Mathematical Image Analysis Group, Saarland University

Dynamical Systems and Image Processing

Winter Term 2007/08

Dynamical Systems and Image Processing

Dr. Martin Welk (bld. E11, room 3.10.1, phone 0681-302-57343)

Winter 2007/08

Lectures (2h) with exercises (1h), winter term 2007/08

Lectures: Fridays 10–12, Bld. E1.3, Lecture hall 003.
Tutorials: Thursdays 16–18 (4 p.m.–6 p.m.) every other week, Bld. E2.5, room H07 (»Zeichensaal«).
Next meeting: February 22 (exam)

Specialised course in mathematical image analysis, suitable for students in mathematics, computer science and visual computing programs.
Participants learn important concepts of dynamical systems and how they can be applied in image processing.

Breaking news – Entrance requirements – Contents – Assessments / Exams – References – Download

Breaking News

Second chance exams: Deviating from earlier announcements, second chance exams will be offered also for those who passed the first exam. Those wishing a second chance exam will have to notify Martin Welk soon after the results of the first exam are known to allow for timely planning.

Entrance requirements

Undergraduate knowledge of mathematics. For computer science students, this requirement is met by successful completion of the Mathematics for Computer Scientists lecture cycle.

Mathematical prerequisites which exceed the basic mathematics courses are provided within the lecture. Previous knowledge in either digital image processing or dynamical systems is therefore helpful but not required.

Contents

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
Autonomous Systems of ordinary differential equations
Stability and bifurcations
Chaos and attractors
Reaction-diffusion systems
Application in texture restoration
Pattern generation
Image segmentation using dynamical systems
Semidiscrete and discrete analysis of image filters
Image compression by iterated function systems
Cellular automata

Assessments / Exams

The written exam takes place on February 22, 1015–1145 (during the regular lecture time slot) in Lecture Hall 003, Bld. E1.3.
Scripts and exercise materials from the course may be used, as well as pocket calculators.
For students who fail on the written exam, there will be second chance exams, to be announced later.

References

F. Verhulst. Nonlinear Differential Equations and Dynamical Systems. 2nd edition, Springer, Berlin, 1996
J. Jost. Dynamical Systems. Springer, Berlin, 2005
Articles from journals and conferences.

Download

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

Lectures

No. Title Date Last update

1 Introduction, Basic Concepts October 26 2007.10.26.1458

2 Ordinary Differential Equations, Basic Stability Analysis November 2 2007.12.12.1317

3 Stability Analysis of Periodic Solutions November 9 2007.12.08.0044

4 Perturbed and Coupled Oscillations November 16 2007.11.29.1917

5 Reaction-Diffusion Systems and Texture Restoration November 30 2007.12.02.2315

6 Pattern Generation and Image Segmentation by Dynamical Systems December 7 2007.12.07.0117

7 Structural Stability, Bifurcations, Attractors, Chaos December 14 2007.12.21.1710

8 Iterated Function Systems and Fractals December 21 2008.01.10.2240

9 Fractal Image Coding January 11 2008.01.11.0318

10 Semidiscrete and Discrete Analysis of Image Filters 1 January 18 2008.01.30.1735

11 Semidiscrete and Discrete Analysis of Image Filters 2 January 25 2008.01.30.1738

12 Semidiscrete and Discrete Analysis of Image Filters 3 February 1 2008.02.01.1854

13 Cellular Automata February 8 2008.02.08.1728

14 Further Cellular Models, Summary February 15 2008.02.15.0437

Exercises (H homework, C classroom)

No. Date posted Date to be submitted Solution

H1 November 2 November 8 H1S

C1 November 8 C1S

H2 November 22 November 29 H2S

C2 November 29 C2S

H3 December 8 December 13 H3S

C3 December 13 C3S

H4 January 4 January 10 H4S

C4 January 10 C4S

H5 January 18 January 24 H5S

C5 January 24 C5S

H6 January 25 January 31 H6S

C6 January 31 C6S

H7 February 8 February 14 H7S

Martin Welk / October 12, 2007–February 21, 2008