An Image Processing Tour of College Mathematics

by Yevgeniy V. Galperin

Overview

An Image Processing Tour of College Mathematics aims to provide meaningful context for reviewing key topics of the college mathematics curriculum, to help students gain confidence in using concepts and techniques of applied mathematics, to increase student awareness of recent developments in mathematical sciences, and to help students prepare for graduate studies.

The topics covered include a library of elementary functions, basic concepts of descriptive statistics, probability distributions of functions of random variables, definitions and concepts behind first- and second-order derivatives, most concepts and techniques of traditional linear algebra courses, an introduction to Fourier analysis, and a variety of discrete wavelet transforms – all of that in the context of digital image processing.

Features

• Pre-calculus material and basic concepts of descriptive statistics are reviewed in the context of image processing in the spatial domain.
• Key concepts of linear algebra are reviewed both in the context of fundamental operations with digital images and in the more advanced context of discrete wavelet transforms.
• Some of the key concepts of probability theory are reviewed in the context of image equalization and histogram matching.
• The convolution operation is introduced painlessly and naturally in the context of naïve filtering for denoising and is subsequently used for edge detection and image restoration.
• An accessible elementary introduction to Fourier analysis is provided in the context of image restoration.
• Discrete wavelet transforms are introduced in the context of image compression, and the readers become more aware of some of the recent developments in applied mathematics.
• This text helps students of mathematics ease their way into mastering the basics of scientific computer programming.

Product Details

ISBN-13:	9780367694487
Publisher:	CRC Press
Publication date:	08/01/2022
Series:	Chapman & Hall/CRC Mathematical and Computational Imaging Sciences Series
Pages:	348
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Yevgeniy V. Galperin is Associate Professor of Mathematics at East Stroudsburg University of Pennsylvania. He holds a PhD in mathematics and has published several papers in the field of time-frequency analysis and related areas of Fourier analysis. His research and academic interests also include numerical methods, simulation of stochastic processes for real-life applications, and mathematical pedagogy. He has given numerous conference presentations on instructional and course-design approaches directed at increasing student motivation and awareness of societal value of mathematics and on incorporating signal and image processing into the undergraduate mathematics curriculum.

Table of Contents

Preface ix

1 Introduction to Basics of Digital Images 1

1.1 Grayscale Digital Images 1

1.2 Working with Images in MATLAB® 3

1.3 Images and Statistical Description of Quantitative Data 7

1.3.1 Image Histograms 7

1.3.2 Measures of Center and Spread 10

1.4 Color Images and Color Spaces 16

2 A Library of Elementary Functions 21

2.1 Introduction 21

2.2 Power Functions and Gamma-Correction 22

2.3 Exponential Functions and Image Transformations 30

2.4 Logarithmic Functions and Image Transformations 35

2.5 Linear Functions and Contrast Stretching 41

2.6 Automation of Image Enhancement 50

3 Probability, Random Variables, and Histogram Processing 55

3.1 Introduction 55

3.2 Discrete and Continuous Random Variables 56

3.3 Transformation of Random Variables 63

3.4 Image Equalization and Histogram Matching 69

4 Matrices and Linear Transformations 75

4.1 Basic Operations on Matrices 75

4.2 Linear Transformations and Their Matrices 85

4.3 Homogeneous Coordinates and Projective Transformations 96

5 Convolution and Image Filtering 103

5.1 Image Blurring and Noise Reduction 103

5.2 Convolution: Definitions and Examples 112

5.2.1 Discrete Linear Convolution 113

5.2.2 Circular Convolution 118

5.2.3 Algebraic Properties of Convolution 121

5.2.4 Convolution as a Linear Transformation 122

5.2.5 Convolution in Two Dimensions 124

5.3 Edge Detection 132

5.3.1 Partial Derivatives and the Gradient Edge Detector 133

5.3.2 Directional Derivatives and the Roberts Cross Operator 136

5.3.3 The Prewitt and Sobel Edge Detectors 137

5.3.4 Laplacian Edge Detection 140

5.3.5 Edge Detection in Noisy Images 143

5.3.6 Boolean Convolution and Edge Dilation 145

5.4 Chapter Summary 150

6 Analysis and Processing in the Frequency Domain 151

6.1 Introduction 151

6.2 Frequency Analysis of Continuous Periodic Signals 153

6.2.1 Trigonometric Fourier Coefficients of 1-Periodic Signals 154

6.2.2 A Refresher on Complex Numbers 169

6.2.3 Complex Fourier Coefficients 176

6.2.4 Properties of Fourier Coefficients 182

6.2.5 T-Periodic Signals 189

6.3 Inner Products, Orthogonal Bases, and Fourier Coefficients 196

6.4 Discrete Fourier Transform 208

6.4.1 Discrete Periodic Sequences 208

6.4.2 DFT: Definition, Examples, and Basic Properties 215

6.4.3 Placing the DFT on a Firm Foundation 231

6.4.4 Linear Time-Invariant Transformations and the DFT 234

6.5 Discrete Fourier Transform in 2D 242

6.5.1 Definition, Examples, and Properties 242

6.5.2 Frequency Domain Processing of Digital Images 245

6.6 Chapter Summary 257

7 Wavelet-Baaed Methods in Image Compression 259

7.1 Introduction 259

7.2 Naive Compression in One Dimension 260

7.3 Entropy and Entropy Encoding 263

7.4 The Discrete Haar Wavelet Transform 267

7.5 Haar Wavelet Transforms of Digital Images 272

7.6 Discrete-Time Fourier Transform 281

7.7 From the Haar Transform to Daubechies Transforms 286

7.8 Biorthogonal Wavelet Transforms 297

7.8.1 Biorthogonal Spline Filters 300

7.8.2 Daubechies Theorem for Biorthogonal Spline Filters 309

7.8.3 The CDF97 Transform 311

7.9 An Overview of JPEG2000 316

7.10 Other Applications of Wavelet Transforms 318

7.10.1 Wavelet-Based Edge Detection 318

7.10.2 Wavelet-Based Image Denoting 321

Bibliography 329

Index 333

Editorial Reviews

"Each chapter includes a mixture of theoretical exercises and computational exercises to be done using MATLAB. [. . .] This textbook should be of interest to instructors who want to teach an introductory course on the mathematics Fourier and wavelet transform methods for image processing at the advanced undergraduate level."
– Mathematical Association of America

From the Publisher