Wavelets
Theory and Applications
Wiley Series in Pure and Applied Mathematics
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
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.
