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 newsEntrance requirementsContentsAssessments / ExamsReferencesDownload

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.

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.

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

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.

  • 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.

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

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

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