Saarland University | Mathematical Image Analysis Group | Teaching

Dynamical Systems and Image Processing

Lectures, winter term 2004/2005

Dr. Martin Welk (Bld. 27.1, room 410, phone 0681-302-64383)

Dynamical systems are mathematical models which describe time-dependent processes. Depending on whether the time variable is restricted to integers or real numbers, a dynamical system is called either discrete or continuous. Discrete dynamical systems are mathematically described by iterated functions or iterated function systems while continuous dynamical systems take the form of differential equations. Dynamical systems display a wide variety of different behaviours, from convergence to a steady state via oscillations and limit cycles up to bifurcations and chaos. Their ability to model complex phenomena of self-organisation or pattern formation has made them an attractive tool in many fields of research ranging from mathematical biology to economics.

The course focusses on the application of dynamical systems in image processing, analysis and understanding. The necessary mathematical theory will be provided in the course. Topics include

The lecture is designed for advanced students of mathematics as well as of computer science. It requires undergraduate knowledge in mathematics. Previous knowledge in image processing or dynamical systems is useful but not required.

Time: Monday 14–16 (2–4 p.m.) starting Oct. 25, 2004
Location: Bld. 45, Lecture hall 003

Participants of the course can download the slides here (access password-protected):

Oct. 25 Lecture 1 1 Introduction
2 Basic notions
Nov. 8 Lecture 2 3 Ordinary Differential Equations
4 Critical Points
Nov. 15 Lecture 3 4 (End)
5 Periodic Solutions
Nov. 22 Lecture 4 5 (End)
6 Perturbed Oscillations
7 Coupled Oscillators
Nov. 29 Lecture 5 8 Reaction-Diffusion Systems and Texture Restoration
Dec. 6 Lecture 6 9 Numerical Implementation of Reaction-Diffusion Systems
10 Pattern Generation
11 M-Lattice Systems
Dec. 13 Lecture 7 12 Reaction-Diffusion Segmentation
13 Relaxation Oscillations, Image Segmentation by Oscillation
Dec. 20 Lecture 8 14 Fixed Points for Discrete Dynamical Systems; Bifurcations
15 Fractals
Jan. 10 Lecture 9 15 (End)
16 Fractal Image Coding  
Jan. 17 Lecture 10 17 Semidiscrete and Discrete Analysis of Image Filters: TV Flow
Jan. 24 Lecture 11 17 (End)
Jan. 31 Lecture 12 18 Semidiscrete and Discrete Analysis of Image Filters: 1D Shock Filter
Feb. 14 Lecture 13 19 Structural Stability, Bifurcations, Catastrophe Theory
20 Summary

Additional stuff for download (password-protected):

Example problems, Part I for exam preparation referring to Lectures 1–9 Jan. 19
updated Jan. 27
Example problems, Part II for exam preparation referring to Lectures 10–12 Feb. 12

Martin Welk / August 30, 2004–February 15, 2005 / back to MIA home.
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