗hk (ξ, τ ),

with ck 0 and

k

The constraint expressed in the time-delay kernel is: ΦTR (t, τ ) = ck Rh∗ k (t, τ )

+∞

ck < +∞

(5.20)

k=1

(5.21)

k

where Rh is the ACF of window h. 5.3.3. TFR with real values In the absence of interpreting the TFR as an energy density, it could be desirable to interpret it as a distribution of energy. It is easier to interpret energy with real values rather than complex values. Moreover, from a practical point of view, TFR with real values are simpler to visualize. The constraint on the time-delay kernel is that it is of Hermitian symmetry on its delay dimension: ∀(θ, τ ),

ΦTR (θ, τ ) = Φ∗TR (θ, −τ )

(5.22)

5.3.4. Conservation of energy This theoretical property tries to ensure that the energy of the image is distributed completely in time-frequency space without any loss or addition. It translates into: dω 2 = |x(θ)| dθ (5.23) Cx (θ, ω) dθ (2π)N

226


This property is valid when the Doppler-delay kernel checks the constraint: ΦDR (0, 0) = 1. The same constraint on the time-delay kernel is: ΦTR (θ, 0) dθ = 1 (5.24) This condition, which is necessary and sufficient, does not impose any constraint locally on the kernel ΦTR . On the other hand, sufficient conditions can easily be derived from it. It is immediately apparent that kernels of the form: ΦTR (θ, 0) = αi δ(θ − θ i ) (5.25) i

with

i

αi = 1, satisfy (5.24).

This constraint becomes much stronger and the expression much simpler if we also lay down that the TFR must be compatible with central symmetries, equivalent to the temporal reversal of 1-D signals. A necessary and sufficient condition which ensures this property is: ΦDR (ξ, τ ) = ΦDR (−ξ, −τ ). It turns out that sufficient condition (5.25) is too slack on this subject. To be sure, it is enough to transpose this condition in the Doppler-delay spaces: ΦDR (ξ, 0) =

t αi δ(θ − θ i ) e−j ξ θ dθ =

i

t αi e−j ξ θ i

(5.26)

i

The necessary condition ΦDR (ξ, 0) = ΦDR (−ξ, 0) for every ξ then imposes: ΦTR (θ, 0) = δ(θ)

(5.27)

which is a condition sufficient for the TFR to preserve energy.

5.3.5. Spectral estimation The TFR used for spectral analysis must have good spectral estimation properties. We can show that it is possible, under certain conditions, to estimate by detection of the spectral maximum – which is very practical for numerical implementation – the instantaneous frequency of a second order stationary image or an image at bilinear frequency modulation: ∀ t, ωinst (t) = arg max Cx (t, ω) ω∈R

(5.28)

In what follows, a hypothesis will be made from a TFR with real values to give an unambiguous direction to the spectral maximum. In light of the above, this requires a time-delay kernel with Hermitian symmetry on the delay dimension.


227

5.3.5.1. Case of second order stationary images For an application of spectral analysis, spectral localization of the stationary signals is the least that can be required from the TFR. There is a (sufficient) condition on the kernel which ensures this property [GRA 97, HLA 01]: ∀(τx , τy ) ∈ R2 , ΦTR (x, y, τx , τy ) dx dy 0 (5.29) R2

In particular, all the kernels ΦTR with positive values are appropriate. 5.3.5.2. Case of signals with bilinear frequency modulation Under certain conditions, to check a posteriori, the instantaneous frequency of an image with linear frequency modulation on each of the two dimensions is estimable exactly. First, the image model, for which this principle is valid, is: x(θ) = exp jφ(θ)

(5.30)

The magnitude is a unit constant. The phase is defined by: 1 φ(θ) = α + ω t0 θ + θ t Λθ 2

(5.31)

where α and ω 0 are the phase and the instantaneous frequency at the origin, respectively, while Λ is the matrix which characterizes the bilinear modulation: Λ=

ωxx ωxy

ωxy ωyy

(5.32)

The coefficients on the secondary diagonal occur only by their mean value; there is thus no loss of generality to make them equal. The (sufficient) condition for which the TFR indicates the instantaneous frequency at the maximum of the spectral division is [GRA 97]:

∀θ ∈ R2 , ∀τ ∈ R2 , ΦTR (θ, τ ) = ΦTR (−θ, τ ) . ∀τ ∈ R2 , ΦTR (θ, τ ) cos(θ t Λτ ) dθ 0

(5.33)

In practice, integration is done only on the kernel support and it is important to select a finite support for kernel ΦTR . For a given kernel ΦTR it is possible to check if

228


a certain modulation of matrix Λ is sufficiently smooth so that the spectral maximum corresponds to instantaneous frequency. It may be simply that factor cos(θ t Λτ ) remains positive on all the kernel support ΦTR . As in 1-D, the acceptable modulation range relies heavily on kernel dimensions; the more limited the support, the stronger the estimable bilinear frequency modulation.

5.3.6. Evolution of properties of a modified kernel In order to be implementable, a time-delay kernel of the TFR must have finite support on each one of its dimensions. If this is not so, the kernel must be truncated. This operation is a multiplication by a rectangular window function: the properties of the kernel could be altered. The concern of improving a representation also results in modifying kernels. Smoothing can thus reduce the harmful interferences in TFR readability. The three basic transformations described below in detail can lead to more elaborate forms of kernel. The evolution of major properties of spectral analysis of images is given (see [GRA 97] for more detailed developments). 5.3.6.1. Pseudo-smoothed version This transformation acts by weighting the kernel according to the delay variable: ΦPTR (θ, τ ) = h(τ ) ΦTR (θ, τ )

(5.34)

In practice, it will be judicious to take for function h a function having the character of a window with finite support. The advantage in this case is to make it possible to numerically calculate the AFC-I with a finite number of terms, which is equal to double the number of samples covered by window h. The frequency resolution of the TFR is degraded, since this weighting, following delay dimension, corresponds to a filtering, according to the frequency dimension: dΩ (5.35) ΦPTF (θ, ω) = H(Ω) ΦTF (θ, ω − Ω) 2 (2π) 2 R The temporal resolution remains, on the other hand, unchanged (the fundamental difference with the spectrogram where the temporal and frequency resolutions are linked should be noted). In order to preserve at transformed kernel ΦPTF the eventual properties of frequency estimation and conservation of energy of kernel ΦTR , it is sufficient that the weight function is symmetric and of positive real values with h(0, 0) = 1.


229

5.3.6.2. Simply smoothed version This version modifies the kernel only by filtering in x and y. The only effect is thus to degrade the spatial localization of the TFR, which has, in turn, the advantage of smoothing the interferences: ΦSL TR (θ, τ ) =

R2

g(θ ) ΦTR (θ − θ , τ ) dθ

(5.36)

This results in a weighting of Doppler frequencies: ΦSL DR (ξ, τ ) = G(ξ) ΦDR (ξ, τ )

(5.37)

If the original kernel ΦTR has infinite delay support, then this transformation will not be enough to result in a numerically implemented version because the summation, which occurs in the calculation of local autocorrelation function, remains infinite. If not, this kernel, thus transformed, leads to version that could be implemented, provided its support is finite in θ. Taking symmetric functions with real values for g(θ) and G(ξ) with G(0) = 1 makes it possible to maintain the energy conservation properties of spectral estimation with a reserve for the chirps which require checking a posteriori. 5.3.6.3. Simply masked version This version modifies the kernel only by temporal weighting. The effect is to degrade the resolution in Doppler frequency: ΦSM DR (ξ, τ ) =

RN

G(Ω) ΦDR (ξ − Ω, τ )

ΦSM TR (θ, τ ) = g(θ) ΦTR (θ, τ )

dΩ (2π)2

(5.38)

(5.39)

The remark made for the simply smoothed transformation in connection with infinite summation following delay variable τ is still valid. The masked version has a practical benefit only when, on the one hand, the kernel support is already limited in delay (if not, a “pseudo-smoothed” transformation should be linked) and, on the other hand, when the masked kernel has finite support in θ. A symmetric function with positive real values shall be taken for g to best keep the properties. However, the energy conservation property is systematically lost.

230


5.4. Notion of analytic images 5.4.1. Formulation of the problem for images For the 1-D signal with real values f (t) = cos ω1 t cos ω2 t, 0 < ω1 < ω2 , a complex analytic signal can be associated, determined in a unique manner and equal, in this case, to: fa (t) = cos ω1 t exp jω2 t

(5.40)

The good characteristic of this signal is not to have negative frequencies. This has the advantage of giving meaning to the frequency moments, but most of all to calculate instantaneous frequency, in a unique manner. In 1-D, the analytic signal can be obtained simply by canceling the negative frequencies and by doubling the magnitude of positive frequencies (to preserve total energy). In 2-D, the concept of negative and positive frequencies is not established, which justifies the fact that the extension of the analytic signal concept to the images is not immediate. Let us take the signal with two dimensions: f (x, y) = cos(xωx1 + yωy1 ) cos(xωx2 + yωy2 )

(5.41)

which is also written: f (x, y) =

1 × cos x(ωx1 + ωx2 ) + y(ωy1 + ωy2 ) + 2 cos x(ωx2 − ωx1 ) + y(ωy2 − ωy1 )

(5.42)

This 2-D signal is with real values and has four pure frequency components (ωx1 + ωx2 , ωy1 +ωy2 ), (ωx2 −ωx1 , ωy2 −ωy1 ) and their symmetries with respect to the origin. In the form of a complex exponential sum, this signal is written: 1 j(x(ωx1 +ωx2 )+y(ωy1 +ωy2 )) × e + ej(x(ωx2 −ωx1 )+y(ωy2 −ωy1 )) 4 + ej(x(ωx1 −ωx2 )+y(ωy1 −ωy2 )) + e−j(x(ωx1 +ωx2 )+y(ωy1 +ωy2 )) (5.43)

f (x, y) =

In order to extend the analytic signal principle to the images, it would be naturally tempting to apply the 1-D principle following two dimensions successively. This is tantamount to multiplying the magnitude of the frequencies in the quadrant (ωx > 0, ωy > 0) by four and canceling the remainder.


