An Image Processing Tour of College Mathematics

Yevgeniy V. Galperin

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN:
  9780367694487
  9780367002022
• Edition: 1
• eBook ISBN:
  9780429683213
• Topic: Mathematics
• Subtopic: Data Processing
• Index: Mathematics

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.

Chapter 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 Data
  • 1.3.1 Image Histograms
  • 1.3.2 Measures of Center and Spread
• 1.4 Color Images and Color Spaces

In this chapter, the reader will master the basics of working with grayscale and color images in MATLAB. Also discussed are the fundamental statistical characteristics of digital images. Although quite brief, this chapter provides the necessary technical preliminaries for all the practical exercises in the subsequent chapters.

1.1 Grayscale Digital Images

For our purposes, a grayscale digital image is just an $M\times N$ matrix

$$\begin{array}{l}A=\left[\begin{array}{ccccc}{a}_{1,1}& a_{1,2}& \cdots & a_{1,N-1}& a_{1,N}\\ a_{2,1}& a_{2,2}& \cdots & a_{2,N-1}& a_{2,N}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ a_{M-1,1}& a_{M-1,2}& \cdots & a_{M-1,N-1}& a_{M-1,N}\\ a_{M,1}& a_{M,2}& \cdots & a_{M,N-1}& a_{M,N}\end{array}\right]\hfill \end{array}$$

of discrete pixel values. The position in the upper left corner of the image corresponds to the matrix indices $(1,1)$, and the position in the lower right corner of the image corresponds to the matrix indices $(M,N)$. The maximal light intensity is represented by the pixel value of 255, and the absence of light is represented by the pixel value 0, with the shades of gray thus represented by the values between 0 and 255. For example, the matrix

$$\begin{array}{l}\left[\begin{array}{cccc}223& 111& 159& 47\\ 127& 255& 63& 143\\ 191& 79& 207& 15\\ 95& 175& 31& 239\end{array}\right]\hfill \end{array}$$

represents the image (magnified by the factor of 100) in Figure 1.1 below.

FIGURE 1.1:An illustration of grayscale pixel values.

As the emphasis of this text is on college mathematics, we will not discuss such broad and fascinating topics as visual perception, image formation, and many others. There is a huge body of literature covering all those areas, and we cannot possibly do any of them justice within the scope of this text. We also refer the reader to the fundamental work [11] for a detailed introduction into the topics of image sensing and acquisition, image sampling and quantization, representation of digital images, as well as examples of fields of science and technology where methods of digital image processing are used most extensively.

Section 1.1 Exercises.

1. Enter relevant terms like “image processing” and “grayscale images” into a search engine and compile a list of applications of image processing in the following fields:
   • Art and design
   • Astronomy
   • Medicine
   • Meteorology
   • Photography and motion pictures
   • Transportation and autonomous vehicles
2. Find a few black-and-white photos in your family photo album or in old newspapers. Select the ones that could benefit from image enhancement. Scan them and have them available for exercises in subsequent chapters.

1.2 Working with Images in MATLAB®

Throughout this and subsequent chapters, the reader will be asked to complete exercises that require the use of MATLAB¹. Although this text is by no means intended to serve as an introduction to MATLAB, we will, nevertheless, provide a few basic examples of MATLAB usage, particularly when it comes to basic image manipulations.

¹MATLAB®is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc.

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Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1 Introduction to Basics of Digital Images
• 2 A Library of Elementary Functions
• 3 Probability, Random Variables, and Histogram Processing
• 4 Matrices and Linear Transformations
• 5 Convolution and Image Filtering
• 6 Analysis and Processing in the Frequency Domain
• 7 Wavelet-Based Methods in Image Compression
• Bibliography
• Index