inverse problems and theoretical imaging

inverse problems and theoretical imaging

1.M. Combes A. Grossmann Ph. Tchamitchian (Eds.) Wavelets Time-Frequency Methods and Phase Space Proceedings of the International Conference, Marseille, France, December 14-18, 1987 Second Edition With 98 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona

Professor Jean-Michel Combes Professor Alexander Grossmann Professor Philippe Tchamitchian Centre National de la Recherche Scientifique Luminy - Case 907, F-13288 Marseille Cedex 9, France ISBN-13 :978-3-540-530 14-5 e-isbn-13 :978-3-642-75988-8 DOl: 10.1007/978-3-642-75988-8 Library of Congress Cataloging-in-Publication Data. Wavelets: time-frequency methods and phase space: proceedings of the international conference, Marseille, France, December 14-18, 19871 J. M. Combes, A. Grossmann, Ph. Tchamitchian, (eds.).-2nd rev. and enl. ed. p. cm.-(inverse problems and theoretical imaging) Includes indexes.lsbn-13:978-3-540-53014-5 1. Phase space (statistical physics) Congresses. 2. Time measurements-congresses. 3. Mathematical physics-congresses. I. Combes, J. M. (Jean-Michel), 1941-. II. Grossmann, A. (Alexander), 1930-. III. Tchamitchian, Philippe. IV. Series. QCI74.85.P48W38 1990 530.1'5-dc20 90-10343 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Verlag Berlin Heidelberg 1989 and 1990 The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2157/3140-543210 - Printed on acid-free paper

Preface The last two subjects mentioned in the title "Wavelets, Time Frequency Methods and Phase Space" are so well established that they do not need any explanations. The first is related to them, but a short introduction is appropriate since the concept of wavelets emerged fairly recently. Roughly speaking, a wavelet decomposition is an expansion of an arbitrary function into smooth localized contributions labeled by a scale and a position parameter. Many of the ideas and techniques related to such expansions have existed for a long time and are widely used in mathematical analysis, theoretical physics and engineering. However, the rate of progress increased significantly when it was realized that these ideas could give rise to straightforward calculational methods applicable to different fields. The interdisciplinary structure (R.C.P. "Ondelettes") of the C.N.R.S. and help from the Societe Nationale Elf-Aquitaine greatly fostered these developments. The conference, the proceedings of which are contained in this volume, was held at the Centre National de Rencontres Mathematiques (C.N.R.M) in Marseille from December 14-18, 1987 and bought together an interdisciplinary mix of participants. We hope that these proceedings will convey to the reader some of the excitement and flavor of the meeting. In the preparation of the conference we have benefited from the help and support of the following organizations: the Societe Mathematiquede France and the C.N.R.M.; the Universite Aix-Marseille II, FacuIte de Luminy; the Universite de Toulon et du Var; the Conseil Regional Provence-Alpes-Cote d' Azur; the Laboratoire de Mecanique et Acoustique and Centre de Physique Theorique, both at the C.N.R.S., Marseille. The company DIGILOG kindly provided the signal processor SYSTER for demonstration purposes. The editors are extremely grateful to all of them, to the participants and to all other people who helped in various ways to make this meeting a real success. Marseille, December 1988 1.-M. Combes A. Grossmann Ph. Tchamitchian (final manuscript received: March 16, 1989) v

In Memoriam We have learned with shock the news of the sudden death of Professor Franz B. Tuteur His absence is keenly felt by those of us who had the privilege of knowing him and working with him. VI

Contents Part I Introduction to Wavelet Transforms Reading and Understanding Continuous Wavelet Transforms By A. Grossmann, R. Kronland-Martinet, and J. Morlet (With 23 Figures) 2 Orthonormal Wavelets By Y. Meyer... 21 Orthonormal Bases of Wavelets with Finite Support - Connection with Discrete Filters By 1. Daubechies (With 9 Figures)... 38 Part IT Some Topics in Signal Analysis Some Aspects of Non-Stationary Signal Processing with Emphasis on Time-Frequency and Time-Scale Methods By P. Flandrin... 68 Detection of Abrupt Changes in Signal Processing By M. Basseville (With 1 Figure)... 99 The Computer, Music, and Sound Models By J.-c. Risset (With 2 Figures)... 102 Part m Wavelets and Signal Processing Wavelets and Seismic Interpretation By J.L. Larsonneur and J. Morlet (With 3 Figures) 126 Wavelet Transformations in Signal Detection By F.B. Tuteur (With 4 Figures)... 132 Use of Wavelet Transforms in the Study of Propagation of Transient Acoustic Signals Across a Plane Interface Between Two Homogeneous Media By G. Saracco, A. Grossmann, and Ph. Tchamitchian (With 7 Figures).. 139 VII

Time-Frequency Analysis of Signals Related to Scattering Problems in Acoustics Part I: Wigner-Ville Analysis of Echoes Scattered by a Spherical Shell By J.P. Sessarego, J. Sageloli, P. Flandrin, and M. Zakharia (With 4 Figures)....................................... 147 Coherence and Projectors in Acoustics By J.G. Slama... 154 Wavelets and Granular Analysis of Speech By J.S. Lienard and C. d'alessandro (With 4 Figures)... 158 Time-Frequency Representations of Broad-Band Signals By J. Bertrand and P. Bertrand (With 2 Figures)... 164 Operator Groups and Ambiguity Functions in Signal Processing By A. Berthon... 172 Part IV Mathematics and Mathematical Physics Wavelet Transform Analysis of Invariant Measures of Some Dynamical Systems By A. Arneodo, G. Grasseau, and M. Holschneider (With 15 Figures).. 182 Holomorphlc Integral Representations for the Solutions of the Helmholtz Equation By J. Bros.................................. 197 Wavelets and Path Integrals By T. Paul... 204 Mean Value Theorems and Concentration Operators in Bargmann and Bergman Space By K. Seip... 209 Besov -Sobolev Algebras of Symbols By G. Bohnke... 216 Poincare Coherent States and Relativistic Phase Space Analysis By J.-P. Antoine....................................... 221 A Relativistic Wigner Function Affiliated with the Weyl-Poincare Group By J. Bertrand and P. Bertrand........ 232 Wavelet Transforms Associated to the n-dimensional Euclidean Group with Dilations: Signal in More Than One Dimension By R. Murenzi... 239 Construction of Wavelets on Open Sets By S. Jaffard (With 8 Figures)... 247 Wavelets on Chord-Arc Curves By P. Auscher 253 VIII

Multiresolution Analysis in Non-Homogeneous Media By R.R. Coifrnan... 259 About Wavelets and Elliptic Operators By Ph. Tchamitchian.................................... 263 Towards a Method for Solving Partial Differential Equations Using Wavelet Bases By V. Perrier (With 7 Figures)... 269 Part V Implementations A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform By M. Holschneider, R. Kronland-Martinet, J. Morlet, and Ph. Tchamitchian... 286 An Implementation of the "algorithme a trous" to Compute the Wavelet Transform By P. Dutilleux (With 7 Figures)... 298 An Algorithm for Fast Imaging of Wavelet Transforms By P. Hanusse... 305 Multiresolution Approach to Wavelets in Computer Vision By S.O. Mallat (With 10 Figures)... 313 Subject Index... 329 Index of Contributors................................... 331 IX