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
basic notions for discrete and continuous dynamical systems
overview over important results in the classical theory: stability and convergence analysis, limit cycles and oscillatory behaviour, bifurcations
pattern formation by reaction-diffusion systems
enhancement of periodic image features
image compression using iterated function systems
spatially discrete analysis of image denoising processes by systems of ordinary differential equations
stability analysis of discontinuity-enhancing image filters.
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 |