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Wavelets and Wavelet Transforms by C. Sidney Burrus - HTML preview
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Wavelets and Wavelet Transforms
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Wavelets and Wavelet Transforms
Table of Contents
Chapter
1
.
Preface
1.1
.
Instructions to the Reader
1.2
.
References
Chapter
2
.
Introduction to Wavelets
2.1
.
Wavelets and Wavelet Expansion Systems
What is a Wavelet Expansion or a Wavelet Transform?
What is a Wavelet System?
More Specific Characteristics of Wavelet Systems
Haar Scaling Functions and Wavelets
What do Wavelets Look Like?
Why is Wavelet Analysis Effective?
2.2
.
The Discrete Wavelet Transform
2.3
.
The Discrete-Time and Continuous Wavelet Transforms
2.4
.
Exercises and Experiments
2.5
.
This Chapter
2.6
.
References
Chapter
3
.
A multiresolution formulation of Wavelet Systems
3.1
.
Signal Spaces
3.2
.
The Scaling Function
Multiresolution Analysis
3.3
.
The Wavelet Functions
3.4
.
The Discrete Wavelet Transform
3.5
.
A Parseval's Theorem
3.6
.
Display of the Discrete Wavelet Transform and the Wavelet Expansion
3.7
.
Examples of Wavelet Expansions
3.8
.
An Example of the Haar Wavelet System
3.9
.
References
Chapter
4
.
Filter Banks and the Discrete Wavelet Transform
4.1
.
Analysis – From Fine Scale to Coarse Scale
Filtering and Down-Sampling or Decimating
4.2
.
Synthesis – From Coarse Scale to Fine Scale
Filtering and Up-Sampling or Stretching
4.3
.
Input Coefficients
4.4
.
Lattices and Lifting
4.5
.
Different Points of View
Multiresolution versus Time-Frequency Analysis
Periodic versus Nonperiodic Discrete Wavelet Transforms
The Discrete Wavelet Transform versus the Discrete-Time Wavelet Transform
Numerical Complexity of the Discrete Wavelet Transform
4.6
.
References
Chapter
5
.
Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and unconditional Bases
5.1
.
Bases, Orthogonal Bases, and Biorthogonal Bases
Matrix Examples
Fourier Series Example
Sinc Expansion Example
5.2
.
Frames and Tight Frames
Matrix Examples
Sinc Expansion as a Tight Frame Example
5.3
.
Conditional and Unconditional Bases
5.4
.
References
Chapter
6
.
The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients
6.1
.
Tools and Definitions
Signal Classes
Fourier Transforms
Refinement and Transition Matrices
6.2
.
Necessary Conditions
6.3
.
Frequency Domain Necessary Conditions
6.4
.
Sufficient Conditions
Wavelet System Design
6.5
.
The Wavelet
6.6
.
Alternate Normalizations
6.7
.
Example Scaling Functions and Wavelets
Haar Wavelets
Sinc Wavelets
Spline and Battle-Lemarié Wavelet Systems
6.8
.
Further Properties of the Scaling Function and Wavelet
General Properties not Requiring Orthogonality
Properties that Depend on Orthogonality
6.9
.
Parameterization of the Scaling Coefficients
Length-2 Scaling Coefficient Vector
Length-4 Scaling Coefficient Vector
Length-6 Scaling Coefficient Vector
6.10
.
Calculating the Basic Scaling Function and Wavelet
Successive Approximations or the Cascade Algorithm
Iterating the Filter Bank
Successive approximation in the frequency domain
The Dyadic Expansion of the Scaling Function
6.11
.
References
Chapter
7
.
Regularity, Moments, and Wavelet System Design
7.1
.
K-Regular Scaling Filters
7.2
.
Vanishing Wavelet Moments
7.3
.
Daubechies' Method for Zero Wavelet Moment Design
7.4
.
Non-Maximal Regularity Wavelet Design
7.5
.
Relation of Zero Wavelet Moments to Smoothness
7.6
.
Vanishing Scaling Function Moments
7.7
.
Approximation of Signals by Scaling Function Projection
7.8
.
Approximation of Scaling Coefficients by Samples of the Signal
7.9
.
Coiflets and Related Wavelet Systems
Coifman Wavelet Systems from a Specified Filter Length
7.10
.
Minimization of Moments Rather than Zero Moments
7.11
.
References
Chapter
8
.
Generalizations of the Basic Multiresolution Wavelet System
8.1
.
Tiling the Time–Frequency or Time–Scale Plane
Nonstationary Signal Analysis
Tiling with the Discrete-Time Short-Time Fourier Transform
Tiling with the Discrete Two-Band Wavelet Transform
General Tiling
8.2
.
Multiplicity-M (M-Band) Scaling Functions and Wavelets
Properties of M-Band Wavelet Systems
M-Band Scaling Function Design
M-Band Wavelet Design and Cosine Modulated Methods
8.3
.
Wavelet Packets
Full Wavelet Packet Decomposition
Adaptive Wavelet Packet Systems
8.4
.
Biorthogonal Wavelet Systems
Two Channel Biorthogonal Filter Banks
Biorthogonal Wavelets
Comparisons of Orthogonal and Biorthogonal Wavelets
Example Families of Biorthogonal Systems
Cohen-Daubechies-Feauveau Family of Biorthogonal Spline Wavelets
Cohen-Daubechies-Feauveau Family of Biorthogonal Wavelets with Less Dissimilar Filter Length
Tian-Wells Family of Biorthogonal Coiflets
Lifting Construction of Biorthogonal Systems
8.5
.
Multiwavelets
Construction of Two-Band Multiwavelets
Properties of Multiwavelets
Approximation, Regularity and Smoothness
Support
Orthogonality
Implementation of Multiwavelet Transform
Examples
Geronimo-Hardin-Massopust Multiwavelets
Spline Multiwavelets
Other Constructions
Applications
8.6
.
Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases
Overcomplete Representations
A Matrix Example
Shift-Invariant Redundant Wavelet Transforms and Nondecimated Filter Banks
Adaptive Construction of Frames and Bases
8.7
.
Local Trigonometric Bases
Nonsmooth Local Trigonometric Bases
Construction of Smooth Windows
Folding and Unfolding
Local Cosine and Sine Bases
Signal Adaptive Local Trigonometric Bases
8.8
.
Discrete Multiresolution Analysis, the Discrete-Time Wavelet Transform, and the Continuous Wavelet Transform
Discrete Multiresolution Analysis and the Discrete-Time Wavelet Transform
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