Wavelets A Concise Guide

ISBN-10: 1421404966

ISBN-13: 9781421404967

Edition: 2012

List price: $35.50
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Description:

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.
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Book details

List price: $35.50
Copyright year: 2012
Publisher: Johns Hopkins University Press
Publication date: 4/20/2012
Binding: Paperback
Pages: 304
Size: 6.00" wide x 9.00" long x 0.75" tall
Weight: 0.924
Language: English

List of Figures
List of Acronyms
Preface
Acknowledgments
Analysis in Vector and Function Spaces
Introduction
The Lebesgue Integral
Discrete Time Signals
Vector Spaces
Linear Independence
Bases and Basis Vectors
Normed Vector Spaces
Inner Product
Banach and Hilbert Spaces
Linear Operators, Operator Norm, the Adjoint Operator
Reproducing Kernel Hilbert Space
The Dirac Delta Distribution
Orthonormal Vectors
Orthogonal Projections
Multi-Resolution Analysis Subspaces
Complete and Orthonormal Bases in L<sub>2</sub> (R)
The Dirac Notation
The Fourier Transform
The Fourier Series Expansion
The Discrete Time Fourier Transform
The Discrete Fourier Transform
Band-Limited Functions and the Sampling Theorem
The Basis Operator in L<sub>2</sub>(R)
Biorthogonal Bases and Representations in L<sub>2</sub> (R)
Frames in a Finite Dimensional Vector Space
Frames in L<sub>2</sub> (R)
Dual Frame Construction Algorithm
Exercises
Linear Time-Invariant Systems
Introduction
Convolution in Continuous Time
Convolution in Discrete Time
Convolution of Finite Length Sequences
Linear Time-Invariant Systems and the Z Transform
Spectral Factorization for Finite Length Sequences
Perfect Reconstruction Quadrature Mirror Filters
Exercises
Time, Frequency, and Scale Localizing Transforms
Introduction
The Windowed Fourier Transform
The Windowed Fourier Transform Inverse
The Range Space of the Windowed Fourier Transform
The Discretized Windowed Fourier Transform
Time-Frequency Resolution of the Windowed Fourier Transform
The Continuous Wavelet Transform
The Continuous Wavelet Transform Inverse
The Range Space of the Continuous Wavelet Transform
The Morlet, the Mexican Hat, and the Haar Wavelets
Discretizing the Continuous Wavelet Transform
Algorithm A' Trous
The Morlet Scalogram
Exercises
The Haar and Shannon Wavelets
Introduction
Haar Multi-Resolution Analysis Subspaces
Summary and Generalization of Results
The Spectra of the Haar Filter Coefficients
Half-Band Finite Impulse Response Filters
The Shannon Scaling Function
The Spectrum of the Shannon Filter Coefficients
Meyer's Wavelet
Exercises
General Properties of Scaling and Wavelet Functions
Introduction
Multi-Resolution Analysis Spaces
The Inverse Relations
The Shift-Invariant Discrete Wavelet Transform
Time Domain Properties
Examples of Finite Length Filter Coefficients
Frequency Domain Relations
Orthogonalization of a Basis Set: b1 Spline Wavelet
The Cascade Algorithm
Biorthogonal Wavelets
Multi-Resolution Analysis Using Biorthogonal Wavelets
Exercises
Discrete Wavelet Transform of Discrete Time Signals
Introduction
Discrete Time Data and Scaling Function Expansions
Implementing the DWT for Even Length h0 Filters
Denoising and Thresholding
Biorthogonal Wavelets of Compact Support
The Lazy Filters
Exercises
Wavelet Regularity and Daubechies Solutions
Introduction
Zero Moments of the Mother Wavelet
The Form of H<sub>0</sub>(z) and the Decay Rate of �(�)
Daubechies Orthogonal Wavelets of Compact Support
Wavelet and Scaling Function Vanishing Moments
Biorthogonal Wavelets of Compact Support
Biorthogonal Spline Wavelets
The Lifting Scheme
Exercises
Orthogonal Wavelet Packets
Introduction
Review of the Orthogonal Wavelet Transform
Packet Functions for Orthonormal Wavelets
Discrete Orthogonal Packet Transform of Finite Length Se-quences
The Best Basis Algorithm
Exercises
Wavelet Transform in Two Dimensions
Introduction
The Forward Transform
The Inverse Transform
Implementing the Two-Dimensional Wavelet Transform
Application to Image Compression
Image Fusion
Wavelet Descendants
Exercises
Bibliography
Index
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