Discrete Fourier Analysis and Wavelets : Applications to Signal and Image Processing

by Broughton, S. Allen; Bryan, Kurt M.

• ISBN13: 9780470294666
• ISBN10: 0470294663
• Format: Hardcover
• Copyright: 2008-11-10
• Publisher: Wiley-Interscience

Summary

This book presents a thorough introduction to the mathematical foundations of signal and image processing. Key concepts and applications are addressed in a thought-provoking manner and are implemented using vector, matrix, and linear algebra methods. With a focus on mathematical theory and computational techniques, it equips readers with the knowledge needed to transition smoothly from mathematical models to practical digital data applications. Examples using image compression demonstrate how mathematical theory translates into application, while each chapter provides a series of exercises as well as a MATLABÆ project.

Author Biography

S. Allen Broughton, PhD, is Professor and Head of Mathematics at the Rose-Hulman Institute of Technology. The author or coauthor of over twenty published articles, Dr. Broughton's research interests include finite group theory, Riemann surfaces, the mathematics of image and signal processing, and wavelets.

Kurt Bryan, PhD, is Professor of Mathematics at the Rose-Hulman Institute of Technology. Dr. Bryan has published more than twenty journal articles, and he currently focuses his research on partial differential equations related to electrical and thermal imaging.

Table of Contents

Preface	p. xi
Acknowledgments	p. xv
Vector Spaces, Signals, and Images
Overview	p. 1
Some Common Image Processing Problems	p. 1
Signals and Images	p. 3
Vector Space Models for Signals and Images	p. 9
Basic Waveforms-The Analog Case	p. 17
Sampling and Aliasing	p. 21
Basic Waveforms-The Discrete Case	p. 26
Inner Product Spaces and Orthogonality	p. 29
Signal and Image Digitization	p. 41
Infinite-dimensional Inner Product Spaces	p. 46
Matlab Project	p. 56
Exercises	p. 61
The Discrete Fourier Transform	p. 71
Overview	p. 71
The Time Domain and Frequency Domain	p. 72
A Motivational Example	p. 73
The One-dimensional DFT	p. 78
Properties of the DFT	p. 85
The Fast Fourier Transform	p. 90
The Two-dimensional DFT	p. 93
Matlab Project	p. 97
Exercises	p. 101
The Discrete Cosine Transform	p. 105
Motivation for the DCT-Compression	p. 105
Other Compression Issues	p. 106
Initial Examples-Thresholding	p. 107
The Discrete Cosine Transform	p. 113
Properties of the DCT	p. 117
The Two-dimensional DCT	p. 120
Block Transforms	p. 122
JPEG Compression	p. 124
Matlab Project	p. 132
Exercises	p. 134
Convolution and Filtering	p. 138
Overview	p. 138
One-dimensional Convolution	p. 138
Convolution Theorem and Filtering	p. 145
2D Convolution-Filtering Images	p. 150
Infinite and Bi-infinite Signal Models	p. 156
Matlab Project	p. 171
Exercises	p. 174
Windowing and Localization	p. 182
Overview: Nonlocality of the DFT	p. 182
Localization via Windowing	p. 184
Matlab Project	p. 195
Exercises	p. 197
Filter Banks	p. 201
Overview	p. 201
The Haar Filter Bank	p. 202
The General One-stage Two-channel Filter Bank	p. 210
Multistage Filter Banks	p. 214
Filter Banks for Finite Length Signals	p. 218
The 2D Discrete Wavelet Transform and JPEG 2000	p. 231
Filter Design	p. 239
Matlab Project	p. 251
Alternate Matlab Project	p. 255
Exercises	p. 258
Wavelets	p. 267
Overview	p. 267
The Haar Basis	p. 269
Haar Wavelets versus the Haar Filter Bank	p. 282
Orthogonal Wavelets	p. 292
Biorthogonal Wavelets	p. 314
Matlab Project	p. 318
Exercises	p. 321
References	p. 327
Solutions	p. 329
Index	p. 335
Table of Contents provided by Ingram. All Rights Reserved.