231

This solution has the serious drawback of only being appropriate for certain signals (those whose spectrum is entirely contained within this quadrant and the opposite one) and seriously corrupting the others. For example, under the hypothesis: 0 < ωx1 < ωx2 0 < ωy1 < ωy2

(5.44)

all is well, since the analytic signal would then be: fa (x, y) = ej(x(ωx1 +ωx2 )+y(ωy1 +ωy2 )) + ej(x(ωx2 −ωx1 )+y(ωy2 −ωy1 )) = cos xωx1 + yωy1 × ej(xωx2 +yωy2 ) In contrast, under the hypothesis: 0 < ωx1 < ωx2 0 < ωy2 < ωy1

(5.45)

the analytic signal is no longer representative of the original signal, since a component has disappeared: fa (x, y) = ej(x(ωx1 +ωx2 )+y(ωy1 +ωy2 ))

(5.46)

Preserving any other quadrant while canceling others would not be a better solution. 5.4.2. Traditional solution An alternative to this solution was proposed and solved this problem partially [ZHU 90]. As in 1-D, the analytic signal can be written as the result of a filtering: Z(νx , νy ) = 2 H(νx , νy ) F (νx , νy )

(5.47)

The authors propose to take as the filter: H(νx , νy ) = i.e.:

1 sign(νx ) + sign(νy ) + 2 4

⎧ ⎪ ⎨1 H(νx , νy ) = 0 ⎪ ⎩1 2

if νx > 0, νy > 0 if νx < 0, νy < 0 if νx < 0, νy > 0 or νx > 0, νy < 0

(5.48)

(5.49)

232


This transfer function can be written according to that of a 2-D Hilbert transform K: 1 1 + jK(νx , νy ) H(νx , νy ) = (5.50) 2 which is defined in the frequency domain by: 1 (5.51) K(νx , νy ) = − j sign(νx ) + sign(νy ) 2 and in the temporal domain by: 1 1 1 δ(x) v.p. + δ(y) v.p. (5.52) k(x, y) = 2 πy πx Figure 5.1 shows filter H(νx , νy ) which will make it possible to pass directly from the real signal to the analytic signal (with factor close to 2). The disadvantage of this solution is that all components are not treated equally. For example, the signal: f (x, y) = cos(xωx + yωy )

(5.53) vy

1 2

1 2

1 vx

0

1 2

Figure 5.1. Characteristics of filter H(νx , νy )

If ωx > 0 and ωy > 0, then this solution is coherent with the concept of a 1-D analytic signal, since we get: fa (x, y) = exp j(xωx + yωy ). On the other hand, if the component is in any one of the two adjacent quadrants, for example ωx > 0 and ωy < 0, then the result has no analytic signal, since the signal remains unchanged (and thus remains real): fa (x, y) = f (x, y)

(5.54)


233

5.4.3. Symmetric solution with reference to a hyperplane The solution of dividing the plane into two seems more natural. It is a general solution since it applies to multidimensional signals and involves cutting the total space by a hyperplane [PEY 86]. The Fourier transform of a function with real values is always Hermitian and thus presents a redundancy of information in half the space. Dividing the space into two by a hyperplane thus seems the most appropriate solution, from this point of view. There is thus an entire class of analytic signals for the images and each one depends on the direction of the line which divides the plane into two. Let us take a(a, b) the unit vector perpendicular to this line. The kernel K of equation (5.50) which determines the Hilbert transform corresponding to this analytic signal is, according to the study of Zhu et al. [ZHU 89]: Ka (νx , νy ) = −j sign aνx + bνy

(5.55)

The corresponding impulse response is: ka (x, y) =

1 1 v.p. δ(bx − ay) π ax + by

(5.56)

The transfer function of the filter which makes it possible to transform the real signal into an analytic signal is: Ha (νx , νy ) =

1 1 + sign(aνx + bνy ) 2

(5.57)

A real pure frequency component is made up of two Dirac impulses of symmetric position with respect to the origin of the frequency plane. Unless decided otherwise, the two impulses are always located on both sides of the line. Only one will remain after the multiplication by transfer function H and the other will be canceled. That is true except when the two Dirac impulses are located on the separation line. By admitting that sign(0) = 0, all frequency components located on this line will remain unchanged, just like zero frequency in 1-D. This particular case was, however, not broached by the authors of [ZHU 89].

5.4.4. Solution with a non-symmetric half-plane A solution proposed [GRA 97] to regulate this problem consists of dividing the separation line into three zones: – origin of the frequency plane; – two half-lines issued from the origin.

234


Each half-line is assigned to a half-plane, while the origin of the frequency plane remains isolated. This division of the frequency plane ensures that one of the two impulses out of any pair of Dirac impulses of a pure real component will always be eliminated and the other retained (with its doubled magnitude). The possible continuous component is kept intact. The selection of adherance of the half-lines to a half-plane is not important. In practice, a simple test on the sign of a coordinate of a frequency component can fix on which half-line a component is located. The division, finally being carried out, naturally concurs with the concept of 2-D causality associated with the non-symmetric half-plane. Figure 5.2 shows an example of division where the line νx = 0 is used to make the partition. The choice is made for the half-line (νx = 0, νy > 0) to belong to the half-plane (νx > 0, νy ). The other half-line (νx = 0, νy < 0) thus belongs to the other half-plane (νx < 0, νy ). Finally, we choose to preserve only the part of the spectrum located in the plane (νx > 0, νy ), while the remainder is annuled. This canceled part will thereafter be called the non-symmetric half-plane at zero.

νy

H(νx ,νy) = 1 νx H (0,0) = 12 H (νx ,νy) = 0

Figure 5.2. Example of transfer function for H based on the partition of the spectral plane in two non-symmetric half-planes

In practice, it is desirable to bind the choice of the line to the spectral content of the image to be analyzed. It is necessary to avoid cutting off a spectral component by the line, since this component is then found in two separate pieces. The estimation of frequency moments would not have any significance in this case, but above all,


235

a major phase jump would result from it on the imaginary part of the analytic image. This reason is similar to that which justifies recourse to an analytic signal to calculate the frequency moments of an even order of 1-D signals. The developed solution resolved the ambiguity of the method suggested in the work by Peyrin et al. [PEY 86] well. Let us take the transfer function H presented in Figure 5.2 and the real signal with a single component f (x, y) = cos yωy . Irrespective of the sign of ωy , there remains only one Dirac impulse after multiplication by H. If ωy > 0, then the analytic image is fa (x, y) = exp jyωy , and if ωy < 0 then fa (x, y) = exp −jyωy . The transformation of a 1-D real signal into an analytic signal means considering the highest frequency component as the phase, while the lowest frequency component is admissible in the amplitude category. In 2-D, there is no order relation between the frequencies. What is the mechanism dictating the choice of the frequency component which becomes the phase? To answer this question, we propose to determine the component of the 2-D signal defined by equation (5.41) or (5.42) which will become the phase. Equation (5.43) presents this 2-D signal in the form of a sum of four complex pure frequency components, whose values are: ⎧ ω1 + ω2 ⎪ ⎪ ⎪ ⎨ω − ω 2 1 (ωx , ωy )t = ⎪ ω − ω 1 2 ⎪ ⎪ ⎩ −(ω 1 + ω 2 )

with ω 1 =

ω1x ω1y

and ω 2 =

ω2x ω2y

(5.58) The method finally adopted to calculate the analytic signal of the images will preserve two of these exponentials, while the other two will be eliminated. Figure 5.3 shows four Dirac impulses making up the spectrum of the signal. The line which divides the spectral plane into two non-symmetric half-planes is also plotted. The halfplane on the side of vector a is that of the components, which are retained to calculate the analytic signal. The two frequency components which are preserved to form the analytic signal for the choice of vectors ω 1 and ω 2 , which correspond to the line, are:

ω1 + ω2 ω2 − ω1

(5.59)

236

Two-Dimensional Signal Analysis ␻y

a␻x + b␻y = 0

␻2 2 ␻1 2␻1 d2 ␻2

2(␻1 1 ␻2)

␻x aW

2␻2

␻2 1 ␻1

d1 ␻1 ␻1 2 ␻2

Figure 5.3. Example of distribution of frequency components in the spectral plane. The line of equation aωx + bωy = 0 delimits the half-plane which is preserved from the cancelled half-plane

since their scalar products with vector a are positive: > 0 > 0 while the two others are negative: < 0 < 0

(5.60)

(5.61)

The analytic image expression associated with the choice of division of the spectral plane made in Figure 5.3 is: fa (x, y) = cos xωx1 + yωy1 × ej(xωx2 +yωy2 ) Frequency component ω 2 forms the phase while component ω1 forms the magnitude. The rule of selection of the component which will form the phase is


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deduced from (5.60). In fact, since the scalar product is a linear operation, (5.60) is equivalent to: # # > ## (5.62) However, the value of these scalar products is none other than the distance from the frequency component to the line of division of the plane. In conclusion, the analytic signal that we propose for the images selects the most distant frequency component (in Euclidean distance) from the line of separation of the plane in two half-planes to make it the complex signal phase component. The lowest frequency component is then admissible in the complex signal amplitude component. 5.4.5. Choice of spectral division The division of the spectral plane into two non-symmetric half-planes influences the phase of the analytic image. When this division is poorly selected and when it cuts a spectral component of the image, then the structure of the image is altered. We use two examples to illustrate these points. Both consist of calculating the analytic image of an explicitly known synthetic image in the spectral as well as the spatial domain. Each time, the real part and the imaginary part are known and, by construction, the imaginary part is the quadrature of the real part. For each example, several analytic images are calculated for various divisions, from the only real part. The task is to compare the imaginary part of these analytic images with the real imaginary part of the original complex image. These comparisons are performed just as well on the phase images, which will also be presented for this reason. The first example is a complex synthetic image with linear frequency modulation. The real and imaginary part as well as the spectrum are presented in Figure 5.4. The values of this image are known explicitly: f (x, y) = exp j2π(fx x + fy y + gx x2 + gy y 2 )

(5.63)

where (fx , fy ) is the frequency of the image at the origin (located in the middle of the image) and (gx , gy ) is the gradient of the 2-D linear frequency modulation. The instantaneous frequency is thus known at every pixel of this image: finst (x, y) =

fx + 2gx x fy + 2gy y

(5.64)

The spectral support of the real part of this synthetic image is shown in Figure 5.4c. The examples, presented in Figure 5.5, show that the final result is affected at the level of the phase when the line of separation cuts the spectral package. The division line of

238


(a) Real part

(b) Imaginary part

(c) Spectrum support on the real part

Figure 5.4. Synthetic image with linear frequency modulation. The dimension of the image is 128 × 256. The frequency at the origin (center of the image) is (0.11; 0.0008). The gradient (gx , gy ) of the frequency modulation is (0.0005; 0.001)

the spectral plane in two non-symmetric half-planes is shown at the top of Figure 5.5 for three cases. It is superimposed on the spectrum of the real part of the synthetic image, which makes it possible to appreciate the importance of the separation of the image spectrum in two parts. The effects of the spectral division on the imaginary part of the analytic image and on its phase are shown for each one of the three cases. The first case (first column of the figure) is the example of a good spectral division, which leads to a correct evaluation of an analytic image. The calculated imaginary part proves to be, in fact, very close to the true imaginary part (see Figure 5.4b). The other two columns show two improper spectral divisions of varying degrees. That in the middle shows one just at the theoretical limit. Because of spectral spread due to the limited duration of the image, this line actually cuts a small portion of the spectrum, which is seen on the image of the imaginary part or on the phase image. The third example shows the disastrous effect of a clean division of the spectral package by the line. A division will be correct if the line falls outside this angular sector with sufficient margin to overcome spectral spread due to the window function (see Figure 5.5a). If not, the division will also introduce phase angles in the imaginary part of the analytic image generated, which are very clearly visible on the phase. The importance of phase angles is modulated by the portion of the cut spectral support (see Figures 5.5b and 5.5c). The distortion of the image due to phase angles is all the more visible, as the spectral package is divided into two parts of equal energies.


