Ten Lectures on Wavelets.pdf

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INGRID DAUBECHIES Rutgers University and AT&T Bell Laboratories Ten Lectures on Wavelets SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS PHILADELPHIA, PENNSYLVANIA 1992

Copyright 1992 by the Society for Industrial and Applied Mathematics All rights reserved. No part of this book may be reproduced, transmitted For information, 3600 University write the Society for Industrial City Science Center, Philadelphia, in any manner without the written permission of the Publisher. and Applied Mathematics, Pennsylvania stored, or 19104-2688. Second Printing 1992 Third Printing 1994 Fourth Printing Fifth Printing 1997 Sixth Printing 1995 1999 Library of Congress Cataloging-in-Publication Data Daubechies, Ingrid. Ten lectures on wavelets / Ingrid Daubechies. p. cm. - (CBMS-NSF regional conference series in applied mathematics; 61) Includes bibliographical ISBN 0-89871-274-2 1. Wavelets (Mathematics)-Congresses. references and index. I. Title. II. Series. QA403.3.D38 1992 515' .2433-dc20 92-13201 sianL. is a registered trademark.

To my mother, To my father, in science. who gave me the will to be independent. who stimulated my interest iii

Contents Vll INTRODUCTION Density and Orthononnal Bases xi PRELIMINARIES AND NOTATION CHAPTER 1: The What, Why, and How of Wavelets 17 CHAPTER 2: The Continuous Wavelet Transfonn 53 CHAPTER 3: Discrete Wavelet Transfonns: Frames 107 CHAPTER 4: Time-Frequency 129 CHAPTER 5: Orthononnal Bases of Wavelets and Multiresolution 167 CHAPTER 6: Orthononnal Bases of Compactly Supported 215 CHAPTER 7: More About the Regularity 251 CHAPTER 8: Symmetry for Compactly Supported Wavelet Bases 289 CHAPTER 9: Characterization 313 CHAPTER 10: Generalizations 341 REFERENCES 353 SUBJECT INDEX 355 AUTHOR INDEX and Tricks for Orthononnal Wavelet Bases of Functional Spaces by Means of Wavelets Analysis Wavelets of Compactly Supported Wavelets v

I ntrod uction tool with development rate. There are several origins and in applied ago (Morlet ics. Their mathemat (1984)); in the last ten , Arens, Fourgeau, On the one hand, the concept coding), physics (coherent (study of Calder6n rea of wavelets can be or thirty the last twenty states, renormalization s). As a operator -Zygmund appeal of these interdisciplinary to scientists , wavelets are a fairly hand, wavelets backgrounds. On the other Already of possible a great variety applications. (sound, images) (some applications analysis in signal Martinet , Morlet and Grossmann (1987), Mallat and numerical , (1989c); more recent references analy in Beylkin, Coifman, and Rokhlin also Wavelets are a relatively recent name itself was coined approximately a decade and Giard (1982), Morlet and Morlet (1983), Grossmann in them has grown at an explosive years interest success. present sons for their as a synthesis of ideas which originated viewed during years in engineering (subband group), and pure mathematics consequence of many different engineers mathematical simple they have led to exciting early references are Kronland- (1989b) sis (fast algorithms for integral (1991)); many other contributes I delivered This book contains CBMS conference of Lowell, Massachuset at the University partment ces, other format of the CBMS conferen R. Coifman, K. Grochenig, A. Cohen, J. Liandrat lectures and A. Willsky) provided and inverse problems three were organized, workshops analysis (H. Feichtinger), by B. DeFacio (chaired ), group theory of researchers and signal analysis and other scientists in the field of wavelets as well as of mathematicians who knew little engineers group constituted on wavelets a tutorial for more recent work grounding exposed about two thirds sequently, (G. Battle, work related to physics on applications consisted active to learn more. This second I saw it as my task to provide which would then be a solid as the principal speaker De ts. According , S. Mallat, B. Torresani, to the usual G. Beylkin, C. Chui, to the interest on wavelets they generate. ten lectures organized about wavelets and hoped part of the audience. to this part of the audience, by the of my lectures other lecturers Con of "basic wavelet consisted in June 1990 by the Mathematics applications transforms are being studied on their to wavelets. Moreover, and myself. theory," (M. Vetterli). The audience and harmonic are given later) . This wide applicability at the speakers and the largest vii

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