'7 A WAVELET TOUR OF SIGNAL PROCESSING A WAVELET TOUR OF SIGNAL PROCESSING Second Edition Stephane Mallat &cole Polytechnique, Paris Courant Institute, New York University W ACADEMIC PRESS A Harcourt Science and Technology Company San Diego San Francisco New York Boston London Sydney Tokyo This book is printed on acid-free paper. @ Copyright 0 1998,1999, Elsevier (USA) All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing fi-om the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777. Academic Press An imprint of Elsevier 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press 84 Theobald’s Road, London WClX 8RR, UK http ://m .academicpress. corn ISBN: 0-12-466606-X A catalogue record for this book is available from the British Library Produced by HWA Text and Data Management, Tunbridge Wells Printed in the United Kingdom at the University Press, Cambridge PRINTED IN THE UNITED STATES OF AMERICA 03 04 05 06 987654 A mes parents, Alexandre et Francine Contents PREFACE xv PREFACE TO THE SECOND EDITION xx NOTATION xxii INTRODUCTION TO A TRANSIENT WORLD I. I Fourier Kingdom I .2 Time-Frequency Wedding I .2. I I .2.2 Wavelet Transform Bases of Time-Frequency Atoms I .3. I I .3.2 I .4. I Approximation I .4.2 Estimation I .4.3 Compression I .5. I I .5.2 Road Map Windowed Fourier Transform I .3 Wavelet Bases and Filter Banks Tilings of Wavelet Packet and Local Cosine Bases I .4 Bases for What? I .5 Travel Guide Reproducible Computational Science vii 2 2 3 4 6 7 9 11 12 14 16 17 17 18 [...]... “optimal” linear procedures, and fast algorithms are available PREFACE xvii WAVELAB LASTWAVE and Toolboxes Numerical experimentations are necessary to fully understand the algorithms and theorems in this book To avoid the painful programming of standard procedures, nearly all wavelet and time-frequency algorithms are available in the WAVELAB package, programmed in M~TLAB WAVELAB is a freeware software... 577 585 587 Appendix A MATHEMATICAL COMPLEMENTS A I Functions and Integration A 2 Banach and Hilbert Spaces A 3 Bases of Hilbert Spaces A. 4 A. 5 Linear Operators Separable Spaces and Bases 59 1 593 595 596 598 XiV CONTENTS A 6 Random Vectors and Covariance Operators A. 7 Dims 599 60 1 Appendix B SOFTWARE TOOLBOXES 603 609 610 B.1 WAVELAB B.2 LASTWAVE B.3 Freeware Wavelet Toolboxes BIBLIOGRAPHY INDEX 6I2... construction of sparse representations with orthonormalbases, and study applicationsof non-linear diagonal operators in these bases It may start in Chapter 10 with a comparison of linear and non-linear operatorsused to estimate piecewiseregular signals contaminatedby a white noise A quick excursion in Chapter 9 introduces linear and non-linear approximations to explain what is a sparse representation Wavelet. .. Yet, classical signal processing has devoted most of its efforts to the design of time-invariant and space-invariant operators, that modify stationary signal properties This has led to the indisputable hegemony of the Fourier transform, but leaves aside many information-processingapplications The world of transients is considerably larger and more complex than the garden of stationary signals The search... discovery of filter banks and wavelet bases has created a popular new sport of basis hunting Families of orthogonal bases are created every day This game may however become tedious if not motivated by applications 0 0 Sparse representations An orthonormal basis is useful if it defines a representation where signals are well approximated with a few non-zero coefficients Applications to signal estimation... VI I WAVELET BASES 7 I 7.2 7.3 7.4 7.5 ' Orthogonal Wavelet Bases 7 I I Multiresolution Approximations 7 I 2 Scaling Function 7 I 3 Conjugate Mirror Filters 7 I 4 In Which Orthogonal Wavelets Finally Arrive Classes of Wavelet Bases 7.2 I Choosing a Wavelet 7.2.2 Shannon, Meyer and Battle-LemariCWavelets 7.2.3 Daubechies Compactly Supported Wavelets Wavelets and Filter Banks 7.3 I Fast Orthogonal Wavelet. .. estimation and compression algorithms It uses deterministic models that can be constructed even for complex signals such as images Chapter 10is rewritten and expanded to explain and compare the Bayes and minimax points of view Bounded Variation Signals Wavelet transforms provide sparse representations of piecewise regular signals The total variation norm gives an intuitive and precise mathematical framework... non-isolated singularities Mandelbrot [43] was the h s t to recognize the existence of multifractals in most corners of nature Scaling one part of a multifractal produces a signal that is statistically similar to the whole This self-similarityappears in the wavelet transform, which modifies the analyzing scale From the global wavelet transform decay, one can measure the singularity distribution of multifractals... found a piecewise linear function $ that also generates an orthonormal basis and gives better approximations of smooth functions Meyer was not aware of this result, and motivated by the work of Morlet and Grossmann he tried to prove that there exists no regular wavelet $ that generates an orthonormal basis This attempt was a failure since he ended up constructing a whole family of orthonormal wavelet bases,... with a scale parameter s, and translated by u: The wavelet transform off at the scale s and position u is computed by correlating f with a wavelet atom Time-Frequency Measurements Like a windowed Fourier transform, a wavelet transform can measure the time-frequency variations of spectral components, but it has a different time-frequency resolution A wavelet transform correlates f with $J~:,.By applying . bases Many orthonormal bases can be designed with fast computational algorithms. The discovery of filter banks and wavelet bases has created a popular new sport of basis hunting. Families of. Classes of Wavelet Bases ' 7.2. I Choosing a Wavelet 7.2.2 7.2.3 Daubechies Compactly Supported Wavelets Wavelets and Filter Banks ' 7.3. I Fast Orthogonal Wavelet Transform. In applied mathematics, this course is an introduction to wavelets but also to signal processing. Signal processing is a newcomer on the stage of legitimate applied mathematics topics.