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

ISBN: 978-1-118-03244-2

Feb 2011

360 pages

Select type: O-Book

## Description

A thorough guide to the classical and contemporary mathematical methods of modern signal and image processing

Discrete Fourier Analysis and Wavelets 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 balanced focus on mathematical theory and computational techniques, this self-contained book equips readers with the essential knowledge needed to transition smoothly from mathematical models to practical digital data applications.

The book first establishes a complete vector space and matrix framework for analyzing signals and images. Classical methods such as the discrete Fourier transform, the discrete cosine transform, and their application to JPEG compression are outlined followed by coverage of the Fourier series and the general theory of inner product spaces and orthogonal bases. The book then addresses convolution, filtering, and windowing techniques for signals and images. Finally, modern approaches are introduced, including wavelets and the theory of filter banks as a means of understanding the multiscale localized analysis underlying the JPEG 2000 compression standard.

Throughout the book, examples using image compression demonstrate how mathematical theory translates into application. Additional applications such as progressive transmission of images, image denoising, spectrographic analysis, and edge detection are discussed. Each chapter provides a series of exercises as well as a MATLAB project that allows readers to apply mathematical concepts to solving real problems. Additional MATLAB routines are available via the book's related Web site.

With its insightful treatment of the underlying mathematics in image compression and signal processing, Discrete Fourier Analysis and Wavelets is an ideal book for mathematics, engineering, and computer science courses at the upper-undergraduate and beginning graduate levels. It is also a valuable resource for mathematicians, engineers, and other practitioners who would like to learn more about the relevance of mathematics in digital data processing.

## Related Resources

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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 of image and signal analysis would do well to read this work. Summing Up: Highly recommended.” (Choice Reviews, June 2009)

"There seems to be a shortage of books that deliver an appropriate mix of theory and applications to an undergraduate math major. I believe that Discrete Fourier Analysis and Wavelets, Applications to Signal and Image Processing helps fill this void...This book is enjoyable to read and pulls together a variety of important topics in the subject at a level that upper level undergraduate mathematics students can understand." (MAA Reviews 2009)