(a) Good division

(b) Bad division

(c) Bad division

Figure 5.5. Influence of the division on the generated analytic image. The angular sector from which the support of the theoretical spectrum is seen is [−79, 78; 79, 76] (in degrees). The middle row presents the imaginary parts of the calculated analytic images and the bottom row shows their phase

239

240


The second example shows that a division which goes over a main spectral component of the image modifies the structure of the image. For that, we chose an image with only one spectral component, pure in theory but spread out in practice due to calculation with limited data. The complex image is shown in Figure 5.6 and is defined by: f (x, y) = exp j2π(fx x + fy y)

(5.65)

In theory, the analytic image calculated with the non-symmetric half-plane method should give good results (within a possible complex conjugation). In practice, it is not true because of the inaccuracies due to calculation methods; calculations are carried out in the spectral field and go through a discrete Fourier transform. At the time of this operation, a pure spectral component is distributed (in most cases) on the neighboring frequency samples. The analytic image calculated will be correct only if the division line does not separate these neighboring samples. In the favorable case, where these neighboring samples are located on the same side of the division line, the spectral structure of the image is preserved (compare the imaginary part and the phase of Figure 5.8 for the case with 45 degrees with true imaginary part and true phase shown in Figure 5.6). The spectrum calculated with this analytic image is shown in Figure 5.9b.

(a) Real part

(b) Imaginary part

(c) Phase

Figure 5.6. Synthetic image with unique pure frequency component (fx ; fy ) = (0.1601560; 0.2382810). The dimension of the image is 128 × 128

In the unfavorable case where the division goes over the group of adjacent samples constituting the spectral component, the spectral structure is modified (see Figure 5.7), which is also seen on the imaginary part and the phase of the calculated analytic image (see Figure 5.8 in the case of a division at –33.9 degrees). The analytic image has two components now (non-symmetric with respect to the origin), which are found practically by evaluating its spectrum (see Figure 5.9c). Although the spectral contents are identical from one solution to another within a phase, the bilinear time-frequency analysis methods are obstructed by the generated


241

phase jump and the result is altered. In fact, in the calculation of the analytic image, the imaginary part is obtained by dephasing, in the same direction, all components located on one side of the division line while the components located on the other side are dephased in the opposite direction.

5.5. Spectral analysis application of SAR images The spectral analysis has at least two applications for the ocean surface images. First, it can be used to characterize the swell fields by giving estimates of essential

fy

Original frequency component

Separation line in two non-symmetric half-planes

A

D

B

C

Canceled half-plane fx

Frequency components of the analytic image C'

B'

D'

A'

Preserved half-plane

Figure 5.7. Division of a frequency component and rebuilt analytic image. The amplitude of the true frequency component is distributed mainly and equally in the closest adjacent discrete plane of the discrete Fourier transform (points A, B, C and D). The spectrum of the real part also comprises symmetric points (A , B , C and D ). The analytic image will have four frequency components (A , B , C and D) for the division indicated

242


(a) good division

(b) imaginary part

(c) phase

(d) bad division

(e) imaginary part

(f) phase

Figure 5.8. Influence of the division on the generated analytic image. Good division (45 degrees) for the top row, bad division for the bottom row (–33.9 degrees)

parameters: wavelength, direction of propagation, energy and dispersion. Second, it can be used for study of internal wave fields. Again, the essential parameters such as wavelength and the direction can be quickly estimated. It should be noted, moreover, that the wavelength of the internal waves is always higher (approximately 1,000 meters) than the wavelength of the swell (200 meters on average). Sometimes the two phenomena have adjacent wavelengths; this is the case for the image shown in Figure 5.17 where the wavelengths of the two phenomena meet at around 400 meters. When the two phenomena have quite distinct wavelengths, two analyses are made on the same image; the analysis of the internal waves is carried out on an image decimated to reduce the quantity of data to be dealt with. The energy of the internal wave is also an important parameter, but its estimation from a spectral analysis of an SAR image does not seem easy. An interesting application would be to detect, localize and delimit the internal wave fields automatically. This problem is not currently resolved in a good way and to solve it we tried to work out criteria which could make it possible to detect the possible presence of an internal wave.


(a) Spectrum of the complex image

(a) Spectrum of the correct analytic image

243

(a) Spectrum of the badly calculated analytic image

Figure 5.9. Spectra estimated with a bilinear time-frequency method. The spectrum calculated from the original complex image is shown on the left for comparison. The spectrum shown in the middle is calculated on the analytic image obtained with a division at 45 degrees. That on the right-hand side corresponds to a (bad) division at −33.9 degrees

Figure 5.10. Analysis of an internal wave with pseudo-smoothed Wigner-Ville TFR (63.16): result of 8 × 10 analyses carried out

5.5.1. Analysis of an internal waveform 5.5.1.1. Characteristics of the image and preprocessing Figure 5.10 shows an internal wave whose characteristic is to curve greatly. The curvature is continuous on the entire surface of the field which means that this internal

244


wave field is not stationary, even locally. The hypothesis usually formulated within the speech analysis framework on the local stationarity (precisely on windows of around 20 milliseconds) is not transposable here. In fact, we could note in experiments that this internal wave field has an instantaneous frequency law, which approximates a bilinear frequency modulation (see below). The area of this image is 13.5 km in width by 19.8 km in height. The image was sub-sampled by a factor 8 on each dimension to reduce the volume of calculation. Let us also specify that an analytic image of this sub-sampled image is calculated so as to remove half the interferential terms of timefrequency representations. Given the directions taken by the internal wave of the image in Figure 5.10, the separation line of the plane in two non-symmetric half-planes was fixed at ωx = 0. In this way, no significant component of the image is cut into two. 5.5.1.2. Analysis method The results shown in Figure 5.11 were extracted from a pseudo-smoothed Wigner-Ville representation. For each analysis, the most energetic wavelength component ranging between 400 meters and 1,800 meters is represented with a segment whose length is identical to the wavelength of the spectral component. The absence of a segment means that the energy of the detected component was considered too weak to correspond to a physical reality. The kernel of this representation is built by separating the variables by dimension. The parameters of a 1-D kernel are as follows: – delay weighting is carried out with a rectangular window of a length of 63 samples; – smoothing is carried out with a triangular window of a width of 16 samples, i.e. 1,600 meters on this image, or what still represents an eighth of the width of the image. Note that a sufficiently broad smoothing window was selected to smoothen the interferences. This ensures that the maxima detected on the spectrum correspond to the integral components and not to the interferential terms. If the analyzed signal was a stationary component, the detected peak would precisely indicate the true value of the frequency (it is the minimum service required). On the other hand, in the event of frequency modulation, even if only bilinear, perfect localization is not guaranteed. In fact, the goal of this analysis is simply to locate the spectral components which benefit us. The precise estimates of those could then be carried out by repeating calculations with a kernel which has a minimal smoothing effect, such as a kernel defined by the product of two triangular windows defined on three points (the width of the matrix corresponds to a sampling period): ⎡

⎤ 1/4 1/2 1/4 ⎣1/2 1 1/2⎦ 1/4 1/2 1/4

(5.66)


245

Figure 5.12 shows components of the greatest energies estimated with such a kernel (delay weighting did not change). The quantity of errors is significant and is presumably due to a lack of smoothing of the interference. In order to effectively use the results provided by this representation, more attention needs to be focused on

Figure 5.11. At the top: spectral division for the analysis point at the top left of Figure 5.10. The internal wave has just begun and nothing significant is detected between 400 and 1,800 meters. The peak corresponds to a very long component and comes from the contour of the start of the internal wave. At the bottom: spectral division for the point of analysis at the bottom left of Figure 5.10. The internal wave is well-detected and its energy is much higher than that of the noise. NB: energy graduations in the two graphs are different

246


the exact localization of only the strongest components detected beforehand with a representation having sufficient smoothing, such as that presented in Figure 5.10. An efficient method exists to precisely localize the maximum [GRA 97].

Figure 5.12. Analysis of an internal wave with pseudo-smoothed Wigner-Ville TFR without sorting the spectral components and with minimum smoothing: the absence of sorting the detected components translates into the presence of many results without significant benefit for us

5.5.1.3. Presentation of the results and interpretation By nature, a time-frequency representation of an image is a function with four dimensions, including two for localization of the 2-D spectrum on the image. It is important in practice to select the calculation points of the representation. To simplify calculations, and also because nothing is lost, the calculation points correspond to positions of image samples. Even while limiting ourselves to these points, calculation of the spectrum of each of them would be superfluous. Indeed, even if the image is not (second order) stationary, the spectral evolution on the image is rather slow and there would only be a few changes from one sample to its neighboring sample. In the case of the image analyzed here and presented in Figure 5.10, we chose to


247

Figure 5.13. At the top: spectral division for the analysis point at the top left of Figure 5.12. The smoothing window being short, this point is located closer to the corner of the image than the corresponding point in Figure 5.10. The analyzed area does not comprise any significant component, which confirms the noisy aspect of the spectral division. At the bottom: spectral division for the analysis point at the bottom left of Figure 5.12. The internal wave is again well-detected and its energy is much higher than that of the noise

evaluate the representation in 80 points, since we estimate that it is sufficient to describe the evolution of the internal wave. At each point of analysis, an image spectrum is calculated. Independent of the large number of calculated spectra, the volume of information contained in only one of these spectra is larger than needed and hence it is highly desirable to summarize it. The presentation of the results made in Figure 5.10 follows this principle and only the

