Burrus C.S., Gopinath R.A., Guo H. Introduction to wavelets and wavelet transforms: a primer

Prentice Hall Inc., 1998. — 282 p.This book develops the ideas behind and properties of wavelets and shows how they can be used as analytical tools for signal processing, numerical analysis, and mathematical modeling. We try to present this in a way that is accessible to the engineer, scientist, and applied mathematician both as a theoretical approach and as a potentially practical method to solve problems. Although the roots of this subject go back some time, the modern interest and development have a history of only a few years. The goal of most modern wavelet research is to create a set of basis functions (or general expansion functions) and transforms that will give an informative, efficient, and useful description of a function or signal. If the signal is represented as a function of time, wavelets provide efficient localization in both time and frequency or scale. Another central idea is that of multiresolution where the decomposition of a signal is in terms of the resolution of detail.
Wavelet-based analysis is an exciting new problem-solving tool for the mathematician, scientist, and engineer. It fits naturally with the digital computer with its basis functions defined by summations not integrals or derivatives. Unlike most traditional expansion systems, the basis functions of the wavelet analysis are not solutions of differential equations. In some areas, it is the first truly new tool we have had in many years. Indeed, use of wavelets and wavelet transforms requires a new point of view and a new method of interpreting representations that we are still learning how to exploit.Preface.
Introduction to Wavelets.
Wavelets and Wavelet Expansion Systems.
The Discrete Wavelet Transform.
The Discrete-Time and Continuous Wavelet Transforms.
Exercises and Experiments.
This Chapter
A Multiresolution Formulation of Wavelet Systems.
Signal Spaces.
The Scaling Function.
The Wavelet Functions.
The Discrete Wavelet Transform.
A Parseval's Theorem.
Display of the Discrete Wavelet Transform and the Wavelet Expansion.
Examples of Wavelet Expansions.
An Example of the Haar Wavelet System.
Filter Banks and the Discrete Wavelet Transform.
Analysis - From Fine Scale to Coarse Scale.
Synthesis - From Coarse Scale to Fine Scale.
Input Coefficients.
Lattices and Lifting.
Different Points of View.
Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases.
Bases, Orthogonal Bases, and Biorthogonal Bases.
Frames and Tight Frames.
Conditional and Unconditional Bases.
The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients.
Tools and Definitions.
Necessary Conditions.
Frequency Domain Necessary Conditions.
Sufficient Conditions.
The Wavelet.
Alternate Normalizations.
Example Scaling Functions and Wavelets.
Further Properties of the Scaling Function and Wavelet.
Parameterization of the Scaling Coefficients.
Calculating the Basic Scaling Function and Wavelet.
Regularity, Moments, and Wavelet System Design.
K-Regular Scaling Filters.
Vanishing Wavelet Moments.
Daubechies' Method for Zero Wavelet Moment Design.
Non-Maximal Regularity Wavelet Design.
Relation of Zero Wavelet Moments to Smoothness.
Vanishing Scaling Function Moments.
Approximation of Signals by Scaling Function Projection.
Approximation of Scaling Coefficients by Samples of the Signal.
Coiflets and Related Wavelet Systems.
Minimization of Moments Rather than Zero Moments.
Generalizations of the Basic Multiresolution Wavelet System.
Tiling the Time-Frequency or Time-Scale Plane.
Multiplicity-M (M-Band) Scaling Functions and Wavelets.
Wavelet Packets.
Biorthogonal Wavelet Systems.
Multiwavelets.
Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases.
Local Trigonometric Bases.
Discrete Multiresolution Analysis, the Discrete-Time Wavelet Transform, and the Continuous Wavelet Transform.
Filter Banks and Transmultiplexers.
Introduction.
Unitary Filter Banks.
Unitary Filter Banks-Some Illustrative Examples.
M-band Wavelet Tight Frames.
Modulated Filter Banks.
Modulated Wavelet Tight Frames.
Linear Phase Filter Banks.
Linear-Phase Wavelet Tight Frames.
Linear-Phase Modulated Filter Banks.
Linear Phase Modulated Wavelet Tight Frames.
Time-Varying Filter Bank Trees.
Filter Banks and Wavelets-Summary.
Calculation of the Discrete Wavelet Transform.
Finite Wavelet Expansions and Transforms.
Periodic or Cyclic Discrete Wavelet Transform.
Filter Bank Structures for Calculation of the DWT and Complexity.
The Periodic Case.
Structure of the Periodic Discrete Wavelet Transform
More General Structures
Wavelet-Based Signal Processing and Applications.
Wavelet-Based Signal Processing.
Approximate FFT using the Discrete Wavelet Transform Introduction.
Nonlinear Filtering or Denoising with the DWT.
Statistical Estimation.
Signal and Image Compression.
Why are Wavelets so Useful?
Applications.
Wavelet Software.
Summary Overview.
Properties of the Basic Multiresolution Scaling Function.
Types of Wavelet Systems.
References.
Bibliography.
Appendixes.
Derivations for Chapter 5 on Scaling Functions.
Derivations for Section on Properties.
MatLAB Programs.
Index.