Welcome to the homepage of the lecture

Differential Equations in Image Processing and Computer Vision

Winter Term 2021 / 2022

Differential Equations in Image Processing and Computer Vision

Four Teaching Awards (3 in Computer Science, 1 in Mathematics)

Lecturer: Dr. Joachim Weickert

Coordinator of tutorials: Kristina Schaefer

Winter Term 2021 / 2022

Lectures (4h) with tutorials (2h)
(9 ETCS points)

Online lectures based on the Zoom platform (privacy information):
Wednesday, 10:15-12:00
Friday, 10:15-12:00

First lecture: Wednesday, October 20

The permanent Zoom link and the password for downloading the slides and videos have been e-mailed to registered participants on October 22.

Tutorials: 2 hours each week; see below.

NewsSynopsisPrerequisitesTutorialsRegistrationWritten ExamsContentsSelf TestMaterial for the Programming AssignmentsExample Solutions for the AssignmentsReferences

01. 04. 2022: Opportunity for exam inspection:
Wedensday, April 06, Lecture Hall 001, Building E1.3.

01. 04. 2022: The results of the second exam are online.

18. 02. 2022: Opportunity for exam inspection:
Friday, February 25, Lecture Hall 001, Building E1.3.

18. 02. 2022: The results of the first exam are online.

08. 02. 2022: The list of admitted students is online.

04. 11. 2021: Further update of the course contents.

25. 10. 2021: Course contents updated to reflect the final schedule.

22. 10. 2021: The registration is closed. The password for downloading material and the permanent Zoom link have been e-mailed to registered participants.

20. 10. 2021: The registration is open.

20. 10. 2021: Registration will be possible on this webpage between Wednesday, Oct. 20, 14:00 and Friday, Oct. 22, 12:00. See below.

17. 10. 2021: The website is online.

Many modern techniques in image processing and computer vision make use of methods based on partial differential equations (PDEs) and variational calculus. Moreover, many classical methods may be reinterpreted as approximations of PDE-based techniques. In this course we will get an in-depth insight into these methods. For each of these techniques, we will discuss the basic ideas as well as theoretical and algorithmic aspects. Examples from the fields of medical imaging and computer aided quality control illustrate the various application possibilities.

Since this class guides its participants to many research topics in our group, its attendance is required for everyone who wishes to pursue a master thesis in our group.

Equally suited for students of visual computing, mathematics and computer science. Requires undergraduate knowledge in mathematics (e.g. ''Mathematik für Informatiker I-III''). Knowledge in image processing or differential equations is useful, but not required. The lectures will be given in English.

In the online tutorials via Zoom we discuss the homework assignments (theory and programming). The programming assignments give an intuition about the way how image processing and computer vision algorithms work, while the theoretical assigments provide additional insights, also from a mathematical perspective.

For the homework assignments you can obtain up to 24 points per week. To qualify for both exams you need 50 percent of all possible points. Working in groups of up to 3 people is permitted, but all persons must be in the same tutorial group.

By presenting your solution to a homework problem in the tutorials, you can earn 2 bonus points.

If you have questions concerning the tutorials, please do not hesitate to contact Kristina Schaefer.

Two groups are scheduled:

  • Group 1 (in English): Tuesday, 10:15-12:00
    Tutor: Michael Ertel
    Office Hour: Tuesday 16:00-17:00.

  • Group 2 (in English): Tuesday, 14:15-16:00
    Tutor: Michael Ertel
    Office Hour: Tuesday 16:00-17:00

The tutorial group can be reached via the mail address
where x has to be replaced by the group name (t1 or t2).

Registration is now closed. You can still check in which group you are via web form.

It is planned to have two written exams. Changes due to the development of the Covid-19 pandemic cannot be excluded. In this case you will be informed as soon as possible.

The first written exam will take place on
Wednesday, February 16, 2022 from 14:00 to 17:00
in Building E 2.2, Günter Hotz Lecture Theatre and in Building E 1.3, Lecture Hall HS002.

