Wavelets

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.

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

A. K. Louis and D. Maass are the authors of Wavelets: Theory and Applications, published by Wiley. A. K. Louis, Universität des Saarlandes, Germany P. Maass, Universität Potsdam, Germany A. Rieder, Universität des Saarlandes, Germany Wavelets have in recent years brought new approaches to the areas of analysis and synthesis of signals, a few examples of which include pattern recognition, data compression, numerical analysis, quantum field theory and acoustics. In this book the authors take the reader through both the theory of wavelets and their applications. Chapter one is devoted to the theoretical background of the wavelet transform and to some of its properties, moving on to the discrete transform. The second chapter addresses the functions of wavelets within mathematics and their construction is introduced. Finally, chapter three presents a selection of the broad variety of applications of wavelets, including examples from signal analysis, quality control, data compression in digital image processing, the regularlization of ill posed problems and numerical analysis of boundary value problems. This book provides an invaluable resource for researchers and professionals in applied mathematics, particularly in numerical analysis and signal processing, as well as for engineers and physicists with a strong mathematical background. Contents Notations Introduction

1. The Continuous Wavelet Transform
2. The Discrete Wavelet Transform
3. Applications of the Wavelet Transform

Appendix The Fourier Transform References Index