Introduction to Wavelets and Wavelet Transforms A Primer

by Burrus, C. Sidney; Gopinath, Ramesh A.; Guo, Haitao

• ISBN13: 9780134896007
• ISBN10: 0134896009
• Edition: 1st
• Format: Paperback
• Copyright: 1997-08-14
• Publisher: Pearson

Summary

This book is the only source available that presents a unified view of the theory and applications of discrete and continuous- time signals.This is the only book to present the mathematical point of view, as well as the discrete-time signal processing perspective. It brings together information previously available only in research papers, in engineering and applied mathematics.Appropriate for researchers and practitioners in signal processing and applied mathematics.

Table of Contents

Preface	xi
1 Introduction to Wavelets	1	(9)
1.1 Wavelets and Wavelet Expansion Systems	2	(5)
What is a Wavelet Expansion or a Wavelet Transform?	2	(1)
What is a Wavelet System?	2	(1)
More Specific Characteristics of Wavelet Systems	3	(2)
Haar Scaling Functions and Wavelets	5	(1)
What do Wavelets Look Like?	5	(1)
Why is Wavelet Analysis Effective?	6	(1)
1.2 The Discrete Wavelet Transform	7	(1)
1.3 The Discrete-Time and Continuous Wavelet Transforms	8	(1)
1.4 Exercises and Experiments	9	(1)
1.5 This Chapter	9	(1)
2 A Multiresolution Formulation of Wavelet Systems	10	(21)
2.1 Signal Spaces	10	(1)
2.2 The Scaling Function	11	(3)
Multiresolution Analysis	12	(2)
2.3 The Wavelet Functions	14	(3)
2.4 The Discrete Wavelet Transform	17	(1)
2.5 A Parseval's Theorem	18	(1)
2.6 Display of the Discrete Wavelet Transform and the Wavelet Expansion	18	(2)
2.7 Examples of Wavelet Expansions	20	(3)
2.8 An Example of the Haar Wavelet System	23	(8)
3 Filter Banks and the Discrete Wavelet Transform	31	(10)
3.1 Analysis - From Fine Scale to Coarse Scale	31	(5)
Filtering and Down-Sampling or Decimating	32	(4)
3.2 Synthesis - From Coarse Scale to Fine Scale	36	(1)
Filtering and Up-Sampling or Stretching	36	(1)
3.3 Input Coefficients	37	(1)
3.4 Lattices and Lifting	38	(1)
3.5 Different Points of View	38	(3)
Multiresolution versus Time-Frequency Analysis	38	(1)
Periodic versus Nonperiodic Discrete Wavelet Transforms	38	(1)
The Discrete Wavelet Transform versus the Discrete-Time Wavelet Transform	39	(1)
Numerical Complexity of the Discrete Wavelet Transform	40	(1)
4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases	41	(9)
4.1 Bases, Orthogonal Bases, and Biorthogonal Bases	41	(4)
Matrix Examples	43	(1)
Fourier Series Example	44	(1)
Sinc Expansion Example	44	(1)
4.2 Frames and Tight Frames	45	(3)
Matrix Examples	46	(1)
Sinc Expansion as a Tight Frame Example	47	(1)
4.3 Conditional and Unconditional Bases	48	(2)
5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients	50	(23)
5.1 Tools and Definitions	50	(3)
Signal Classes	50	(1)
Fourier Transforms	51	(1)
Refinement and Transition Matrices	52	(1)
5.2 Necessary Conditions	53	(1)
5.3 Frequency Domain Necessary Conditions	54	(2)
5.4 Sufficient Conditions	56	(2)
Wavelet System Design	57	(1)
5.5 The Wavelet	58	(1)
5.6 Alternate Normalizations	59	(1)
5.7 Example Scaling Functions and Wavelets	59	(3)
Haar Wavelets	60	(1)
Sinc Wavelets	60	(2)
Spline and Battle-Lemarie Wavelet Systems	62	(1)
5.8 Further Properties of the Scaling Function and Wavelet	62	(3)
General Properties not Requiring Orthogonality	63	(1)
Properties that Depend on Orthogonality	64	(1)
5.9 Parameterization of the Scaling Coefficients	65	(2)
