Wavelets and Their Applications PDF
Preview Wavelets and Their Applications
Wavelets and Their Applications NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NA TO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an intemational board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics London and New York C Mathematical Kluwer Academic Publishers and Physical Sciences Dordrecht, Boston and London D Behavioural and Social Sciences E Applied Sciences F Computer and Systems Sciences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, London, H Cell Biology Paris and Tokyo I Global Environmental Change NATO-PCO-DATA BASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than 30000 contributions from international scientists published in all sections of the NATO ASI Series. Access to the NATO-PCO-DATA BASE is possible in two ways: - via online FILE 128 (NATO-pea-DATA BASE) hosted by ESRIN, Via Galileo Galilei, 1-00044 Frascati, Italy. - via CD-ROM "NATo-pea-DATA BASE" with user-friendly retrieval software in English, French and German (©WTV GmbH and DATAWARE Technologies Inc. 1989). The CD-ROM can be ordered through any member of the Board of Publishers or through NATO-PCO, Overijse, Belgium. Series C: Mathematical and Physical Sciences -Vol. 442 Wavelets and Their Applications edited by J. S. Byrnes Prometheus Inc., Newport, Rhode Island and University of Massachusetts, Boston, Massachusetts, U.S.A. Jennifer L. Byrnes Harvard University, Cambridge, Massachusetts, U.S.A. Kathryn A. Hargreaves University of Massachusetts, Boston, Massachusetts, U.S.A. and Karl Berry Interleaf, Cambridge, Massachusetts and University of Massachusetts, Boston, Massachusetts, U.S.A. SPRINGER SCIENCE+BUSINESS MEDIA, B.V. Proceedings of the NATO Advanced Study Institute on Wavelets and Their Applications II Ciocco, Italy 16-29 August 1992 A C.I.P. Catalogue record forthis book is available from the Ubrary of Congress. ISBN 978-94-010-4448-6 ISBN 978-94-011-1028-0 (eBook) DOI 10.1007/978-94-011-1028-0 AII Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1 st edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo copying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Acknowledgements Support for the conference and this volume, from the following organizations and individuals, is gratefully acknowledged: NATO, Dr. Louis V. da Cunha NSF, Dr. Richard P. Metcalf O~, Dr.~eil(;err ONREUR, Dr. Robert Ryan Prometheus Inc. University of Massachusetts at Boston This work relates to Department of the ~avy (;rant ~OOOl4-92-J-9033 issued by the Office of ~aval Research European Office and to Department of the ~avy (;rant ~00014-92-J-1286 issued by the Office of ~aval Research. The United States has a royalty-free license throughout the world in all copyrightable material contained herein. Colophon This book was designed by Kathryn A. Hargreaves and Karl Berry. They typeset it using TEX, developed by Donald E. Knuth. The text is set in Palatino, and the mathematics in Euler, both designed by Hermann Zapf. The heads are set in bold Helvetica, designed by Max Miedinger. The camera-ready copy was produced in PostScript on a 600 dpi Hewlett-Packard LaserJet 4. All non-PostScript figures were scanned using a Xerox 7650 1200 dpi black-and-white scanner. Cross-referencing, indexing, and the various tables of contents were done automatically, using the Eplain macros written by Karl Berry and oth ers, as well as (many) additional macros. § Preface It is probably true quite generally that in the history of human thinking the most fruitful developments frequently take place at those points where two different lines of thought meet. Hence, if they actually meet, that is, if they are at least so much related to each other that a real interaction can take place, then one may hope that new and interesting developments may follow. Werner Heisenberg This volume contains papers presented at the August 1992 NATO Advanced Study Institute on Wavelets and Their Applications. The conference was held at the beautiful Il Ciocco resort near Lucca, in the glorious Tuscany region of northern Italy. Once again we gathered at this idyllic spot to explore and extend the reciprocity between mathematics and engineering. The dynamic interaction between world-renowned scientists from the usu ally disparate communities of pure mathematicians and applied scientists, which occurred at our 1989 and 1991 ASI's, continued at this meeting. Wavelet theory and technology is in an important growth stage at which theoretical and practical results are being compared with existing methods. There have been spectacular wavelet successes and sobering comparisons with traditional ideas-but still there is a wide expanse of scientific problems to explore. Since these problems lie at the forefront of both pure mathematics and applied science, our NATO ASI was especially pertinent at this time. Wavelets and wavelet packets provide a theory analogous to Fourier analysis and tools analogous to coherent state methods. Among their nu merous applications, wavelets have been used for data compression in both image and sound processing. Further, they are intimately and significantly related to splines. Wavelets have become a tool in analyzing fractals and iterative schemes associated with dynamical systems. Signal processing methods such as quadrature mirror filters go hand in hand with wavelet techniques in studying a host of communications problems. The profound issues of classical turbulence are being studied using wavelet packets. Both wavelet packet software and wavelet transform microchips are now avail able. There are also applications of wavelet theory in theoretical physics, oil exploration, irregular sampling, and singular integral operators. Many of the world's experts in the field of wavelets were principal speakers at the ASI, and their papers appear in this