Wavelets

by Najmi, Amir-homayoon

• ISBN13: 9781421404967
• ISBN10: 1421404966
• Format: Paperback
• Copyright: 2012-02-24
• Publisher: Johns Hopkins Univ Pr

Summary

Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction properties. Amir-Homayoon Najmi's introduction to wavelet theory explains this mathematical concept clearly and succinctly. Wavelets are used in processing digital signals and imagery from myriad sources. They form the backbone of the JPEG2000 compression standard, and the Federal Bureau of Investigation uses biorthogonal wavelets to compress and store its vast database of fingerprints. Najmi provides the mathematics that demonstrate how wavelets work, describes how to construct them, and discusses their importance as a tool to investigate and process signals and imagery. He reviews key concepts such as frames, localizing transforms, orthogonal and biorthogonal bases, and multi-resolution. His examples include the Haar, the Shannon, and the Daubechies families of orthogonal and biorthogonal wavelets. Our capacity and need for collecting and transmitting digital data is increasing at an astonishing rate. So too is the importance of wavelets to anyone working with and analyzing digital data. Najmi's primer will be an indispensable resource for those in computer science, the physical sciences, applied mathematics, and engineering who wish to obtain an in-depth understanding and working knowledge of this fascinating and evolving field.

Author Biography

Amir-Homayoon Najmi completed the Mathematical Tripos at Cambridge University and obtained his D.Phil. at Oxford University. He is with the Johns Hopkins University's Applied Physics Laboratory and is a faculty member of the Whiting School of Engineering CE programs in applied physics and electrical engineering.

Table of Contents

List of Figures	p. xiii
List of Acronyms	p. xix
Preface	p. xxi
Acknowledgments	p. xxix
Analysis in Vector and Function Spaces	p. 1
Introduction	p. 1
The Lebesgue Integral	p. 3
Discrete Time Signals	p. 5
Vector Spaces	p. 5
Linear Independence	p. 6
Bases and Basis Vectors	p. 7
Normed Vector Spaces	p. 8
Inner Product	p. 9
Banach and Hilbert Spaces	p. 12
Linear Operators, Operator Norm, the Adjoint Operator	p. 13
Reproducing Kernel Hilbert Space	p. 15
The Dirac Delta Distribution	p. 18
Orthonormal Vectors	p. 20
Orthogonal Projections	p. 21
Multi-Resolution Analysis Subspaces	p. 22
Complete and Orthonormal Bases in L2 (R)	p. 25
The Dirac Notation	p. 28
The Fourier Transform	p. 31
The Fourier Series Expansion	p. 34
The Discrete Time Fourier Transform	p. 36
The Discrete Fourier Transform	p. 37
Band-Limited Functions and the Sampling Theorem	p. 38
The Basis Operator in L2(R)	p. 41
Biorthogonal Bases and Representations in L2 (R)	p. 43
Frames in a Finite Dimensional Vector Space	p. 45
Frames in L2 (R)	p. 50
Dual Frame Construction Algorithm	p. 54
Exercises	p. 56
Linear Time-Invariant Systems	p. 59
Introduction	p. 59
Convolution in Continuous Time	p. 59
Convolution in Discrete Time	p. 60
Convolution of Finite Length Sequences	p. 61
Linear Time-Invariant Systems and the Z Transform	p. 63
Spectral Factorization for Finite Length Sequences	p. 66
Perfect Reconstruction Quadrature Mirror Filters	p. 68
Exercises	p. 73
Time, Frequency, and Scale Localizing Transforms	p. 75
Introduction	p. 75
The Windowed Fourier Transform	p. 79
The Windowed Fourier Transform Inverse	p. 81
The Range Space of the Windowed Fourier Transform	p. 81
The Discretized Windowed Fourier Transform	p. 83
Time-Frequency Resolution of the Windowed Fourier Transform	p. 88
The Continuous Wavelet Transform	p. 90
The Continuous Wavelet Transform Inverse	p. 93
The Range Space of the Continuous Wavelet Transform	p. 95
The Morlet, the Mexican Hat, and the Haar Wavelets	p. 96
Discretizing the Continuous Wavelet Transform	p. 101
Algorithm A' Trous	p. 104
The Morlet Scalogram	p. 107
Exercises	p. 110
The Haar and Shannon Wavelets	p. 111
Introduction	p. 111
Haar Multi-Resolution Analysis Subspaces	p. 112
Summary and Generalization of Results	p. 119
The Spectra of the Haar Filter Coefficients	p. 122
Half-Band Finite Impulse Response Filters	p. 124
The Shannon Scaling Function	p. 125
The Spectrum of the Shannon Filter Coefficients	p. 130
Meyer's Wavelet	p. 131
Exercises	p. 133
General Properties of Scaling and Wavelet Functions	p. 135
Introduction	p. 135
Multi-Resolution Analysis Spaces	p. 135
The Inverse Relations	p. 140
The Shift-Invariant Discrete Wavelet Transform	p. 143
Time Domain Properties	p. 145
Examples of Finite Length Filter Coefficients	p. 149
Frequency Domain Relations	p. 150
Orthogonalization of a Basis Set: b1 Spline Wavelet	p. 157
The Cascade Algorithm	p. 159
Biorthogonal Wavelets	p. 163
Multi-Resolution Analysis Using Biorthogonal Wavelets	p. 167
Exercises	p. 170
Discrete Wavelet Transform of Discrete Time Signals	p. 173
Introduction	p. 173
Discrete Time Data and Scaling Function Expansions	p. 174
Implementing the DWT for Even Length h0 Filters	p. 179
Denoising and Thresholding	p. 185
Biorthogonal Wavelets of Compact Support	p. 187
The Lazy Filters	p. 191
Exercises	p. 191
Wavelet Regularity and Daubechies Solutions	p. 193
Introduction	p. 193
Zero Moments of the Mother Wavelet	p. 194
The Form of H0(z) and the Decay Rate of  (É)	p. 199
Daubechies Orthogonal Wavelets of Compact Support	p. 200
Wavelet and Scaling Function Vanishing Moments	p. 204
Biorthogonal Wavelets of Compact Support	p. 207
Biorthogonal Spline Wavelets	p. 211
The Lifting Scheme	p. 215
Exercises	p. 217
Orthogonal Wavelet Packets	p. 221
Introduction	p. 221
Review of the Orthogonal Wavelet Transform	p. 221
Packet Functions for Orthonormal Wavelets	p. 224
Discrete Orthogonal Packet Transform of Finite Length Se-quences	p. 231
The Best Basis Algorithm	p. 236
Exercises	p. 239
Wavelet Transform in Two Dimensions	p. 241
Introduction	p. 241
The Forward Transform	p. 242
The Inverse Transform	p. 247
Implementing the Two-Dimensional Wavelet Transform	p. 248
Application to Image Compression	p. 249
Image Fusion	p. 257
Wavelet Descendants	p. 258
Exercises	p. 259
Bibliography	p. 261
Index	p. 267
Table of Contents provided by Ingram. All Rights Reserved.