Wavelet Theory and Its Applications

232 Pages·1993·11.16 MB·English

by  Randy K. Young (auth.)

WAVELET THEORY AND ITS
APPLICATIONS
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
VLSI, COMPUTER ARCHITECTURE AND
DIGITAL SIGNAL PROCESSING
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WAVELETTHEORY AND ITS
APPLICATIONS
by
Randy K. Young
Pennsylvania State Unzversity
~.
"
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library ofCongress Cataloging-in-Publication Data
Young, Randy K., 1963-
Wavelet theory and its applications / Randy K. Young
p. cm. -- (Kluwer international series in engineering and
computer science ; SECS 189)
Includes bibliographical references and index.
ISBN 978-1-4613-6593-8 ISBN 978-1-4615-3584-3 (eBook)
DOI 10.1007/978-1-4615-3584-3
1. Signal processing--Mathematics. 2. Wavelets (Mathematics)
3. Linear time invariant systems. l. Title. II. Series
TK5102.5 . Y67 1992
621 . 382 ' 2--dc20 92-23362
CIP
Copyright © 1993 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint ofthe hardcover 1st edition 1993
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmi tted in any form or by any means, mechanical, photo-copying, recording,
or otherwise, without the prior written permission of the publisher, Springer Science+
Business Media, LLC.
Printed on acid-free paper.
To: Rita, Jason, and Dad
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi
Preface ........................................... xiii
Chapter 1: lntroductionnBackground
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
Wavelet Theory Basics - Scaling and Translation ............ 4
Appendix I-A: Time Referencing . . . . . . . . . . . . . . . . . . . . .. 16
Chapter 2: The Wavelet Transfonn
· .................................... 19
Wavelet Transform Definitions and Operators .............. 19
Wavelet Transform Examples ........................ 22
The Resolution of Identity .......................... 27
Continuous Inverse Wavelet Transform Theorem ............ 30
Energy Distribution in the Wavelet Transform Domain. . . . . . . .. 34
Discrete Wavelet Transform (Continuous Time Wavelet Series) ... 35
Wideband Matched Filter Interpretation of the Lattice Density .... 38
Review of the Inverse Discrete Wavelet Transform: Scale-
translation Lattice Density and Mother Wavelet
Constraints .............................. 41
Discrete Wavelet Mathematics - Rigorous Justification . . . . . . . .. 44
Discrete Time Wavelet Series ........................ 50
Multiresolution, Orthogonal, and Biorthogonal Wavelet
Transforms (and PR-QMFs) . . . . . . . . . . . . . . . . . . .. 51
Discrete Time Wavelet Series - A Specific Structure .......... 52
Multiresolution ................................. 55
Orthogonal and Biorthogonal Wavelet Transforms ............ 57
An Image Processing Example . . . . . . . . . . . . . . . . . . . . . . .. 60
Appendix 2-A: Nonunique Wavelet Domain Representations ..... 64
Chapter 3: Practical Resolution, Gain, and Processing Structures
· .................................... 71
Introduction ................................... 71
Fourier/Narrowband Gain and Resolution Comparisons. . . . . . . .. 72
Multiple Mother Wavelets - Gain and Resolution Properties
Only .................................. 77
Mother Wavelet Properties - Relationships to Established
Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79
Signal Analysis - Time-frequency or time-scale Applications ..... 80
A Physical Interpretation of Scale-translation Resolution:
Wideband Ambiguity Function Conditions ........... 81
Wideband Conditions ............................. 82
Wideband Signals and the Analytic Signal Model ............ 83
Effective rms Time-bandwidth Product . . . . . . . . . . . . . . . . . ., 85
Wideband Systems and Signals . . . . . . . . . . . . . . . . . . . . . . .. 87
Active and Passive Sensing . . . . . . . . . . . . . . . . . . . . . . . . .. 92
Ambiguity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
Reformulation of the WBCAF with Wavelet Transforms . . . . . .. 101
Properties of the Reformulated WBCAF . . . . . . . . . . . . . . . .. 105