volume. These renowned scientists addressed their talks to an audience which consisted of a broad spectrum of pure and applied mathematians, as well as a diverse group of engineers and scientists. Thus, the reader has the opportunity to learn ( Preface or reinforce fundamental concepts from the individuals who have created and developed the flourishing field of wavelets, and to see them discuss in accessible terms their profound contributions and ideas for future research. The papers to be found herein include: two interesting works by Stephane Jaffard and Patrick Flandrin, exploring the applications of wavelets to random processes, including fractional Brownian motion; two illuminat ing discussiond of time-frequency analysis in general, and Gabor represen tations in particular, by A.J.E.M. Janssen and Richard Tolimieri; and an ex ceptionally clear description by Victor Wickerhauser et al., of the application of wavelet packets to data compression in signal processing. In addition, by considering processes indexed by the nodes of a homogeneous tree, Bernhard Claus gives a comprehensive description of his particular wavelet approach: multiscale statistical modeling. Two faScinating applications of frame-like expansions are presented. John Benedetto uses this idea, together with his irregular sampling theory and weighted Fourier transform norm inequalities, to formulate and imple ment his wavelet auditory model for data compression in speech. Louis Aus lander and Frank Geshwind describe an approximation method whereby almost" frames yield methods for analyzing and synthesizing signals of II interest in radar problems. Two premier applications of wavelets are described by Bjorn Jawerth and Joshua Zeevi. Jawerth, by discussing Clifford wavelets and Hardy spaces offers a clear introduction to the use of wavelets for solving par tial differential equations. Zeevi, generalizing the concept of wavelets to group representations other than the standard two-parameter affine group, presents a wavelet-type approach to image representation in vision systems. Harold Shapiro clearly and concisely shows how the generalized mod ulus of continuity, which he introduced almost twenty-five years ago, is intimately related to what is now called the continuous wavelet transform. He applies this to deduce the regularity of a function from appropriate de crease of its wavelet coefficients. In a somewhat similar vein, Christopher Heil examines how smoothness properties of the wavelet or scaling function are affected by changes in the coefficients in the associated dilation equations. The cooperation of many individuals and organizations was required in order to make the conference the success that it was. First and foremost I wish to thank NATO, and especially Dr. Louis V. daCunha and his staff, for the initial grant and subsequent help. Financial support was also received from the Office of Naval Research (Dr. Neil Gerr), the European office of ONR (Dr. Robert Ryan), the National Science Foundation (Dr. Richard P. Metcalf), the University of Massachusetts at Boston, and Prometheus Inc. This additional support is gratefully acknowledged. I wish to express my sincere appreciation to my assistants, Marcia Byrnes and Alan and Leslie Fisher, for their invaluable aid. I am also grateful ( Preface) to Kathryn Hargreaves and Karl Berry, our T£Xnicians, for their superlative work on all printed and emailed aspects of the conference, from the initial application to this volume. Their extraordinary effort in T£Xing these pro ceedings, resulting in one of the very few NATO Proceedings where all papers are identically typeset, deserves special acclamation. Finally, my heartfelt thanks to the 11 Ciocco staff, especially Bruno Giannasi and Alberto Suffre dini, for offering an ideal setting, not to mention the magnificent meals, that promoted the productive interaction between the participants of the con ference. All of the above, the other speakers, and the remaining conferees, made it possible for our Advanced Study Institute, and this volume, to ful fill the stated NATO objectives of disseminating advanced knowledge and fostering international scientific contacts. January 4,1994 J.S. Byrnes, Newport, Rhode Island § Contents John J. Benedetto From a wavelet auditory model to definitions of the Fourier transform 1»1>1 Christopher Heil Some stability properties of wavelets and scaling functions 1»1>19 Susan Eo Kelly, Mark A. Kon, Louise Arakelian Raphael Convergence: Fourier series vs. wavelet expansions 1»1>39 Stephane Jaffard Regularity analysis of functions and random processes using wavelets 1»1>51 Harold S. Shapiro The generalized modulus of continuity and wavelets 1»1>83 Louis Auslander, Frank Geshwind Approximate frames and the narrowband multitarget radar problem 1»1>95 Patrick Flandrin Time-scale analyses and self-similar stochastic processes 1»1>121 B. Claus Multiscale statistical modeling: isotropic processes on a homogeneous tree 1»1>143 Joseph Segman, Yehoshua Y. Zeevi A wavelet-type approach to image analysis and vision 1»1>169 Walter Schempp, Joseph Segman Analog VLSI network models, cortical linking neural network models, and quantum holographic neural technology 1»1>213 Bjorn Jawerth, Marius Mitrea Clifford wavelets, Hardy spaces, and elliptiC boundary value problems 1»1>261 Ronan Le Bras, George Mellman Wavelet transforms and downward continuation 1»1>291 A.J.E.M. Janssen Bilinear time-frequency distributions 1»1>297 Jayakumar Ramanathan, Pankaj Topiwala Time-frequency localization operators of Cohen's class 1»1>313 Serge Collineau Some remarks about the scalograms of wavelet transform coefficients 1»1>325 Richard Tolimieri Problems in Gabor representation 1»1>331 Joseph Segman, Walter Schempp On the extension of the Heisenberg group to incorporate multiscale resolution 1»1>347
The list of books you might like
