Dynamical Systems and Image Processing

Summer Term 2009

Dynamical Systems and Image Processing

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

Summer 2009

Lectures (2h) with exercises (1h), summer term 2009

Lectures and Tutorials: Tuesdays 14–16, Fridays 14–16, Bld. E1.3, Lecture hall 001.
Lectures and tutorials will alternate in these time slots after prior notice, with some interruptions, see schedule.
Next meeting: Second exam on October 9.

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 / ExamsReferencesScheduleDownload

Second exam: October 9, 14–16, Bld. E1.3, Lecture Hall 1. The second exam is open for all students who were qualified to the first exam, independent on whether the first exam has been passed or failed, or not attended.

Exam results: All participants of the first exam have been notified of their results by e-mail.

Summary now available, including Lectures 1–12.

Homework assignments: Problem Sheet H5 is online, see below! Submission deadline is July 24.

HISPOS: Students enrolled in study programmes attached to the HISPOS system must register for the course also in the HISPOS system.
Computer Science and Bioinformatics students: Registration in HISPOS must be completed before June 3.
Mathematics: BSc/MSc students must register; a different timeline than for Computer Science students may apply. Diploma students are not included in HISPOS so far.
Visual Computing students are still exempt from HISPOS.
Other programmes: please check with your department.

General notice: Please be aware that this is not a remote study course. This web page does not (and is not intended to) replace regular attendance of lectures and tutorials.

Regular participation in classroom exercises and regular submission of homework assignments is a prerequisite for admission to the exam (as announced in the lecture). Homework submissions are accepted in writing only (no e-mail submissions) and only in the lecture hall.

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 July 31, 1415–1600 (during the regular lecture time slot) in Lecture Hall 001, Bld. E1.3.
Second chance exams: There will be a second chance exam (October 9, 14–16h, Bld. E1.3, seminar room 016). Deviating from earlier announcements, the second chance exam is open also for those who passed the first exam.

  • J. Jost. Dynamical Systems. Springer, Berlin, 2005
  • L. Perko. Differential Equations and Dynamical Systems. Third edition. Texts in Applied Mathematics, 7. Springer, New York 2001
  • F. Verhulst. Nonlinear Differential Equations and Dynamical Systems. Second edition, Springer, Berlin, 1996
  • Articles from journals and conferences.

This schedule is preliminary and subject to change!

Date (Tuesday)Scheduled event Date (Friday)Scheduled event
April 21Lecture 1 April 24Lecture 2
April 28Tutorial 1 May 1– (public holiday!)
May 5 May 8
May 12Lecture 3 May 15Lecture 4
May 19Tutorial 2 May 22Lecture 5
May 26Lecture 6 May 29Tutorial 3
June 2 June 5
June 9Lecture 7 June 12Lecture 8
June 16Lecture 9 June 19Tutorial 4
June 23Lecture 10 June 26
June 30Lecture 11 July 3Tutorial 5
July 7Lecture 12 July 10
July 14 July 17
July 21 July 24Lecture 13
July 28Tutorial 6 July 31Written Exam

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


No. Title Date Last update
1 Introduction, Basic Concepts April 21 2009.04.21.1608
2 Ordinary Differential Equations, Basic Stability Analysis April 24 2009.07.28.1541
3 Stability Analysis of Fixed Points and Periodic Solutions May 12 2009.05.20.1214
4 Periodic Solutions in 2D; Perturbed and Coupled Oscillations May 15 2009.07.28.1543
5 Reaction-Diffusion Systems and Texture Restoration May 22 2009.07.28.1544
6 Pattern Generation; Bifurcations, Catastrophes and Chaos May 26 2009.05.26.1248
7 Iterated Function Systems and Fractals June 9 2009.06.12.1715
8 Fractal Image Coding June 12 2009.07.28.1547
9 Semidiscrete and Discrete Analysis of Image Filters 1 June 16 2009.07.28.1549
10 Semidiscrete and Discrete Analysis of Image Filters 2 June 23 2009.06.23.1850
11 Semidiscrete and Discrete Analysis of Image Filters 3 June 30 2009.06.30.1329
12 Semidiscrete and Discrete Analysis of Image Filters 4 July 7 2009.07.07.1336
13 Cellular Models July 24 2009.07.24.1708

When preparing for the exam, you might also consider the Summary (PDF file). Note that this text is intended as a key to the lecture scripts, not as a replacement.

Exercises (H homework, C classroom)

No. Date posted Date to be submitted Solution
H1 April 24 May 12 H1S
C1 April 28   C1S
H2 May 15 May 26 H2S
C2 May 19   C2S
H3 May 28 June 12 H3S
C3 May 29   C3S
H4 June 16 June 30 H4S
C4 June 19   C4S
H5 July 3 July 24 H5S
C5 July 3   C5S
C6 July 28   C6S

Martin Welk / March 30–October 9, 2009

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