 |
Wavelets and Multiwavelets |
Back Cover
(with description of contents)
Errata
Known errors are listed in file errata.pdf. If you find any others, please report to the author.
Software
A toolbox of Matlab subroutines is made available with the book. Documentation (in both .pdf and .ps format) is included in the archive.
| mw.tar | Unix tar format, 960 KB |
| mw.tar.Z | compressed Unix tar format, 604 KB |
| mw.zip | PC zip archive of Matlab subroutines, 460 KB |
Table of Contents
Contents
Part I - Scalar Wavelets
1 Basic Theory
1.1 Refinable Functions
1.2 Orthogonal MRAs and Wavelets
1.3 Wavelet Decomposition
1.4 Biorthogonal MRAs and Wavelets
1.5 Moments
1.6 Approximation Order
1.7 Symmetry
1.8 Point Values and Normalization
2 Practical Computation
2.1 Discrete Wavelet Transform
2.2 Pre- and Postprocessing
2.3 Handling Boundaries
2.3.1 Data Extension Approach
2.3.2 Matrix Completion Approach
2.3.3 Boundary Function Approach
2.3.4 Further Comments
2.4 Putting It All Together
2.5 Modulation Formulation
2.6 Polyphase Formulation
2.7 Lifting
2.8 Calculating Integrals
2.8.1 Integrals with Other Refinable Functions
2.8.2 Integrals with Polynomials
2.8.3 Integrals with General Functions
3 Creating Wavelets
3.1 Completion Problem
3.1.1 Finding Wavelet Functions
3.1.2 Finding Dual Scaling Functions
3.2 Projection Factors
3.3 Techniques for Modifying Wavelets
3.4 Techniques for Building Wavelets
3.5 Bezout Equation
3.6 Daubechies Wavelets
3.6.1 Bezout Approach
3.6.2 Projection Factor Approach
3.7 Coiflets
3.7.1 Bezout Approach
3.7.2 Projection Factor Approach
3.7.3 Generalized Coiflets
3.8 Cohen Wavelets
3.9 Other Constructions
4 Applications
4.1 Signal Processing
4.1.1 Detection of Frequencies and Discontinuities
4.1.2 Signal Compression
4.1.3 Denoising
4.2 Numerical Analysis
4.2.1 Fast Matrix--Vector Multiplication
4.2.2 Fast Operator Evaluation
4.2.3 Differential and Integral Equations
5 Existence and Regularity
5.1 Distribution Theory
5.2 L^1-Theory
5.3 L^2-Theory
5.3.1 Transition Operator
5.3.2 Sobolev Space Estimates
5.3.3 Cascade Algorithm
5.4 Pointwise Theory
5.5 Smoothness and Approximation Order
5.6 Stability
Part II - Multiwavelets
6 Basic Theory
6.1 Refinable Function Vectors
6.2 MRAs and Multiwavelets
6.2.1 Orthogonal MRAs and Multiwavelets
6.2.2 Biorthogonal MRAs and Multiwavelets
6.3 Moments
6.4 Approximation Order
6.5 Point Values and Normalization
7 Practical Computation
7.1 Discrete Multiwavelet Transform
7.2 Pre- and Postprocessing
7.2.1 Interpolating Prefilters
7.2.2 Quadrature-Based Prefilters
7.2.3 Hardin--Roach Prefilters
7.2.4 Other Prefilters
7.3 Balanced Multiwavelets
7.4 Handling Boundaries
7.4.1 Data Extension Approach
7.4.2 Matrix Completion Approach
7.4.3 Boundary Function Approach
7.5 Putting It All Together
7.6 Modulation Formulation
7.7 Polyphase Formulation
7.8 Calculating Integrals
7.8.1 Integrals with Other Refinable Functions
7.8.2 Integrals with Polynomials
7.8.3 Integrals with General Functions
7.9 Applications
7.9.1 Signal Processing
7.9.2 Numerical Analysis
8 Two-Scale Similarity Transforms
8.1 Regular TSTs
8.2 Singular TSTs
8.3 Multiwavelet TSTs
8.4 TSTs and Approximation Order
8.5 Symmetry
9 Factorizations of Polyphase Matrices
9.1 Projection Factors
9.1.1 Orthogonal Case
9.1.2 Biorthogonal Case
9.2 Lifting Steps
9.3 Raising Approximation Order by Lifting
10 Creating Multiwavelets
10.1 Orthogonal Completion
10.1.1 Using Projection Factors
10.1.2 Householder-Type Approach
10.2 Biorthogonal Completion
10.3 Other Approaches
10.4 Techniques for Modifying Multiwavelets
10.5 Techniques for Building Multiwavelets
11 Existence and Regularity
11.1 Distribution Theory
11.2 L^1-Theory
11.3 L^2-Theory
11.3.1 Transition Operator
11.3.2 Sobolev Space Estimates
11.3.3 Cascade Algorithm
11.4 Pointwise Theory
11.5 Smoothness and Approximation Order
11.6 Stability
A.1 Scalar Orthogonal Wavelets
A.2 Scalar Biorthogonal Wavelets
A.3 Orthogonal Multiwavelets
A.4 Biorthogonal Multiwavelets
Appendix B: Mathematical Background
B.1 Notational Conventions
B.2 Derivatives
B.3 Functions and Sequences
B.4 Fourier Transform
B.5 Laurent Polynomials
B.6 Trigonometric Polynomials
B.7 Linear Algebra
Appendix C: Computer Resources
C.1 Wavelet Internet Resources
C.2 Wavelet Software
C.3 Multiwavelet Software
References
Index