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Wavelets: Theory and Applications

ISBN: 978-0-471-96792-7
342 pages
October 1997
Wavelets: Theory and Applications (0471967920) cover image

Description

With applications in pattern recognition, data compression and numerical analysis, the wavelet transform is a key area of modern mathematics that brings new approaches to the analysis and synthesis of signals. This book presents the central issues and emphasizes comparison, assessment and how to combine method and application. It reviews different approaches to guide researchers to appropriate classes of techniques.
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Table of Contents

Preface ix

Notation xi

Introduction xv

1 The Continuous Wavelet Transform 1

1.1. Definition and Elementary Properties 1

1.2 Affine Operators 10

1.3 Filter Properties of the Wavelet Transform 12

1.4 Approximation Properties 22

1.5 Decay Behaviour 32

1.6 Group-Theoretical Foundations and Generalizations 36

1.7 Extension of the One-Dimensional Wavelet Transform to Sobolev Spaces 59

Exercises 69

2 The Discrete Wavelet Transform 73

2.1 Wavelet Frames 73

2.2 Multiscale Analysis 97

2.3 Fast Wavelet Transform 121

2.4 One-Dimensional Orthogonal Wavelets 131

2.5 Two-Dimensional Orthogonal Wavelets 203

Exercises 226

3 Applications of the Wavelet Transform 231

3.1 Wavelet Analysis of One-Dimensional Signals 231

3.2 Quality Control of Texture 235

3.3 Data Compression in Digital Image Processing 239

3.4 Regularization of Inverse Problems 251

3.5 Wavelet – Galerkin Methods for Two-Point boundary Value Problems 259

3.6 Schwarz Iterations Based on Wavelet Decompositions 278

3.7 An Outlook on Two-Dimensional Boundary Value Problems 300

Exercises 306

Appendix The Fourier Transform 309

References 313

Index 321

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