Wavelets: Time-Frequency Methods and Phase Space Proceedings of the International Conference, Marseille, France, December 14–18, 1987

336 Pages·1990·17.513 MB·English

by  A. Grossmann, R. Kronland-Martinet, J. Morlet (auth.), Professor Jean-Michel Combes, Professor Alexander Grossmann, Professor Philippe Tchamitchian (eds.)

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: pro
ceedings of the international conference, Marseille, France, December 14-18, 19871 J. M. Combes, A. Gross
mann, 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
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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 pa
rameter. 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 par
ticipants. 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 sup
port 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 Labora
toire 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
m
Part 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
a
An Implementation of the "algorithme 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
Part I
Introduction to Wavelet Transforms
Reading and Understanding Continuous Wavelet Transforms
A. Grossmann 1, R. Kronland-Martinet 2, and J. M orlet 3
lCentre de Physique Theorique, Section II, e.N.R.S.,
Luminy Case 907, F-13288 Marseille Cedex 09, France
2Faculte des Sciences de Luminy and Laboratoire de Mecanique
et d' Acoustique, C.N.R.S., 31, Chemin I. Aiguier,
F-13402 Marseille Cedex 09, France
3TRAVIS, c/o O.R.Le. 371 bis, Rue Napoleon Bonaparte,
F-92500 Rueil-Malmaison, France
1. Introduction
One of the aims of wavelet transforms is to provide an easily interpretable visual
representation of signals. This is a prerequisite for applications such as selective
modifications of signals or pattern recognition.
This paper contains some background material on continuous wavelet transforms and a
description of the representation methods that have gradually evolved in our work. A related
topic, also discussed here, is the influence of the choice of the wavelet in the interpretation
of wavelets transforms. Roughly speaking, there are many qualitative features (in particularly
concerning the phase) which are independent of the choice of analyzing wavelet; however, in
some situations (such as detection of "musical chords") an appropriate choice of wavelet is
essential.
We also briefly discuss the finite interpolation problem for wavelet transforms with
respect to a given analyzing wavelet, and give some details about analyzing wavelets of
gaussian type.
2. Definitions
The continuous wavelet transform of a real signal s(t) with respect to the analyzing
wavelet g(t) (in general, g(t) is complex) may be defined as a function:
fa-fg
((t~b))S(t)
(2.1) S(b,a)= dt (gdenotes the complex conjugate of g)
defined on the open "time and scale" half-plane H (b E R, a>O). We shall find it convenient to
use a somewhat unusual coordinate system on H, with the b-axis ("dimensionless time") facing
to the right and the a-axis ("scale") facing downward (Fig 2.1).
The a-axis faces downward since small scales correspond, roughly speaking, to high
frequencies, and we are used to seeing high frequencies above low frequencies.
2