Length-2 Scaling Coefficient Vector	65	(1)
Length-4 Scaling Coefficient Vector	66	(1)
Length-6 Scaling Coefficient Vector	66	(1)
5.10 Calculating the Basic Scaling Function and Wavelet	67	(6)
Successive Approximations or the Cascade Algorithm	67	(1)
Iterating the Filter Bank	68	(1)
Successive approximations in the frequency domain	68	(2)
The Dyadic Expansion of the Scaling Function	70	(3)
6 Regularity, Moments, and Wavelet System Design	73	(25)
6.1 K-Regular Scaling Filters	73	(2)
6.2 Vanishing Wavelet Moments	75	(1)
6.3 Daubechies' Method for Zero Wavelet Moment Design	76	(7)
6.4 Non-Maximal Regularity Wavelet Design	83	(1)
6.5 Relation of Zero Wavelet Moments to Smoothness	83	(3)
6.6 Vanishing Scaling Function Moments	86	(1)
6.7 Approximation of Signals by Scaling Function Projection	86	(1)
6.8 Approximation of Scaling Coefficients by Samples of the Signal	87	(1)
6.9 Coiflets and Related Wavelet Systems	88	(9)
Generalized Coifman Wavelet Systems	93	(4)
6.10 Minimization of Moments Rather than Zero Moments	97	(1)
7 Generalizations of the Basic Multiresolution Wavelet System	98	(50)
7.1 Tiling the Time-Frequency or Time-Scale Plane	98	(4)
Nonstationary Signal Analysis	99	(1)
Tiling with the Discrete-Time Short-Time Fourier Transform	100	(1)
Tiling with the Discrete Two-Band Wavelet Transform	100	(1)
General Tiling	101	(1)
7.2 Multiplicity-M (M-Band) Scaling Functions and Wavelets	102	(8)
Properties of M-Band Wavelet Systems	103	(6)
M-Band Scaling Function Design	109	(1)
M-Band Wavelet Design and Cosine Modulated Methods	110	(1)
7.3 Wavelet Packets	110	(4)
Full Wavelet Packet Decomposition	110	(1)
Adaptive Wavelet Packet Systems	111	(3)
7.4 Biorthogonal Wavelet Systems	114	(8)
Two-Channel Biorthogonal Filter Banks	114	(2)
Biorthogonal Wavelets	116	(1)
Comparisons of Orthogonal and Biorthogonal Wavelets	117	(1)
Example Families of Biorthogonal Systems	118	(1)
Cohen-Daubechies-Feauveau Family of Biorthogonal Spline Wavelets	118	(1)
Cohen-Daubechies-Feauveau Family of Biorthogonal Wavelets with Less Dissimilar Filter Length	118	(1)
Tian-Wells Family of Biorthogonal Coiflets	119	(1)
Lifting Construction of Biorthogonal Systems	119	(3)
7.5 Multiwavelets	122	(6)
Construction of Two-Band Multiwavelets	123	(1)
Properties of Multiwavelets	124	(1)
Approximation, Regularity and Smoothness	124	(1)
Support	124	(1)
Orthogonality	125	(1)
Implementation of Multiwavelet Transform	125	(1)
Examples	126	(1)
Geronimo-Hardin-Massopust Multiwavelets	126	(1)
Spline Multiwavelets	127	(1)
Other Constructions	127	(1)
Applications	128	(1)
7.6 Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases	128	(6)
Overcomplete Representations	129	(1)
A Matrix Example	129	(3)
Shift-Invariant Redundant Wavelet Transforms and Nondecimated Filter Banks	132	(1)
Adaptive Construction of Frames and Bases	133	(1)
7.7 Local Trigonometric Bases	134	(7)
Nonsmooth Local Trigonometric Bases	136	(1)
Construction of Smooth Windows	136	(1)
Folding and Unfolding	137	(2)
Local Cosine and Sine Bases	139	(2)
Signal Adaptive Local Trigonometric Bases	141	(1)
7.8 Discrete Multiresolution Analysis, the Discrete-Time Wavelet	141	(7)
Transform, and the Continuous Wavelet Transform	141	(2)
Discrete Multiresolution Analysis and the Discrete-Time Wavelet Transform	143	(1)
Continuous Wavelet Transforms	144	(1)
Analogies between Fourier Systems and Wavelet Systems	145	(3)
8 Filter Banks and Transmultiplexers	148	(40)
8.1 Introduction	148	(7)
