Stuart Hoggar is a research fellow and a former senior mathematics lecturer at the University of Glasgow.  He writes that the book is based on graduate course notes for PhD students which he developed at Ohio State University.  Mathematics of Digital Images is a huge book (854 pages) consisting of 18 chapters that are divided into six main sections. 

This book covers a lot of ground!  An extensive range of image related topics are included in the areas of compression, restoration and recognition.  As the title suggests, a large number of theorems, proofs and examples are included for each topic.  A 13-page introduction provides a roadmap for the book and describes dependencies between chapters.

The first part of the book, concerned with plane geometry and pattern generation, is 112 pages long and is based on the author’s 1992 book, Mathematics for Computer Graphics.  One of the stated aims of this part is to provide the reader with enough background to write pattern-generating software.  Isometries (transformations of the plane which preserve distance such as rotations, translations and reflections) are introduced followed by their composition and classification.  This is followed by a short chapter on 1-dimensional (braid) patterns.  Isometries which send a pattern onto itself (such as floor tiles) are discussed in the next chapter and classified into five types.  These are then extended to 17 plane pattern types and a flow chart on deciding which type a pattern belongs to is presented.  The final chapter in this part of the book covers Coxeter graphs and Wythoff’s construction and presents an algorithm to generate patterns based on a small fundamental region.       

Part two (“Matrix structures”) is an 88-page review of fundamental linear algebra, covering vectors and matrices, transformations, eigenvalues, rank, and the Singular Value Decomposition (SVD).  This material is covered rapidly, and although good for review, these chapters are probably not ideal as an introduction.   

The third part of the book (185 pages in length) is a useful review of mathematical statistics with an emphasis on digital images.  Topics covered include probability, random vectors, correlation, Principal Component Analysis, inference, maximum likelihood estimates, regression, hypothesis testing, Bayes pattern classifiers, simulation, Markov chains, Monte Carlo methods and Bayesian networks.  I found these chapters well written and interesting; however, I think an additional mathematical statistics course text would be required for someone new to these concepts.

 I thought part four (125 pages), which covers information theory, was a highlight of this book.  In the first of two chapters. Hoggar presents the fundamentals of entropy, Shannon’s noiseless coding theorem, Huffman text compression, arithmetic codes, prediction by partial matching and LZW compression.  The second chapter in part four (actually chapter 13) covers channel capacity, error-correcting codes (such as the Reed-Solomon method used in CD players), and a section on probabilistic decoding (including an overview of turbocodes).

In part five Mathematics of Digital Images addresses image processing.  Although not comprehensive (for example, color images, morphological operations and image registration aren’t mentioned), in 162 pages the author covers a lot of important material.  The Discrete Fourier Transform (DCT) is covered in depth in 1-D and then 2-D, before convolution and various grayscale transforms (such as Gaussian and edge detection filters) are discussed.  Image restoration and compression are then presented (including an overview of JPEG).  The final chapter in part five covers fractals and wavelets (including the discrete wavelet and Gabor transforms).  It was nice to see fractal compression included. 

The final part of the book (145 pages) includes two chapters.  The first is concerned with B-Splines (important for representing curves and surfaces in computer graphics) and B-Spline wavelets.  The final book chapter presents a useful introduction to neural networks and self organizing nets before integrating these topics with information theory.  In the final section, tomography (imaging by sections) is discussed, including a concise overview of the Hough and Radon transforms

This book provides an excellent review of image related topics.  Each chapter includes questions, a list of related methods and alternate references.  The pseudo-code algorithms presented throughout the book are a useful complement to the math.   The author has a simple web site at which contains answers to most exercises, some image processing examples, and an Apple Mac program for producing plane patterns and showing their symmetry operations.    

I did have a couple of thoughts after reading the book.  Firstly, although image creation is mentioned on the cover, I don’t think the book covers this (for example, image acquisition is not discussed).  Secondly, I thought that the topic of image texture could have been introduced in the first part of the book (after pattern generation).  Also, although a large amount of topics are covered, the book is not exhaustive (for example, JPEG2000 or digital image watermarking are not mentioned).  Finally, I think this book could be a difficult read for people without a reasonable mathematical background; however there is no mention of pre-requisite reading or courses in the introduction. 

In summary, I was very impressed with Mathematics of Images.  The writing style, editing and figures and overall presentation are excellent.  I found it a pleasure to read and will keep it handy as a reference book.  This is a book that I would have found extremely useful at the start of my PhD, and I would recommend it particularly for graduate students who need a robust reference on mathematical foundations.  This book would also be of interest to students and researchers in image processing, computer vision, computer graphics and information theory.        



Mathematics of Digital Images



Stuart Hoggar

Cambridge University Press, September, 2006


Reviewed by:  Jason Dowling

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