An Image Processing Tour of College Mathematics

Description

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. Cover Half Title Series Page Title Page Copyright Page Contents Preface 1. Introduction to Basics of Digital Images 1.1. Grayscale Digital Images 1.2. Working with Images in MATLAB® 1.3. Images and Statistical Description of Quantitative 1.3.1. Image Histograms 1.3.2. Measures of Center and Spread 1.4. Color Images and Color Spaces 2. A Library of Elementary Functions 2.1. Introduction 2.2. Power Functions and Gamma-Correction 2.3. Exponential Functions and Image Transformations 2.4. Logarithmic Functions and Image Transformations 2.5. Linear Functions and Contrast Stretching 2.6. Automation of Image Enhancement 3. Probability, Random Variables, and Histogram Processing 3.1. Introduction 3.2. Discrete and Continuous Random Variables 3.3. Transformation of Random Variables 3.4. Image Equalization and Histogram Matching 4. Matrices and Linear Transformations 4.1. Basic Operations on Matrices 4.2. Linear Transformations and Their Matrices 4.3. Homogeneous Coordinates and Projective Transformations 5. Convolution and Image Filtering 5.1. Image Blurring and Noise Reduction 5.2. Convolution: De nitions and Examples 5.2.1. Discrete Linear Convolution 5.2.2. Circular Convolution 5.2.3. Algebraic Properties of Convolution 5.2.4. Convolution as a Linear Transformation 5.2.5. Convolution in Two Dimensions 5.3. Edge Detection 5.3.1. Partial Derivatives and the Gradient Edge Detector 5.3.2. Directional Derivatives and the Roberts Cross Operator 5.3.3. The Prewitt and Sobel Edge Detectors 5.3.4. Laplacian Edge Detection 5.3.5. Edge Detection in Noisy Images 5.3.6. Boolean Convolution and Edge Dilation 5.4. Chapter Summary 6. Analysis and Processing in the Frequency Domain 6.1. Introduction 6.2. Frequency Analysis of Continuous Periodic Signals 6.2.1. Trigonometric Fourier Coecients of 1-Periodic Signals 6.2.2. A Refresher on Complex Numbers 6.2.3. Complex Fourier Coe cients 6.2.4. Properties of Fourier Coe cients 6.2.5. T-Periodic Signals 6.3. Inner Products, Orthogonal Bases, and Fourier Coefficients 6.4. Discrete Fourier Transform 6.4.1. Discrete Periodic Sequences 6.4.2. DFT: De nition, Examples, and Basic Properties 6.4.3. Placing the DFT on a Firm Foundation 6.4.4. Linear Time-Invariant Transformations and the DFT 6.5. Discrete Fourier Transform in 2D 6.5.1. Definition, Examples, and Properties 6.5.2. Frequency Domain Processing of Digital Images 6.6. Chapter Summary 7. Wavelet-Based Methods in Image Compression 7.1. Introduction 7.2. Naive Compression in One Dimension 7.3. Entropy and Entropy Encoding 7.4. The Discrete Haar Wavelet Transform 7.5. Haar Wavelet Transforms of Digital Images 7.6. Discrete-Time Fourier Transform 7.7. From the Haar Transform to Daubechies Transforms 7.8. Biorthogonal Wavelet Transforms 7.8.1. Biorthogonal Spline Filters 7.8.2. Daubechies Theorem for Biorthogonal Spline Filters 7.8.3. The CDF97 Transform 7.9. An Overview of JPEG2000 7.10. Other Applications of Wavelet Transforms 7.10.1. Wavelet-Based Edge Detection 7.10.2. Wavelet-Based Image Denoising Bibliography Index