Mathematical Methods of Image Processing  (MA439-01)  -  S. Allen Broughton, Fall 2006

The answer is image compression.  Modern mathematics (developed in the last 15-20 years), electrical engineering (DSP), and computer science have been responsible for the development of good image compression algorithms. The course will cover the mathematical basis of many of the ideas behind image processing such as filtering, filter banks, the discrete Fourier and cosine transforms and the discrete wavelet transform. All of this will be balanced by concrete applications of  these ideas to various image processing problems with a special emphasis on image compression. The images may the mathematics all concrete as the two pictorial illustrations of Discrete Cosine Transform (DCT) based JPEG compression  and Discrete Wavelet Transform (DWT) based JPEG 2000 compression show.
 

Discrete Cosine Transform Based Compression Original picture Picture of first phase of DCT - based  compression using 8x8 blocks

Discrete Wavelet Transform Based Compression Original picture Picture of first phase of  DWT compression using 3 levels and db2 wavelets

The course is aimed at juniors and seniors, though any mathematically well-prepared student will benefit from the course. The basic mathematical ideas can be built upon matrix algebra and some Fourier series. Previous exposure to Digital Signal Processing is helpful but not essential. The course work will consist of homework assignments, exams and one or two projects. The computational tools of the course will be MATLAB, and, to some extent, MAPLE . It is not necessary to know these tools beforehand, though some computational expertise is expected (e.g. one of MATLAB, MAPLE, or C++).

This course is based on many previous offerings of the course, including a joint development effort by Allen Broughton and Ed Doering years ago, a section by Roger Lautzenheiser in Winter 01-02, and recent offereings by Kurt Bryan.  As the course ahs devleoped we have concentrated on image compression topics, filter banks and the discrete wavelet transform, so that enough time can be spent on concrete application of these topics. In particular we expect to spend some time understanding the mathematics behind the current DCT-based JPEG (see picture above) compression method and the new wavelet-based JPEG 2000 compression method (see picture above) that is expected to debut in the next few years of years.  By learning the mathematical foundations as well as their application, we expect that students will gain background to further their learning of image processing methods.

I particularly welcome any student who wants to do a related Imaging Certificate project after taking the course.

Any questions? Send me  email: allen.broughton@rose-hulman.edu
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