248


Figure 5.14. Analysis of an internal wave with pseudo-smoothed Wigner-Ville TFR without sorting the spectral component. The parameters are identical to those used for the representation of Figure 5.10. The difference comes from the absence of sorting, which reflects the presence of many results without significant benefit for us as, for example, in the top left corner or the right edge of the image

component considered to be interesting was shown for each analyzed area. Of course, the problem is to distinguish what is interesting and what is not. In fact, the benefit, which is attributed to a detected component comes simply from what we seek. In our case, we seek to detect and characterize internal waves. For this reason, only the spectral components between 400 meters and 1,800 meters will be retained. Among these, only those which are sufficiently energetic will be retained and the most energetic is shown. If no component has sufficient energy, nothing is shown for the area being analyzed. This is the case, for example, in the top left corner of the image shown in Figure 5.10 where the absence of any interesting spectral component is easily verifiable visually. The energy threshold considered to be sufficient was empirically decided by certain outcomes. It was noted that a linear law depending on the wavelength was well-suited: ρ(λ) = ρ0 +

λ p

(5.67)


249

where ρ variables indicate the ratio of the energy of a component to the total positive energy of representation and where p is the gradient (in meters). Figure 5.14 shows what is happening to the representation shown in Figure 5.10, when sorting by the energy value is removed. The possibility of automatic evaluation for a component detected by its energy level creates interesting prospects and can make it possible to consider automatic detection of internal wave fields on SAR images. However, the criterion that we used to generate the representation shown in Figure 5.10 is empirical. Even if some theoretical considerations could explain it partly, its validity remains to be verified on other images. This possibility of discriminating the spectral components by an energy criterion is one of the useful characteristics of bilinear time-frequency representations. Parametric methods do not offer this possibility, which makes discrimination difficult, if not impossible.

5.5.2. Analysis of an internal wave field with superimposition In certain tumultuous zones, it happens that internal waves superimpose themselves. Figure 5.15 shows such a zone. The size of the image is 37,275 kilometers wide and 247,625 kilometers high. The scale of the image is 1 centimeter for every 2,485 kilometers. The analysis method is the same as that used to provide the image result of Figure 5.10. The difference is that a second energy component consistently lower than the first was shown (in gray) in order to delineate a second wave system. The quality of the results could be improved, at least, for the calculation of analytic image, since a same division into two half-planes was carried out for the whole image. Here, the division was carried out on the axis ωx = 0, which means that the components oriented by height were cut into two. Figure 5.16 shows the result for the same analysis method when the division line becomes ωy = 0. The components oriented by the height are no longer cut and therefore are analyzed better. On the other hand, the components oriented by the width are cut and are likely to be misconstrued (their energy can already be divided by two because of the division).

5.5.3. Analysis of a small area internal wave field Certain internal wave signatures contain only a restricted number of undulations. The internal wave field shown in Figure 5.17 is a typical example. These internal waves superimpose themselves on the swell whose wavelength lies between 250 and 350 meters. As in certain zones of this image, the wavelength of the internal waves

250


can go down rather low, up to 400 meters, thus occasionally misleading the process of detection which confuses the swell with the presence of internal waves.

Figure 5.15. Analysis of internal waves superimposed with pseudo-smoothed Wigner-Ville TFR. The parameters are identical to those used for the representation of Figure 5.10. Only components of greatest energies and satisfactory wavelengths are shown

5.5.4. Prospects The recourse to bilinear time-frequency methods seems promising as for the possibility of developing robust tools for detection, localization and characterization of internal wave fields. The results obtained on SAR images are encouraging. These methods offer interesting prospects for this field and compete with parametric methods [CAR 91, HAF 94]. The advantage of time-frequency methods over parametric autoregressive or autoregressive moving average methods is the ability to characterize the intensity of periodic phenomena. Thus, this authorizes the development of strategies to sort detected components. As evidenced by given examples, this point is crucial because, more than once, a spectral component which could have corresponded to an internal wave corresponded, in fact, only to noise. However, the advantage of such an application lies in a low rate of false alarms. In any event, further research would need to be carried out on this point to arrive at a robust method.


251

The re-allocation method could be used to improve the readability of the spectral divisions. Some tests have already been carried out with the Wigner-Ville TFR in its pseudo and pseudo-smoothed versions [GRA 96]. The gain in net readability is in the sense where the spectral components become finer and more concentrated. However, the benefit for our application does not appear so obvious, for the following reasons: – the calculation of the re-allocation can be made only on a sampled spectral domain for clear calculational reasons; – the influence of noise is not controlled; – the method is very demanding in calculations when it is applied to the TFR of images. The re-allocation method has been intensely studied [AUG 93, AUG 94, AUG 95]. It brings into use some techniques for time-frequency representations, which were formerly introduced to improve the spectrogram [KOD 76, KOD 78]. Figure 5.18 shows how a spectral division of a pseudo-smoothed Wigner-Ville TFR evolves, when time and frequency re-allocation is applied to it.

Figure 5.16. Analysis of internal waves superimposed with pseudo-smoothed Wigner-Ville TFR. The parameters are identical to those used for the representation of Figure 5.15. The spectral division for the calculation of the analytic image is the line ωy = 0. Few changes should be noted

252


Figure 5.17. Analysis of a small area internal wave field with pseudo-smoothed Wigner-Ville TFR. The parameters are identical to those used for the representation of Figure 5.10. The width of the image is 11 kilometers and the length 49.8 kilometers

5.6. Approximation of an internal wave train Here, we show how an internal wave train can be approximated by a bilinear frequency modulation. After having underlined the benefit of such an approximation, the process implemented and the results are successively detailed. 5.6.1. Benefit of approximation of the frequency law A pseudo-smoothed Wigner-Ville representation, in its minimal smoothing version where its smoothing window is defined by (5.66), has the double advantage of precisely estimating bilinear frequency modulations, like the pseudo-version of the same distribution, and of not generating the double spectral repetition. It is, however, necessary in order for the estimate of this modulation to be correct that the modulation is sufficiently weak so that the constraint (5.33) is verified. The purpose of the experiment which follows is to show how an internal wave field can be approximated by an analytic image, whose instantaneous frequency component follows a bilinear modulation. The benefit of the experiment is that it is possible to approximate this internal wave field, particularly curved, using a modulated bilinear frequency field. It would be reasonable to think that several other internal wave fields could also be thus modeled. However, we have a robust tool to calculate the instantaneous frequency of bilinear modulated fields: a pseudo-smoothed Wigner-Ville representation, with minimum smoothing, from which comes the benefit of such an approximation. It remains to be verified that the bilinear modulation is sufficiently smooth so that the constraint (5.33) linked to the positivity is verified. 5.6.2. Problem resolution The modulation parameters must be estimated from the image. Let us suppose that we are attempting to approximate the frequency law of the zone surrounded by the rectangle in Figure 5.19. The instantaneous frequencies must be estimated at the four corners of the rectangle, as well as its center, which will thus be used as the origin for


253

Figure 5.18. Re-allocation of a spectral division. At the top: spectral division calculated with pseudo-smoothed Wigner-Ville TFR. The smoothing and weight functions are Gaussian. The analyzed square has dimensions of 31 × 31 and is located in the bottom left corner of the image of Figure 5.10. At the bottom: spectral division re-allocated in time and frequency. The group delay was limited to three samples in each dimension to limit the volume of calculations related to re-allocation in position. As could be expected, the spectral components are much more concentrated. On the other hand, the calculational cost is particularly high

the coordinate system. The set of all these estimated values makes it possible to fix the frequency law at a bilinear modulated frequency which approximates the real field, as well as possible, within the meaning of a criterion that we will specify. For this, we take a simplified model for the image: f (θ) = exp jφ(θ)

(5.68)

254


where the phase is defined by: 1 φ(θ) = α + ω t0 θ + θ t Λθ 2

(5.69)

with θ = (x, y)t , ω 0 = (ω0x , ω0y )t and: Λ=

ω11 ω21

ω12 ω22

(5.70)

The expressions developed from the phase: 1 1 1 φ(θ) = α + ω0x x + ω0y y + ω11 x2 + (ω12 + ω21 )xy + ω22 y 2 2 2 2

(5.71)

and from the instantaneous frequency: 1 1 ωinst (x, y) = (ω0x + ω11 x + (ω12 + ω21 )y, ω0y + ω22 y + (ω12 + ω21 )x)t (5.72) 2 2 show that modulation indices ω12 and ω21 occur only by their mean and thus the modeling is not changed if the bilinear modulation matrix is replaced by the matrix:

Figure 5.19. Left: image of the internal wave field which we seek to estimate the instantaneous frequency law. Right: image with the section of the field approximated by a bilinear modulation law


Λ=

ω11 ω

ω ω22

255

(5.73)

with: ω = 21 (ω12 + ω21 ). Our goal is to approximate the instantaneous frequency, as far as possible, on the entire surface of the rectangle, within the meaning of a criterion which remains to be fixed. Since the coordinate system is centered in the middle of the rectangle, frequency vector ω 0 will be fixed by the estimate of the instantaneous frequency in this mid-point. The coefficients of the modulation matrix will be fixed so as to approximate instantaneous frequencies at the four corners of the zone to be modeled. It is indeed impossible, with some exceptions, to bilinearly interpolate a field of vectors by imposing its values on the four corners. The system to be solved is, indeed, oversized: ⎛

xB ⎜0 ⎜ ⎜xB ⎜ ⎜0 ⎜ ⎝xD 0

yB xB yD xB yD xD

⎞ ⎛ ⎞ ωBx − ωAx 0 ⎛ ⎞ ⎜ ωBy − ωAy ⎟ yB ⎟ ⎟ ⎜ ⎟ ω11 ⎟ ⎜ 0⎟ ⎟ × ⎝ ω ⎠ = ⎜ωCx − ωAx ⎟ ⎜ ωCy − ωAy ⎟ yD ⎟ ⎟ ⎜ ⎟ ω22 ⎝ωDx − ωAx ⎠ 0⎠ ωDy − ωAy yD

(5.74)

This system is solved by minimizing the mean quadratic error. The three parameters obtained make it possible to build the bilinear modulation matrix. This matrix is then used to create a synthetic image approximating the real field. This field was included in the original image (Figure 5.19, on the right) to assess the approximation and the result seems to be visually correct. We can, however, note a drift in the top left corner, where the curvature of the field is insufficient. This deviation reflects the fact that the bilinear approximation is not sufficient and that it would at least be necessary to add a quadratic component to the modulation in order to better approximate the real field. However, no time-frequency method of Cohen’s class can estimate such a modulation correctly; it is for this reason that we do not seek to approximate the real field by a quadratic frequency modulation. Figure 5.20 represents the frequency components at the four corners of the rectangle for the two fields. The visual differences between the real field and the synthetic field are quite small in wavelength as well as direction.

