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

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

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

Lectures

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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