A second activity undertaken by the Mathematics department is a seminar on Fourier and wavelet methods in imaging, which is currently underway. The talks are based the developed courses and a web version is given below:
Why is geometry often described as 'cold' and 'dry?' One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." So begins Benoit Mandelbrot's classic book, "The Fractal Geometry of Nature." Mandelbrot conceived and developed a new geometry of nature that describes many of the irregular and fragmented patterns around us, by identifying a family of shapes that are now known as 'fractals.' In 1975, Jim Yorke coined the term "chaos" to describe the seemingly random behavior of deterministic systems. It is now understood that chaotic dynamics is inherent in many nonlinear systems. As a result, the last twenty years has seen an explosion in the study of nonlinear dynamical systems. Scientists from all fields have become fascinated with chaos and the closely related field of fractal geometry because of its importance in applications as well as the beauty of the geometric patterns produced. To many, the study of chaotic systems is one of the most important breakthroughs in modern science and is rapidly becoming an instrumental component of many disciplines.
This course will investigate fractals, chaos, and their interrelationship. We will use the book "Chaos and Fractals: New Frontiers of Science" by Heinz-Otto Peitgen, Hartmut Jurgens, and Dietmar Saupe. We will study different types of fractal dimensions, iterated function systems, random fractals, bifurcations, symbolic dynamics, quadratic maps, and Mandelbrot and Julia sets. This will enable us to understand the beautiful relationship between chaos and fractals.
Mathematical Methods of Image Processing
Co-developers: Allen Broughton and Edward Doering
Course web page: MA490A/EC497
Course Abstract:
Modern mathematics (developed in the last 10 years), electrical engineering
(DSP) and fast computers have allowed the development of many tools to solve
these problems, and solve them quickly. The course will cover the mathematical
basis of many of these ideas including filtering, filter banks, the discrete
Fourier and cosine transforms as well as wavelet analysis. All of this will
be balanced by concrete application of these ideas to various image processing
problems such as image
restoration.
Applied Math Seminar: Wavelet Based Methods in Image
Processing
Speaker: Allen Broughton
Seminar web page: Image
Pro Talks
Abstract:
It is obvious to all that image (and signal) processing has become an exceedingly important and hot topic that cuts across many areas: engineering, physical science, computer science and of course mathematics. Less obvious, but becoming increasingly well known, is that signal and image processing are getting great improvements in performance by using wavelet based methods.
In an effort to achieve better understand the wavelet based methods and image processing itself, I will give a series of talks that introduce wavelet based image processing methods in the context of traditional image processing. The plan is to have about four talks that roughly cover the following:
1. Introduction to image processing and image processing
problems: restoration, compression and denoising
2. Filtering and Fourier methods
3. Localization, wavelet and windowed transforms
4. Filter bank implementation of wavelet based methods
The talks will, of course, have an underlying mathematical theme, but I intend
to use lots of examples and computer illustrations that Ed Doering and I have
developed in our joint course last fall. The talks should be accessible to all
faculty and most interested undergraduates.