5.6.3. Adequacy of bilinear modulation with instantaneous frequency estimation 5.6.3.1. Overview of the benefit of this adequacy The fact that a large number of internal wave fields have a frequency law, which could be approximated by a bilinear frequency modulation, is interesting only if the modulations in question are sufficiently small for positivity constraint (5.33)

256


to be verified. It is indeed the condition for which the instantaneous frequency could be estimated by maximum detection on the pseudo-smoothed Wigner-Ville representation, with minimum smoothing. If this condition was verified for a large number of fields, it would mean that this time-frequency representation would be able to precisely calculate the instantaneous frequency on all these fields. 5.6.3.2. Verification of the adequacy hypothesis The adequacy hypothesis is valid when positivity constraint (5.33) is verifiable for all the delays values included in the kernel support on the pseudo-smoothed WignerVille representation, with minimum smoothing. Let us recall that this representation is always compatible with constraint (5.33) related to symmetry independently of the weight function of the delay selected. It is for this reason that we are looking only at the positivity constraint here. The matrix coefficients of bilinear frequency modulation are, generally, low (in absolute value). Thus, for the approximate internal wave field, which is shown in Figure 5.20, the matrix is equal to:

1.37943E−3 −5.50492E−4 (5.75) −5.50492E−4 −7.54109E−5

Figure 5.20. Left: image of original internal wave fields with estimated frequency components. Right: image with the section of synthetic field containing frequency the synthetic field components at the four corners


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In the specific case of the kernel of the pseudo-smoothed Wigner-Ville representation, with minimum smoothing, the integral sum of positivity constraint: ∀(τx , τy ) ∈ R , 2

ΦTR (x, y, τx , τy ) cos (x, y)t Λ+ (τx , τy ) dx dy 0

(5.76)

becomes a simple sum, since the kernel is defined by Dirac impulses and its support is of finite duration: & % & % ∀(kx , ky ) ∈ −kxM , . . . , kxM × −kyM , . . . , kyM ΦTR (x, y, kx Tx , ky Ty ) cos (x, y)t Λ+ (kx Tx , ky Ty ) 0 x

(5.77)

y

In practice, this calculation is very simple to carry out. Indeed, variables x and y take only three values each: − 12 T, 0 and 12 T where T is the sampling period suitable to the dimension concerned. If in a dimension the delay is even, for example kx is even, then the sum on this dimension is to be carried out only for one value, in this case x = 0. If, on the other hand, the delay is odd, the sum is to be carried out on two values, i.e. x = − 12 T and x = 12 T for this example. It is verified in experiments that the expression is always positive for various sizes of windows (31, 63 or 127 samples). This result is not surprising and could be expected, given the low values making up the modulation matrix and the short duration of the kernel in the plane of two coordinates (x, y). The phase of the cosine remains close to zero, which gives it a value close to 1 and finally makes the expression always positive, since the kernel itself has positive values. The range of this satisfactory result increases the stronger the frequency modulation becomes, in spite of the low value of the coefficients. This leads to an expectation that the result would be equally good in many other cases, since the modulation of the internal wave fields analyzed here was particularly strong.

5.7. Bibliography [AUG 93] AUGER F., “Comparaison de la concentration et de la résolution de quelques représentations temps-fréquence et de leurs versions modifiées par la technique de réallocation”, 14th GRETSI Conference, Juan-les-Pins, p. 347–350, September 1993.

258


[AUG 94] AUGER F., F LANDRIN P., “La réallocation : une méthode générale d’amélioration de la lisibilité des représentations temps-fréquence bilinéaires”, Temps-Fréquence, Ondelettes et multirésolution: théorie, modéles et applications, Insa-Lyon, GDR Images Signal, p. 15.1–15.7, March 1994. [AUG 95] AUGER F., F LANDRIN P., “Improving the Readability of Time-Frequency and Time-Scale Representations by the Reassignment Method”, IEEE Transactions on Signal Processing, vol. 43, no. 5, p. 1068–1089, May 1995. [BAR 93a] BARANIUK R., J ONES D., “Signal-dependent time–frequency analysis using a radially Gaussian kernel”, Signal Processing, vol. 32, no. 3, p. 263–284, 1993. [BAR 93b] BARANIUK R., J ONES D., “A signal-dependent time-frequency representation: optimal kernel design”, IEEE Transactions on Signal Processing, vol. 41, no. 4, p. 1589– 1602, April 1993. [CAR 91] C ARIOU C., Analyse spectrale d’images numériques – Application à la télédétection spatiale des océans, PhD Thesis, University of Western Brittany, 1991. [FLA 93] F LANDRIN P., Temps-fréquence, Hermes, coll. Traitement du signal, Paris, 1993. [GRA 96] G RASSIN S., G ARELLO R., “Spectral analysis of the swell using the reassigned Wigner-Ville representation”, Proc. of the IEEE Conf. Oceans’96, Fort Lauderdale, FL, p. 1539–1544, September 1996. [GRA 97] G RASSIN S., Analyse temps-fréquence des signaux discrets – Application aux images, PhD Thesis, University of Rennes, 1997. [HAF 94] H AFEMEISTER L., Détection, estimation de phénoménes d’onde en analyse d’image – Application aux surfaces marines, PhD Thesis, University Paris XI Orsay, 1994. [HLA 01] H LAWATSCH F., AUGER F., J.-P. OVARLEZ (ed.), Temps-Fréquence: Concepts et Outils, Hermes, IC2 Series, Paris, 2001 [KOD 76] KODERA K., V ILLEDARY C. D., G ENDRIN R., “A new method for the numerical analysis of non-stationary signals”, Physics of the Earth and Planetary Interiors, no. 12, p. 142–150, 1976. [KOD 78] KODERA K., G ENDRIN R., V ILLEDARY C. D., “Analysis of time-varying signals with small BT values”, IEEE Transactions on Acoustics, Speech and Signal Processing, no. 1, p. 64–76, February 1978. [PEY 86] P EYRIN F., Z HU Y. M., G OUTTE R., “Extension of the notion of analytic signal for multidimensional signals. Application to images”, in Signal Processing III: Theories and Applications, I. T. Young (ed.), North Holland, Amsterdam, p. 677–680, 1986. [ZHU 89] Z HU Y. M., P EYRIN F., G OUTTE R., “Equivalence between the two-dimensional real and analytic signal Wigner distributions”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 37, p. 1631–1634, October 1989. [ZHU 90] Z HU Y. M., P EYRIN F., G OUTTE R., “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-D real signals”, Signal Processing, vol. 19, p. 205–220, March 1990.

Chapter 6

2-D Wavelet Representation

6.1. Introduction In this chapter, we present a new tool which enables the study of signals or processes with non-stationary behavior: wavelet decomposition. This forms a representation which accounts for the evolution of frequency content of the signal over the course of time. For this reason, it falls under the class of the time-frequency analyses presented in the preceding chapter. However, the mode of calculation is different from Wigner-Ville representation. Though less precise, this analysis achieves an orthogonal representation with a synthesis algorithm. The coefficients of this wavelet representation are obtained by comparing (by means of scalar products) the signal to be analyzed (for the moment, we introduce the 1-D version) with a family of functions ψa,μ . These functions arise from an elementary pattern called mother wavelet ψ, by the action of two operators, one of which is translation in & μ and the other dilation in scale a. The continuous wavelet transform, noted by % time d◦a,μ , of a function x(t) is defined by: d◦a,μ =

1 x(u) √ ψ ∗ a

u−μ a

du

´ Noël RICHARD and Christine FERNANDEZ. Chapter written by Philippe CARRE,

260


The wavelet transform corresponds to a smooth Wigner-Ville transform. This transformation can be inversed if the wavelet verifies the following admissibility condition: #2 # ##ψ(f 8 )## df < +∞ =⇒ ψ(t)dt = 0 (6.1) f In order to represent information, the transform such as that defined above uses many more coefficients than necessary and such a representation is redundant. One of the major advantages in using a redundant transform is to enable a very fine analysis of the development of the signal behavior. Its major drawbacks are high costs of calculation and a very large number of coefficients. During the analysis of a 2-D signal the decomposition is very expensive because of the representation space required for 4 dimensions and its calculation is very tricky: examples of using continuous 2-D decomposition in other works remain anecdotal. This is why we will concentrate on discrete wavelet bases which can be defined in a much simpler way in 2-D. 6.2. Dyadic wavelet transform: from 1-D to 2-D 6.2.1. Multiresolution analysis Direct construction of orthonormal discrete wavelet bases is based on the theory of multiresolution analysis (MRA). This analysis allows the successive study of smooth approximations of a signal in which the details are gradually removed. The “high-pass” components are collected in each iteration of the analysis. We present the simpler 1-D version here. As we will see later, the 2-D version can be obtained by a simple extension. MRA is an increasing sequence of closed sub-spaces {Vs }s∈Z of L2 () (function space with real values of integrable square) which have the following properties [MEY 90]: ⎧ +∞ +∞ 2 ⎪ ⎪ ∩s=−∞ Vs = {0} , ∪s=−∞2Vs is dense in L () and Vs+1 ⊂ Vs −s ⎪ ⎪ ⎨ For any function x (t) of L () and all s ∈ Z, x (t) ∈ V0 ⇔ x(2 t) ∈ Vs For any function x (t) ∈ V0 and t μ ∈ Z, x (t − μ) ∈ V0 ⎪ s ⎪ There is a function ζ (t) = ζ ⎪ s 2s ∈ Vs such that {ζs (t − 2 μ)}μ∈Z ⎪ ⎩ is a base of Vs Then we have a sequence of sub-spaces {Vs }s∈Z such that: ... ⊂ V1 ⊂ V0 ⊂ V−1 ⊂ ... As and when the scale index s increases, meshing becomes coarse and projections of x in spaces Vs contain less and less information relating to x.


