An Image Processing Tour of College Mathematics

Image processing is an important application of mathematics that seldom receives attention in the undergraduate curriculum.  That is unfortunate because image processing is a subject that requires relatively little in the way of prerequisite mathematics- mostly introductory linear algebra.  Because the results of image processing computations are images, students can easily experiment with image processing techniques and immediately see the results.

 

In the first two chapters, the author discusses how images are stored in MATLAB and discusses basic operations such as histogram adjustments to improve image contrast.  The discussion of color images and color spaces is something not found in many other introductions to signal processing.  In the third and fourth chapters the author reviews some linear algebra and probability theory that is applicable to image processing.  This book could not substitute for introductory course in these areas, but the material in these chapters could be useful for review.

 

The material in the final three chapters is the most important in the book.  Chapter five covers convolution and convolutional filtering of one dimensional signals and two dimensional images.  Edge detection is an important practical application.  Chapter 6 introduces the discrete Fourier transform of a one dimensional signal and then extends this to the two dimensional transform of an image.  Images are then filtered in the Fourier domain.  Chapter 7 introduces Haar wavelets, the Debauchies wavelets, and biorthogonal wavelets in one dimension.  The author finishes with the JPEG2000 standard for wavelet compression of images.

 

Each chapter includes a mixture of theoretical exercises and computational exercises to be done using MATLAB.  In the early chapters these are quite modest, but by the end of the book the exercises ask the student to implement sophisticated image processing algorithms and test their methods on real images.  

 

In recent decades, research in mathematical image processing has expanded to include new approaches based on partial differential equations, convex optimization, and neural networks.  The mathematical background required for these approaches goes well beyond the basic background in linear algebra assumed by the author, so the decision to stop after wavelets is appropriate for the intended audience. 

 

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.  The material would be accessible not just to mathematics majors but also to students from electrical engineering and computer science who have some background in linear algebra.

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Brian Borchers is a professor of mathematics at New Mexico Tech and the editor of MAA Reviews.