An Image Processing Tour of College Mathematics

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.

1. Introduction to The Basics of Digital Images. 1.1. Grayscale Digital Images. 1.2. Working with Images in MATLAB. 1.3. Images and Statistical Description of Quantitative Data. 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: Definitions and Examples. 5.3. Edge Detection. 5.4. Chapter Summary. 6. Analysis and Processing in the Frequency Domain. 6.1. Introduction. 6.2. Frequency Analysis of Continuous Periodic Signals. 6.3. Inner Products, Orthogonal Bases, and Fourier Coefficients. 6.4. Discrete Fourier Transform. 6.5 Discrete Fourier Transform in 2D. 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. 7.9. An Overview of JPEG2000. 7.10. Other Applications of Wavelet Transforms.