261

According to the MRA definition, function ζ (t/2) ∈ V1 ⊂ V0 can be expressed by a linear combination of base vectors of V0 , i.e. {ζ0 (t − k)}k∈Z . Then, we have the following biscale relation: 1 √ ζ (t/2) = h(k)ζ0 (t − k) 2 k∈Z

(6.2)

We see the emergence of the concept of discrete filter h. If we take the Fourier transform of the preceding expression, the ζ function is a low-pass function (and thus h is a low-pass filter) [MAL 89]: ζ8 (0) = 0 and

h(k) = 0

k∈Z

The information (details) lost between two successive approximations can be obtained by projecting the signal in a set of spaces {Ws }s∈Z , called wavelet spaces. Ws is a complementary space of Vs in Vs−1: Vs ⊕ Ws = Vs−1 MRA shows that it is possible to create, from scaling function ζ (t), a wavelet function ψ(t) such that the following: /

0 ψs,k (t) = 2−s/2 ψ 2−s t − k

k∈Z

constitutes an unconditional base of Ws [MEY 90]. As W1 is also contained in V0 , ψ (t/2) function can be expressed by a linear combination of base vectors of V0: 1 √ ψ (t/2) = g(k)ζ0 (t − k) 2 k∈Z

(6.3)

Here, the second discrete filter g, present in the discrete transform, appears. By taking the Fourier transform of the preceding expression and according to admissibility condition (6.1), we discover that function ψ is a high-pass function (and thus g a high-pass filter): ψ8 (0) =

k∈Z

g(k) = 0

262


6.2.2. Wavelets and filter banks We will now present the fast decomposition/synthesis algorithm as proposed by Mallat [MAL 89]. The wavelet transform applies to sampled signals c0,k = x(k) corresponding to the approximation of a function x(t) ∈ L2 (R) by: x=

c0,k ζ0,k

k∈Z

Thus, the original signal is represented in the base of V0 . Applying a wavelet transform in S scales leads to the representation of this signal in a base resulting from the direct sum: VS ⊕ WS ⊕ WS−1 ⊕ ... ⊕ W1 The transform algorithm is iterative and consists of replacing the representation of a component in Vs by a representation in Vs+1 + Ws+1 . Therefore, we can rapidly calculate the approximations and details of a signal in scale s from the approximation in scale s − 1. In fact with the help of equations (6.2) and (6.3) we can show that: cs,k = x, ζs,k = n∈Z h (n − 2k) cs−1,n (6.4) ds,k = x, ψs,k = n∈Z g (n − 2k) cs−1,n with {cs,k }k∈Z the low-pass approximation and {ds,k }k∈Z the wavelet coefficients in scale s. If we note by ↓2 the decimation operation of a factor 2, which consists of eliminating one coefficient out of two of a sequence, we can write equation (6.4) in the form: cs =↓2 (h− ⊗ cs−1 ) ds =↓2 (g − ⊗ cs−1 ) with x− reversal such that x− (k) = x(−k). We note that the calculation of the low-pass approximation and the wavelet coefficients in scale s is summarized in the convolution (filtering) of coefficients of low-pass approximation in scale s − 1 followed by a decimation operation of factor 2. By iterating this process from sequence {c0,k }0≤k 0 The calculation of Meyer WP coefficients consists of applying the restriction operator TIis to the signal transform with interval Iis as defined in the preceding two sections. The outcome of projection TIis is identical to the restriction operator of Malvar s , decomposition introduced previously with a division point i2−s , overlapping αi−1 −s s and a division point (i + 1)2 , overlapping αi . We are left with determining the polarities of these folding operators and thus corresponding trigonometric functions. From equation (6.29) and the scaling properties and translation of the Fourier transs : form4, we can write the Fourier transform of function ψi,k s ψ8i,k (f ) = 2−s/2 wIis (f )

cos(2π(k + 12 )2s f − i2−s−1 ) −i sin(2π(k + 21 )2s f − i2−s−1 )

= usi,k (f ) − iesi,k (f )

4. F T [ψ(at)] =

1 ψ(f /a) a

)e−2jπkf . and F T [ψ(t − k)] = ψ(f

(6.32)

300


s We observe that the contributions of the real and imaginary parts of ψ8i,k (f ) respectively are odd/even and even/odd in extreme points of the interval Iis . In s order to obey these contribution symmetries of ψ8i,k , we infer polarities from it, which must be used for the operator TIis during the restriction of real and imaginary parts.

These various symmetries also state that the Meyer WP can be represented in the frequency field by using local bases of polarities (−, +) for the real part, and local bases of polarities (+, −) for the imaginary part. Such symmetries correspond respectively to the trigonometric transforms of DST-III and DCT-III types [KOL 94]. The calculation of discrete WP coefficients dsi,k is written as:

dsi,k

=

3

s x, ψi,k

4

N/2

=

N/2

x1 (l) usi,k

(l) −

l=0

x2 (l) esi,k (l)

(6.33)

l=0

In order to be able to combine the coefficients corresponding to the real and imaginary parts, we deduce two relations from equation (6.33) and from symmetries highlighted at the bounds of intervals:

dsi,k + dsi,−(k+1) 2 dsi,k − dsi,−(k+1) 2

N/2

=

x1 (l) usi,k (l)

l=0 N/2

=

(6.34) x2 (l) esi,k

(l)

l=0

Since the signal is of limited duration (with the edge problems that this poses), a periodic Meyer wavelet decomposition is defined. We use the same principle by substituting dsi,−(k+1) by dsi, Ns −k−1 5 in equation (6.34). 2

5. Knowing that in scale s we have a coefficient vector of length coefficients are periodic, then dsi,−(k+1) = dsi, N −k−1 . 2s

N 2s

and if we consider that the


301

In order to reveal the local trigonometric transforms on the right side of relations, the left part of equation (6.34) is modulated by 2(−1)k+1 . More precisely6: " (−1)k+1 ! s di,k + dsi, Ns −k−1 2 2 A N/2−1 π 2 −s−1 s (l) sin (k + 1/2) w (l − i2 ) = x1 (l) I lgs i lgs

d s,i,k =

(6.35)

l=0

with: lgs = 2−s and where: " (−1)k+1 ! s di,k − dsi, Ns −k−1 2 2 A N/2 π 2 = x2 (l) wIis (l) cos (k + 1/2) (l − i2−s−1 ) lgs lgs

d s,i,k =

(6.36)

l=0

are the corresponding coefficients in projections of real and imaginary positive parts of x 8 on interval Iis . wIis is a support window [N i2−s , N (i + 1)2−s − 1] having an overlapping defined by equations (6.30) and (6.31) based on whether i is even ! −s−1 " or odd −s−1 (and by substituting in these equations 2 6 by the entire part of 2 6 N ). We deduce7 from the preceding equation the operation combining projections of the real and imaginary part which makes it possible to calculate the WP coefficients in scale s for projection i:

dsi,k

⎧ ⎨

! " −s−1 + d −1 (−1)k+1 d s,i,k s,i,k , si k = 0, .., 2 ! " = −s−1 ⎩ (−1)k+1 d −s , .., 2−s − 1 s,i,2 −k−1 − ds,i,2−s −k−1 , si k = 2 (6.37)

6. We use the property (−1)n cos(f ) = cos(f + nπ). and 7. This concept comes from two observations. First, the number of points of d s,i,k −s−1 −s (and not to 2 like wavelet coefficients). This is due to the symds,i,k is equal to 2 metry around 2−s−1 symbolized by 2−s − k − 1. It results in d s,i,k = ds,i,k = 0 when = d = 0 when k = 0...2−s−1 − 1. k = 2−s−1 ...2−s − 1, and d s,i,2−s−1 −k−1 s,i,2−s−1 −k−1 k+1 k+1 (−1) = 1. Moreover, we observe the simple relation (−1)

302


In equations (6.35) and (6.36), we find the formulation which we had introduced while studying Malvar wavelets: the decomposition in Meyer WP corresponds to a 8 (with adapted polarities) restriction on intervals Iis of the real and imaginary parts of x followed by a local trigonometric decomposition.

6.4.3.6. Calculation of wavelet packet coefficients related to ψ80s+1 We are going to be interested in the case of frame (i = 0). The function ψ80s+1 is defined in each scale by: hs0 ψ80s = ζ8s+1 ψ80s+1 = 8 # # #8 # Let us suppose that w BIe0 (f ) = #ζ(f )#; if we reconsider the terminology of Malvar wavelet decomposition, function w B e (.) is a window defined in interval IB0 = − 1 , 1 . I0

According to equation (6.22), we have a of the interval.

1 6

overlapping on the left and

1 6

2 2

on the right

More generally, the scaling function in position k and in scale s is such that BIes (.) = w ζs,k (x) = 2s/2 ζ(2s x − k) corresponds to a window w BIes (2s .) defined in −s−1 −s−1 −s −s the interval IBs = −2 with an overlapping of 2 6 on the left and 2 6 on ,2 the right [KOL 94]. As for the calculation of wavelet coefficients, operator TIes of the Malvar decomposition allows the restriction of x 8 in interval IBs with, in division point −2−s−1 , α =

2−s 6 ,

and in division point 2−s−1 , α =

2−s 6 .

By following the same reasoning as done for equation (6.33) (translation and scaling of Fourier transform), we can write the Fourier transform of function ζs,k by:

ζ8s,k (f ) = 2−s/2 w BIes (f ) (cos(2πk2s f ) − j sin(2πk2s f )) =u Bs,k (f ) − iB es,k (f )

In this case, the contributions of real and imaginary parts of ζ8s,k (f ) respectively are even/even and odd/odd in extreme points of the intervals IBs . From this, we deduce that the scaling function can be represented by using local bases of polarities (+, +) for the real part, and local bases of polarities (−, −) for the imaginary part, thus respectively corresponding to DCT-I and DST-I transforms [KOL 94].