The Filter Bank	148	(2)
Transmultiplexer	150	(1)
Perfect Reconstruction--A Closer Look	150	(1)
Direct Characterization of PR	150	(2)
Matrix characterization of PR	152	(1)
Polyphase (Transform-Domain) Characterization of PR	153	(2)
8.2 Unitary Filter Banks	155	(5)
8.3 Unitary Filter Banks--Some Illustrative Examples	160	(2)
8.4 M-band Wavelet Tight Frames	162	(2)
8.5 Modulated Filter Banks	164	(4)
Unitary Modulated Filter Bank	167	(1)
8.6 Modulated Wavelet Tight Frames	168	(1)
8.7 Linear Phase Filter Banks	169	(7)
Characterization of Unitary H(p)(z) -- PS Symmetry	173	(1)
Characterization of Unitary H(p)(z) -- PCS Symmetry	174	(1)
Characterization of Unitary H(p)(z) -- Linear-Phase Symmetry	174	(1)
Characterization of Unitary H(p)(z) -- Linear Phase and PCS Symmetry	175	(1)
Characterization of Unitary H(p)(z) -- Linear Phase and PS Symmetry	175	(1)
8.8 Linear-Phase Wavelet Tight Frames	176	(1)
8.9 Linear-Phase Modulated Filter Banks	177	(1)
DCT/DST I/II based 2M Channel Filter Bank	178	(1)
8.10 Linear Phase Modulated Wavelet Tight Frames	178	(1)
8.11 Time-Varying Filter Bank Trees	179	(7)
Growing a Filter Bank Tree	182	(1)
Pruning a Filter Bank Tree	182	(1)
Wavelet Bases for the Interval	183	(1)
Wavelet Bases for L(2)([0,XXX))	183	(1)
Wavelet Bases for L(2)((-XXX,0])	184	(1)
Segmented Time-Varying Wavelet Packet Bases	185	(1)
8.12 Filter Banks and Wavelets--Summary	186	(2)
9 Calculation of the Discrete Wavelet Transform	188	(8)
9.1 Finite Wavelet Expansions and Transforms	188	(2)
9.2 Periodic or Cyclic Discrete Wavelet Transform	190	(1)
9.3 Filter Bank Structures for Calculation of the DWT and Complexity	191	(1)
9.4 The Periodic Case	192	(2)
9.5 Structure of the Periodic Discrete Wavelet Transform	194	(1)
9.6 More General Structures	195	(1)
10 Wavelet-Based Signal Processing and Applications	196	(28)
10.1 Wavelet-Based Signal Processing	196	(1)
10.2 Approximate FFT using the Discrete Wavelet Transform	197	(8)
Introduction	197	(1)
Review of the Discrete Fourier Transform and FFT	198	(2)
Review of the Discrete Wavelet Transform	200	(1)
The Algorithm Development	201	(2)
Computational Complexity	203	(1)
Fast Approximate Fourier Transform	203	(1)
Computational Complexity	203	(1)
Noise Reduction Capacity	204	(1)
Summary	204	(1)
10.3 Nonlinear Filtering or Denoising with the DWT	205	(6)
Denoising by Thresholding	206	(1)
Shift-Invariant or Nondecimated Discrete Wavelet Transform	207	(2)
Combining the Shensa-Beylkin-Mallat-a trous Algorithms and Wavelet Denoising	209	(1)
Performance Analysis	209	(1)
Examples of Denoising	210	(1)
10.4 Statistical Estimation	211	(1)
10.5 Signal and Image Compression	212	(5)
Fundamentals of Data Compression	212	(1)
Prototype Transform Coder	213	(2)
Improved Wavelet Based Compression Algorithms	215	(1)
10.6 Why are Wavelets so Useful?	216	(1)
10.7 Applications	217	(1)
Numerical Solutions to Partial Differential Equations	217	(1)
Seismic and Geophysical Signal Processing	217	(1)
Medical and Biomedical Signal and Image Processing	218	(1)
Application in Communications	218	(1)
Fractals	218	(1)
10.8 Wavelet Software	218	(1)
11 Summary Overview	219	(5)
11.1 Properties of the Basic Multiresolution Scaling Function	219	(2)
11.2 Types of Wavelet Systems	221	(2)
12 References	223	(1)
Bibliography	224	(42)
Appendix A. Derivations for Chapter 5 on Scaling Functions	246	(7)
Appendix B. Derivations for Section on Properties	253	(5)
Appendix C. Matlab Programs	258	(8)
Index	266