Discrete Fourier Analysis and Wavelets (eBook)

S. Allen Broughton, PhD, is Professor and Head ofMathematics at the Rose-Hulman Institute of Technology. The authoror coauthor of over twenty published articles, Dr. Broughton'sresearch interests include finite group theory, Riemann surfaces,the mathematics of image and signal processing, and wavelets. Kurt Bryan, PhD, is Professor of Mathematics at theRose-Hulman Institute of Technology. Dr. Bryan has published morethan twenty journal articles, and he currently focuses his researchon partial differential equations related to electrical and thermalimaging.

PREFACE xi

ACKNOWLEDGMENTS xv

1 VECTOR SPACES, SIGNALS, AND IMAGES 1

1.1 Overview 1

1.2 Some Common Image Processing Problems 1

1.3 Signals and Images 3

1.4 Vector Space Models for Signals and Images 9

1.5 Basic Waveforms--The Analog Case 17

1.6 Sampling and Aliasing 21

1.7 Basic Waveforms--The Discrete Case 26

1.8 Inner Product Spaces and Orthogonality 29

1.9 Signal and Image Digitization 41

1.10 Infinite-dimensional Inner Product Spaces 46

1.11 Matlab Project 56

Exercises 61

2 THE DISCRETE FOURIER TRANSFORM 71

2.1 Overview 71

2.2 The Time Domain and Frequency Domain 72

2.3 A Motivational Example 73

2.4 The One-dimensional DFT 78

2.5 Properties of the DFT 85

2.6 The Fast Fourier Transform 90

2.7 The Two-dimensional DFT 93

2.8 Matlab Project 97

Exercises 101

3 THE DISCRETE COSINE TRANSFORM 105

3.1 Motivation for the DCT--Compression 105

3.2 Other Compression Issues 106

3.3 Initial Examples--Thresholding 107

3.4 The Discrete Cosine Transform 113

3.5 Properties of the DCT 117

3.6 The Two-dimensional DCT 120

3.7 Block Transforms 122

3.8 JPEG Compression 124

3.9 Matlab Project 132

Exercises 134

4 CONVOLUTION AND FILTERING 138

4.1 Overview 138

4.2 One-dimensional Convolution 138

4.3 Convolution Theorem and Filtering 145

4.4 2D Convolution--Filtering Images 150

4.5 Infinite and Bi-infinite Signal Models 156

4.6 Matlab Project 171

Exercises 174

5 WINDOWING AND LOCALIZATION 182

5.1 Overview: Nonlocality of the DFT 182

5.2 Localization via Windowing 184

5.3 Matlab Project 195

Exercises 197

6 FILTER BANKS 201

6.1 Overview 201

6.2 The Haar Filter Bank 202

6.3 The General One-stage Two-channel Filter Bank 210

6.4 Multistage Filter Banks 214

6.5 Filter Banks for Finite Length Signals 218

6.6 The 2D Discrete Wavelet Transform and JPEG 2000 231

6.7 Filter Design 239

6.8 Matlab Project 251

6.9 Alternate Matlab Project 255

Exercises 258

7 WAVELETS 267

7.1 Overview 267

7.2 The Haar Basis 269

7.3 Haar Wavelets versus the Haar Filter Bank 282

7.4 Orthogonal Wavelets 292

7.5 Biorthogonal Wavelets 314

7.6 Matlab Project 318

Exercises 321

REFERENCES 327

SOLUTIONS 329

INDEX 335

"Anyone seeking to understand the process and problems ofimage and signal analysis would do well to read this work. SummingUp: Highly recommended." (Choice Reviews, June 2009)

"There seems to be a shortage of books that deliver anappropriate mix of theory and applications to an undergraduate mathmajor. I believe that Discrete Fourier Analysis and Wavelets,Applications to Signal and Image Processing helps fill thisvoid...This book is enjoyable to read and pulls together a varietyof important topics in the subject at a level that upper levelundergraduate mathematics students can understand." (MAAReviews 2009)