303

The calculation of screen coefficients ds0,k in the case of a Meyer WP decomposition is defined by:

N/2

ds0,k

= x, ζs,k =

N/2

x1 (l) u Bs,k (l) −

l=0

x2 (l) eBs,k (l)

(6.38)

l=0

By following the same reasoning as that of wavelet coefficients we place ourselves in the case of discrete signals with limited duration. Then the wavelets are modulated / 02−s+1 −1 in order to obtain local trigonometric transforms. Coefficients d corres,0,k k=0 sponding to the real part are then written as:

d s,0,0 = ds,0,0 , ds,0,2k+1 = 0,

(−1)k √ (ds,0,k + ds,0,2−s −k ) 2 A N/2−1 2 π = w BIes (l) cos k (l + 2−s−1 ) x1 (l) lgs lgs

d s,0,2k =

l=0

02−s+1 −1 / and coefficients d corresponding to the imaginary part are given by: s,0,k k=0

d s,0,0 = ds,0,2k+1 = 0,

(−1)k+1 √ (ds,0,k − ds,0,2−s −k ) 2 A N/2−1 2 π −s−1 = w B e (l) sin k (l + 2 ) x2 (l) lgs Is lgs

d s,0,2k =

l=0

From this we deduce an operation combining coefficients d s,0,k and ds,0,k allowing us to calculate screen coefficients in scale s: ⎧ d ⎪ s,0,0 , if k ⎪ ! "= 0 ⎨ k (−1) √ ds,0,2k − ds,0,2k , if k = 1, .., 2−s−1 (6.39) ds0,k = 2 ! " ⎪ k ⎪ −s−1 −s ⎩ (−1) √ d , if k = 2 −d + 1, .., 2 − 1 −s −s 2

s,0,2

−k

s,0,2

−k

304


6.4.3.7. Algorithm and 1-D reconstruction We have extracted all the “trigonometric” parameters associated with Meyer WP functions and we can then express the Meyer WP decomposition in terms of Malvar decomposition. For a 1-D signal of finite length equal to N , the algorithm of periodic Meyer WP decomposition is presented in the following way. For s varying from 1 to s. For i varying from 0 to 2s−1 . Folding positive real part of x 8 around the point (2i + 1) 2Ns with positive polarity on the !left and"negative on the right and an overlapping zone equal to the −s . integer part of N 26 Folding positive imaginary part of x 8 around the point (2i + 1) 2Ns with negative polarity on the left and positive on the right and an overlapping zone equal to ! −s " N2 . the integer point 6 Calculation of local trigonometric transforms: d s,0,k = DCT I(TIes x1 ) and

d s,0,k = DST I(TIes x2 ).

0

0

Combination of coefficients d s,0,k and ds,0,k following relation (6.39) in order s to get d0,k .

For i varying from 1 to 2s . s Calculation of local trigonometric transforms: d s,i,k = DST III(TIi x1 ) and = DCT III(TIis x2 ).

d s,i,k

Combination of coefficients d s,i,k and ds,i,k according to relation (6.37) in s order to get di,k .

The algorithm described above is summarized in Figure 6.22. In order to be able to define the reconstruction algorithm in a simple way, it is s+1 s+1 ⊕ W2i+1 . The proof is necessary that Meyer wavelet packets verify Wis = W2i s s immediate if it is considered that filters hi and gi are quadrature mirrors, but the properties of the restriction operators can also constitute a different manner of showing it in the case of Meyer wavelet packets. The study of this second approach enables us to highlight the reconstruction algorithm.


305

Figure 6.22. Decomposition algorithm in Meyer wavelet packets. We indicate the points discussed regarding folding in each scale (denoted by the symbol U ), as well as the size of the associated overlapping zone. The indicated trigonometric transforms correspond to the treatment of the real part of x

Projections TIs x1 , TIs x2 , TIes x1 and TIes x2 can be reconstructed from wavelet coefficients by the reverse trigonometric transform (after separation of the wavelet coefficients between the real and imaginary parts):

TIis x1 (l) =

⎧7 ⎨ lg2

s

⎩ d√s,0,0 +

2s−1 −1 k=0

7

lgs

⎧7 ⎨ TIis x2 (l) =

⎩

2 lgs

π s−1 d ) if i > 0 s,i,k sin (k + 1/2) lgs (l − i2 2s+1 −1 π s−1 c cos k (l + 2 ) if i = 0 s,0,k k=1 lg s

2s−1 −1 ds,i,k cos 7 k=0 s+1 2 −1 2 cs,0,k k=1 lgs

2 lgs

(k + 1/2) lgπ (l − i2s−1 ) if i > 0 s (6.40) π sin k lg (l + 2s−1 ) if i = 0 s

Then, we calculate terms TI∗s TIis x1 (l) and TI∗s TIis x2 (l) (corresponding to a i i deployment of the function around division points) with the help of adjunct folding operator U ∗ introduced in the context of Malvar wavelets. Since projection is orthogonal, i.e., TI∗s TIis TI∗s TI s = 0 for any s = s × i = i and the adjacent intervals have i

i

i

opposite polarities, we can write that: TI∗is TIis xn (l) = TI∗s+1 TI s+1 xn (l) ⊕ TI∗s+1 TI s+1 xn (l) with n = 1, 2 2i

2i

2i+1

2i+1

We thus have the relation allowing the reconstruction of space in scale s from the superposition of two children spaces in a scale s + 1.

306


6.4.4. Extension of Meyer wavelet packets to 2-D For 2-D, a transform-synthesis algorithm in Meyer wavelets was created [KOL 94]. It is based on the same principle as the extensions to the 2-D field of the bases associated with time support filters, such as the Daubechies base. In L2 (2 ) the orthonormal Meyer base is made up of all the translations and dyadic dilatations of three base functions ζ(x)ψ(y), ζ(y)ψ(x) and ψ(x)ψ(y), where ζ and ψ are mother and father functions of the 1-D Meyer decomposition. Because of the separable character of this base and nature of the decomposition algorithm which applies to a vector, it is easy to create a 2-D algorithm from calculation methods explained in 1-D cases. Since the 2-D WP base is the separable product of 1-D Meyer WP functions, we use the 1-D algorithm in each line and each column of the image. The decomposition method in 2-D Meyer wavelet packets is presented as follows (if we denote by 1-D fwt-packet-meyer the decomposition of a Meyer WP vector) [CAR 99]. For s varying from 1 to s, Res-inter ←− application of 1-D fwt-packet-meyer in the scale s on each line of the source image. W s ←− application of 1-D fwt-packet-meyer in the scale s on each column of Res-inter. After the selection of the best representation base by the Best-Basis algorithm, a 2-D reconstruction method can be applied using a similar principle to that of decomposition, i.e., depending on the method of 1-D reconstruction used according to the lines and columns. In order to illustrate the decomposition in 2-D Meyer WP, we use the test image made up of five 2-D FT atoms (Figure 6.5). As we saw, the direct calculation of wavelet coefficients in the frequency plane enables us to use filters with finite frequency support. This results in a better frequency localization, and thus a better separation of the various bands. In order to illustrate this property, we break up the test image on a scale (the following scales have similar results) with the Daubechies WP and the Meyer WP. The result is indicated in Figure 6.23. For Daubechies wavelets we note that the atoms are in frequency bands which do not correspond. For example, the depth of the image which has frequency coordinates equal to (0.3 ; 0.3) is distributed in all the bands, because its frequency position is close to the division points. In the same way atom 3 of coordinates (0.35, 0.05) has


307

Figure 6.23. Decomposition in a scale of a test image with five 2-D FT atoms: a) Daubechies wavelets; b) Meyer wavelets

its energy dispersed in the bloc [0, 0.25] × [0.25, 0.5] but also [0, 0.25] × [0, 0.25]. In the Meyer WP decomposition each atom is entirely localized in the corresponding frequency block. There are only some spurious points at the borders of various atoms due to discontinuities created by the change of model. If we select the best base according to an entropic criterion with the 2-D BestBasis algorithm, an optimal 2-D FT representation can be obtained. In Figure 6.24 we present two representations extracted from Daubechies and Meyer WP. Once again we notice the dispersion of energy of atoms between several frequency bands in the case of Daubechies wavelets. On the other hand, the representation associated with Meyer wavelets extracts more precisely the various atoms present in the signal (symbolized by circles) with a concentration of energy in the frequency theoretical position of the atoms. This example shows the superiority of 2-D Meyer WP for analysis or separation problems of the components of an image. Let us finally note that because of the energy dispersion, the frequency segmentation associated with Daubechies wavelets is more significant. There is a phenomenon of supersegmentation, i.e., a selection of very fine windows which, knowing the atoms present in the image, do not bring any information. The frequency segmentation selected with the Meyer decomposition is less significant and more adapted. We use the properties of excellent frequency separation of Meyer WP to detect various directions of the swell present in a SPOT image. For this we decompose the SPOT image in two WP scales. Then we select, for each spatial coordinate, the frequency band containing maximum coefficient for this spatial position. We indicate the result of this process in Figure 6.25a. This frequency maximum thus indicates to us the direction already observed for each point of the 2-D signal. For each frequency band, we indicate the maximum energy zones by a white contour. Figure 6.25b illustrates the fact that a simple treatment associated with WP with Meyer enables us to correctly detect the main directions of the swell.

308


Figure 6.24. Projection of a test image with five 2-D FT atoms in four scales of wavelet packets; selection of the best base a) Daubechies wavelets; b) Meyer wavelets

6.5. Conclusion This chapter presented some space-frequency-scale methods in a non-exhaustive way. All the techniques developed here have a common point; they seek to translate information present in the analyzed 2-D signal into a new space of representation: – wavelet packets make a selective and adaptive 2-D filtering; – Malvar wavelets seek to spatially divide the 2-D signal into zones having quasistationary frequency behavior; – Meyer wavelets are similar to technical wavelet packets but seek to directly divide the quasi-stationary 2-D signal in zones with spatial behavior in the frequency plane. Generally, all these techniques result from 1-D treatment with a generally marked specialization within 2-D framework. However, the latter problem still remains unresolved. The space-frequency-scale methods are similar to FT representations seen in Chapter 5, and are not directly applicable for a given problem, but, as we noted in different examples, they simply serve as a pre-treatment preceding the analysis and if possible improve it. In this chapter, some other multi-scale decompositions like the lifting scheme and multi-wavelets were not covered. For these particular bases, see [STR 96, SWE 96, SWE 97].