Cross Wavelet Transforms and Signal Commonalities . . . . . . . .. 110
Appendix 3-A: Wideband/Narrowband Ambiguity Functions:
Assumptions, Tradeoffs and Efficiencies ........... 113
Appendix 3-B: Narrowband Ambiguity Function Theory ...... 119
Chapter 4: Wavelet Theory Extensions and Ambiguity Functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123
Introduction .................................. 123
Multiresolution/Orthogonal Wavelets versus Unconstrained
Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123
Sampling Grids .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 126
Unconstrained Wavelet Transforms - Mother Wavelet Freedom. .. 127
The Mother Mapper Operator ....................... 128
Unconstrained Wavelet Transforms/Mother Mapper Operator
Properties and Applications ................... 133
Mother Mapper Operator Applications . . . . . . . . . . . . . . . . .. 137
Mother Mapper Operator - Final Considerations . . . . . . . . . . .. 139
Further Research and Applications of the Mother Mapper
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 140
Chapter 5: Linear Systems Modelling with Wavelet Theory
141
Introduction and Motivation ....................... . 141
Wideband/Nonstationary/Time-varying System Modelling ..... . 143
The Wideband Signal Reflection Process . . . . . . . . . . . . . . . . . 146
Common Framework of System or Channel Characterizations: ... 151
The STY Wavelet Operator - Space-time-varying System Model .. 156
STY Wavelet Operator's Energy Distribution ............. . 159
Estimation of the Wideband System Characterization ........ . 160
Properties of the STY Wavelet Operator ................ . 160
Examples of the STY Wavelet Operator ................ . 164
Wideband Reflection: Comparing the LTI and STY Models .... . 169
Justification for the STY Operator Instead of Convolution for
Signals Represented by Wavelet Transforms ........ . 174
The STY Wavelet Operator in the Wavelet Transform Domain .. . 175
Space-time-varying System Identification Problem with
Wideband/Nonstationary Input/Outputs ........... . 180
viii
Limitations of the STV Wavelet Operator -Time Referencing and
Nonlinear Time Variations. . . . . . . . . . . . . . . . . . .. 181
Bi-wavelet System Representation: Time variation of the Time-
varying System Model ...................... 184
Chapter 6: Wideband Scattering and Environmental Imaging
189
Introduction ................................. . 189
Scattering Theory ............................. . 192
General Scattering Function Background . . . . . . . . . . . . . . . . . 193
Narrowband Scattering Theory ...................... . 194
Wideband Correlation Receiver and its Output . . . . . . . . . . . . . 198
Wideband Point Scatterer Example ................... . 201
Wideband Scattering Functions and Resolutions ........... . 203
Physical Form of the WBCAF ...................... . 205
Time Delay and Scale Estimation . . . . . . . . . . . . . . . . . . . . . 208
WBAAF Moments and Assumptions .................. . 209
Wideband Scattering Review . . . . . . . . . . . . . . . . . . . . . . . . 210
Related Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 211
References ........................................ 215
Subject Index ...................................... 221
ix
Foreword
The continuous wavelet transform has deep mathematical roots in the work
of Alberto P. Calderon. His seminal paper on complex method of interpolation and
intermediate spaces provided the main tool for describing function spaces and their
approximation properties. The Calderon identities allow one to give integral
representations of many natural operators by using simple pieces of such operators,
which are more suited for analysis. These pieces, which are essentially spectral
projections, can be chosen in clever ways and have proved to be of tremendous
utility in various problems of numerical analysis, multidimensional signal processing,
video data compression, and reconstruction of high resolution images and high
quality speech. A proliferation of research papers and a couple of books, written in
English (there is an earlier book written in French), have emerged on the subject.
These books, so far, are written by specialists for specialists, with a heavy
mathematical flavor, which is characteristic of the Calderon-Zygmund theory and
related research of Duffin-Schaeffer, Daubechies, Grossman, Meyer, Morlet, Chui,
and others.
Randy Young's monograph is geared more towards practitioners and even
non-specialists, who want and, probably, should be cognizant of the exciting proven
as well as potential benefits which have either already emerged or are likely to
emerge from wavelet theory. For example, the theory of multidimensional
processing of video signals is a subject of very current interest and a Special Issue
has, recently (in May 1992), been devoted to the topic in the journal on
Multidimensional Systems and Signal Processing, also published by Kluwer
Academic Publishers. The multiresolution method decomposes high resolution
images into a hierarchy of pieces or components, each more detailed than the next.
Thus, the multiresolution approach, which is based on the wavelet theory of
successive approximations provides the mechanism for transmitting various grades
of images depending upon the quality of reconstruction sought and the limitation set
by channel capacity as well as the need for compatibility with existing receivers
when high definition television, for example, enters the market. Certainly, the single
resolution approach is incapable of adjusting to the practical constraints described.
The myriad of applications which wavelet theory has spawned will, undoubtedly,
stimulate the writing of many books which will emphasize applications and attempt
to widen the circle of readers. I am glad that Randy Young has taken the initiative
to do just that, and I believe that others will follow his footsteps.
N. K. Bose
The Pennsylvania State University
University Park
Preface
This book extends wavelet theory, both theoretically and practically. But
even the theoretical extensions are directed toward practical applications. The theory
and derivations require only college senior level mathematics. Primarily, signal
processing applications are addressed, but these applications are quite general (i.e.,
signal and system models) and can be employed in many diverse fields.
Initially, in Chapter 1, wavelet theory is introduced and terminology is
defmed. The time scaling action of the wavelet transform is emphasized. A
pictorial demonstration of the time scaling action is presented by creating several
scaled and translated "mother wavelets." After some brief motivational comments,
the structure of the wavelet transform is provided along with a pictorial example.
Chapter 2 considers both continuous and discrete wavelet transforms and
reviews established wavelet theory. The review is not detailed but references are
cited if additional details are desired. Since this book concentrates on nonorthogonal
and "sophisticated" wavelets, the special cases of orthogonal, biorthogonal, and
multiresolution wavelet transforms are presented, but with an emphasis on their
constraints and limitations. These special wavelet transforms are extremely efficient
and very powerful (and these special wavelet transforms are considered "wavelet
theory" for many people). This book addresses the wavelet theory that is not
included within these special wavelet transforms and relaxes the constraints
introduced by these special transforms. Very general mother wavelets will be
required for the wideband or wavelet system models (the input signal will be
employed as the mother wavelet).
After justifying these more general wavelet transforms, the properties of
these transforms are investigated. Chapter 3 primarily examines the gain and
resolution of the wavelet transforms by relating the processing to matched filtering
concepts. The wideband or wavelet processing is also compared to the Fourier or
narrowband processing. Throughout this book, the parameterization of the wavelet
transform by the mother wavelet is of critical concern. The properties of these
mother wavelets are most easily described by ambiguity functions and, thus,
ambiguity functions are described first. After discussing ambiguity functions,
wideband ambiguity functions are efficiently formulated with wavelet transforms.
This reformulation produces a cross wavelet transform that extracts the common
features of the two signals (or systems, channels, etc.) being cross processed.
Then, in Chapter 4, the "sophisticated" mother wavelets are compared to
their tone-like counterparts from an ambiguity function point of view. By
considering multiple mother wavelets, efficient wavelet domain representations can
be formed. The computation of multiple wavelet transforms is efficiently
reformulated by a new operator, the mother mapper operator. This operator
efficiently maps a wavelet transform with respect to one mother wavelet to a new
wavelet transform with respect to a different mother wavelet.
Next, in Chapter 5, an original model for space-time-varying (STV)
systems, including linear time-invariant systems as a special case, is formulated with