309

Time (frequency)


Time (frequency)

Figure 6.25. Detection of main directions of a swell on a SPOT image by Meyer WP decomposition on two scales: a) for each frequency band, the white zones indicate the zones of maximum energy (at the bottom, the original image is represented in the form of a thumbnail image in order to spatially locate various major directions); b) major deduced directions

6.6. Bibliography [AUS 92] AUSCHER P., W EISS G., W ICKERHAUSER V., “Local Sine and Cosine Bases of Coifman and Meyer and the Construction of Smooth Wavelets”, Wavelets: A Tutorial in Theory and Applications, C. Chui Ed., Academic Press, San Diego, p. 237–256, 1992. [BIJ 94] B IJAOUI A., S TARCK J., M URTAGH F., “Restauration des images multi-echelles par l’algorithme a trous”, Traitement du Signal, vol. 11, p. 232–243, 1994. [CAR 99] C ARR E´ P., F ERNANDEZ C., “Paquets d’ondelettes de Meyer pour le traitement d’images”, 17th GRESTI colloquium, Vannes, France, 1999. [CAR 00] C ARR E´ P., Méthodes numériques temps-échelles et temps-fréquence pour le traitement du signal et des images, PhD Thesis, University of Poitiers, January 2000. [COH 92] C OHEN A., Ondelettes et Traitement Numérique du Signal, Masson, 1992. [COH 93] C OHEN A., DAUBECHIES I., “Non-separable bidimensional wavelet bases”, Rev. Math. Iberoamer, vol. 9, p. 51–137, 1993. [COI 91] C OIFMAN R., M EYER Y., “Remarque sur l’analyse de Fourier a` fenêtres”, Comptes Rendus de l’Académie des Sciences, vol. 312, p. 259–261, 1991. [COI 92] C OIFMAN R., W ICKERHAUSER M., “Entropy-based Algorithms for Best Basis Selection”, IEEE Transactions on Information Theory, vol. 38, no. 2, p. 719–746, 1992. [DAU 92] DAUBECHIES I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

310


[EST 77] E STEBAN D., G ALAND C., “Application of quadrature mirror filters to split-band voice coding schemes”, IEEE Transactions on Acoustics, Speech and Signal Processing, 1977. [FEA 90] F EAUVEAU J., “Analyse multirésolution pour les images avec un facteur de résolution racine de 2”, Traitement du Signal, vol. 7, no. 2, p. 117–128, 1990. [FRE 91] F REEMAN W., A DELSON E., “The Design and Use of Steerable Filters”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 9, p. 891–906, September 1991. [HOL 89] H OLSCHNEIDER M., K RONLAND -M ARTINET R., M ORLET J., T CHAMITCHIAN P., “A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform”, in Wavelet, Time-Frequency Methods and Phase Space, J.M. Combes, A. Grossmann, P. Tchamitchian Eds., Springer-Verlag, Berlin, p. 289–297, 1989. [KOL 94] KOLACZYK E., “Wavelet methods for the inversion of certain homogeneous linear operators in the presence of noisy data”, PhD thesis, Stanford University, October 1994. ˇ ´ J., V ETTERLI M., “Non-separable multidimensional perfect reconC [KOV 92] KOVA CEVI stuction filter banks and wavelet bases for Rn”, IEEE Transactions on Information Theory, vol. 38, p. 533–555, March 1992. ˇ ´ J., V ETTERLI M., “Non-separable two- and three-dimensional C [KOV 95] KOVA CEVI wavelets”, IEEE Transactions on Signal Processing, vol. 43, p. 1269–1273, May 1995. ˇ ´ J., “Local cosine bases in two dimensions, IEEE Transactions on C [KOV 97a] KOVA CEVI Image Processing”, vol. 6, no. 11, p. 1580–1583, 1997. ˇ ´ J., S WELDENS W., “Wavelet Families of Increasing Order in ArbiC [KOV 97b] KOVA CEVI trary Dimensions”, report, Bell Laboratories, Lucent Technologies, 1997. [MAL 89] M ALLAT S., “A theory for multiresolution signal decomposition: the Wavelet Transform”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 7, p. 674–693, 1989. [MAL 92] M ALLAT S., Z HONG S., “Characterization of Signals from Multiscale Edges”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 7, p. 710– 732, July 1992. [MAL 98] M ALLAT S., A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998. [MAL 90] M ALVAR H., “Lapped Transform for Efficient Transform/Subband Coding”, IEEE Transactions on Signal Processing, vol. 38, no. 6, p. 969–978, June 1990. [MEY 90] M EYER Y., Ondelettes et opérateurs, vol. I, Hermann, Paris, 1990. [MEY 94] M EYER Y., Les ondelettes; algorithmes et applications, Armand Colin, Paris, 1994. [NAS 95] NASON G., S ILVERMAN B., “The Stationary Wavelet Transform and some Statistical Applications”, Wavelets and Statistics, Anestis Antoniadis and Georges Oppenheim Eds., Springer-Verlag, LNS 103, New York, p. 281–299, 1995.


311

[PES 96] P ESQUET J., K RIM H., C ARFANTAN H., “Time-Invariant Orthonormal Wavelet Representation”, IEEE Transactions on Signal Processing, vol. 44, no. 8, p. 1964–1970, August 1996. [RAO 90] R AO K., Y IP P., Discrete Cosine Transform, Academic Press, New York, 1990. [SHE 92] S HENSA M., “Wedding the a` trous and Mallat algorithms”, IEEE Transactions on Signal Processing, vol. 40, no. 10, p. 2464–2482, 1992. [SIM 90] S IMONCELLI E., A DELSON E., “Subband Transforms”, J. Woods Ed., Kluwer Academic Press, p. 143–192, 1990.

in Subband Coding,

[STA 94] S TARCK J., M URTAGH F., “Image Restoration with Noise Suppression Using the Wavelet Transform”, Astronomy and Astrophysics, 1994. [STR 96] S TRELA V., “Multiwavelets: Theory and applications”, PhD thesis, Massachusetts Institute of Technology, 1996. [SWE 96] S WELDENS W., “The Lifting Sheme: A Custom-design Construction of Biorthogonal Wavelets”, Applied Comput. Harmon. Anal., vol. 3, no. 2, p. 186–200, 1996. [SWE 97] S WELDENS W., “The Lifting Sheme: A Construction of Second Generation Wavelets”, SIAM J. Math. Anal., vol. 29, no. 2, p. 511–546, 1997. [VET 84] V ETTERLI M., “Multidimensional subband coding: some theory and algorithms”, Signal Processing, vol. 6, p. 97–112, 1984. ˇ ´ J., Wavelets and Subband Coding, Prentice Hall, C [VET 95] V ETTERLI M., KOVA CEVI New Jersey, 1995. [WIC 94] W ICKERHAUSER M., Adapted Wavelet Analysis from Theory to Software, A. K. Peters, Massachusetts, 1994.


List of Authors

Olivier ALATA IRCOM-SIC University of Poitiers Futuroscope Chasseneuil France Claude CARIOU LASTI ENSSAT, Lannion University of Rennes 1 France Philippe CARRÉ IRCOM-SIC University of Poitiers Futuroscope Chasseneuil France Christine FERNANDEZ IRCOM-SIC University of Poitiers Futuroscope Chasseneuil France René GARELLO IT – TELECOM Bretagne Brest France

314


Stéphane GRASSIN Thales Systèmes Aéroportés Elancourt France Jean-Marc LE CAILLEC IT – TELECOM Bretagne Brest France Noël RICHARD IRCOM-SIC University of Poitiers Futuroscope Chasseneuil France Stéphanie ROUQUETTE Motorola Gif-sur-Yvette France

Index

1–9

A

2-D bispectral estimation direct method 195, 199, 204 indirect method 196, 197, 199 parametric 195, 196 2-D causality 37, 234 full plane 38 non-symmetric half plane 38, 67 quarter plane 20, 38 symmetric half plane 38 2-D convolution, product of 33 2-D ergodism 26 2-D spectral estimation autoregressive 117, 122 autoregressive moving average 117, 122 bias 116, 120 classical 118 correlogram 119, 143 Gauss-Markov 117, 122 maximum entropy 117, 122, 129 maximum probability 127, 133 minimum vairance 117, 122 modern 115 non-separability zone 156 parametric 31, 121 periodogram 118, 143 pseudo-spectral 138 variance 116, 119, 120

algorithm Best-Basis 291, 306, 307 Levinson 76, 85, 89, 91, 123 autocorrelation function 24–28, 55–57, 79– 82, 91, 95, 100, 116, 218, 220, 222, 229 estimation 27 biased 120 non-biased 119 first 24, 79 instantaneous 224 second 24, 168 semi-defined positive 25 theoretical 141, 142

C coefficients 1-D reflection 87 direct transverse 86 PARCOR 87 criteria Akaike 78, 105 entropy 78 information 78 Kullback-Leibler 78, 79 least squares 81, 94 maximum a posteriori 75 maximum posterior mode 75

316


maximum probability 78 recursive least squares 95 cylindrical connectivity 93 fields Gibbs 74 L-Markovian 74, 82, 129 Markov 73, 101

F filter 31 analysis 80, 88 finite impulse response 39 frequency response 40 impulse response 32 infinite impulse response 39 linear time-invariant 33, 66 minimal phase 54 non-minimal phase 54 quadrature mirror 272 synthesis 89

H, I Hamersley-Clifford theorem 76 harmonic component 39, 116, 141 Hilbert space 23, 59, 89 Iterative Conditional Mode 75, 102

M, N model autoregressive 65, 66, 122 autoregressive 2-D causal 77 autoregressive moving average 65, 66, 126 Gauss-Markov 65, 74, 128 moving average 65, 66 Volterra 2-D 194 multinomial distribution 104 Multiple Signal Classification (MUSIC) 118, 136, 161 noise 142 colored 128 sub-space 136–139

white 28, 123, 168 normalized frequency 39, 141

P polynomials 2-D orthogonal 91 scalar product 87 Szegö orthogonal 87 power spectral density 55, 65, 66, 116, 121, 130, 135, 141, 155, 171 AR 123, 124 AR-HM 125 ARMA 127 ARMA-HM 128 irregular 122 maximum entropy 116, 117, 121, 128, 138, 169 maximum probability 76, 104, 105, 124, 126, 128, 132, 153, 161 minimum phase factorization 67, 83 minimum variance 171 parametric 121, 169 regular 122, 162, 165, 166, 171, 172 simulations 143, 162 theoretical 116, 117, 120, 140, 142, 161, 162, 165, 171 prediction 1-D support 86 2-D support 66, 80 backward errors 86 coefficients 80 direct errors 86 errors 77, 80, 81 vectors 97 probability density 23, 29, 76

S, T stochastic 2-D process 23 synthetic aperture radar 109, 134, 166, 209 time-frequency representations 215, 217, 219, 244, 249, 251 Toeplitz matrix 30, 85

Index transfer function 51 pole 52 rational 52 zero 52 transform continuous wave 267 Fourier 17, 41 Discrete 25, 43–45, 63, 240, 288 Discrete inverse 44 Rapid 118 rapid wave 266 Wigner-Ville 244, 251, 252, 256, 257

317

W, Y wavelets 2-D multiresolution analysis 282 Filter bank 271 Malver 282, 290 multiresolution analysis 268 packets 264 Yule-Walker equations 66, 67, 76, 79, 80, 82, 85, 95, 100, 124, 206

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Discrete-Signal Analysis and Design

Time frequency signal analysis and processing

Evolutionary Image Analysis and Signal Processing

Signal Analysis: Time, Frequency, Scale, and Structure

Wavelets: A Mathematical Tool for Signal Analysis

Introduction to Applied Statistical Signal Analysis

Biomedical Signal Analysis: Contemporary Methods and Applications

Evolutionary image analysis and signal processing

Signal Analysis: Time, Frequency, Scale, and Structure

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