The second written exam will take place on
Wednesday, March 30, 2022 from 14:00 to 17:00
in Building E 2.2, Günter Hotz Lecture Theatre.

In order to qualify for the exams you need a total amount of 50 percent of all possible points from the homework assignments. In case of qualification, you are allowed to take part in both exams. The better grade counts, but each exam will count as an individual attempt. individually.

Please check here whether you are admitted to the written exam. Additionally, you have to be registered for the exam in the HISPOS system. If you think that there is an error, please contact Kristina Schaefer as soon as possible.

Please do not forget to bring your student ID card with you.

The exams will be closed book. These are the rules during the exams:

  • You are allowed and obliged to bring three things to your desk only: Your student ID card (Studierendenausweis), a ball-pen that has no function other than writing, and a so-called cheat sheet. This cheat sheet is a A4 page with formulas or important equations from the lecture. Please note that the cheat sheet has to be handwritten by yourself. It will be collected at the end of the exam, and you can get it back at the exam inspection.
  • In particular, electronic devices (including your cell phone), bags, jackets, briefcases, lecture notes, homework and classroom work solutions, additional handwritten notes, books, dictionaries, and paper are not allowed at your desk.
  • Please keep your student ID card ready for an attendance check during the exam.
  • Do not use pencils or pens that are erasable with a normal rubber.
  • You are not allowed to take anything with you that contains information about the exam. A violation of this rule means failing the DIC course.
  • You must stay until the exam is completely over.
  • You are expected to wear a face mask. If you have a medical condition that does not allow this, we need a medical certificate and expect you to wear a face shield instead.

The results of the first written exam can be found here, and the corresponding distribution of points and grades here.

Each student who has participated in the first written exam has the opportunity to inspect his/her graded solutions in room Lecture Hall 001 in Bldg. E1.3 on Friday, February 25, 2022, from 14:00 to 16:00.

The results of the second written exam can be found here, and the corresponding distribution of points and grades here.

Each student who has participated in the second written exam has the opportunity to inspect his/her graded solutions in Lecture Hall 001 in Bldg. E1.3 on Wednesday, April 06, 2022, from 2:00 p.m. to 4:00 p.m.

Course material will be made available on this webpage in order to support the classroom teaching and the tutorials, not to replace them. Additional organisational information, examples and explanations that may be relevant for your understanding and the exam are provided in the lectures and tutorials. It is solely your responsibility - not ours - to make sure that you receive this infomation.

The following table shows a preliminary list of topics that will be covered during the semester.

20/10 Introduction, Overview
22/10 Homogeneous Diffusion I: Basic Concepts
(contains preparatory assignment P1 and homework H1)
27/10 Homogeneous Diffusion II: Algorithms, Limitations, Alternatives
29/10 Nonlinear Isotropic Diffusion I: Modelling and Continuous Theory
(contains preparatory assignment P2 and homework H2)
03/11 Nonlinear Isotropic Diffusion II: Discrete Theories and the Diffusion Echo
05/11 Nonlinear Isotropic Diffusion III: Efficient Algorithms
(contains preparatory assignment P3 and homework H3)
10/11 Nonlinear Anisotropic Diffusion I: Modelling
12/11 Nonlinear Anisotropic Diffusion II: Continuous and Discrete Theory
(contains preparatory assignment P4 and homework H4)
17/11 Nonlinear Diffusion: Parameter Selection
19/11 Variational Methods I: Basic Ideas
(contains preparatory assignment P5 and homework H5)
24/11 Variational Methods II: Discrete Aspects
26/11 Variational Methods III: TV Regularisation and Primal-Dual Methods
(contains preparatory assignment P6 and homework H6)
01/12 Variational Methods IV: Functionals of Two Variables
03/12 Vector- and Matrix-Valued Images
(contains preparatory assignment P7 and homework H7)
08/12 Image Sequence Analysis I: Models for the Smoothness Term
10/12 Image Sequence Analysis II: Models for the Data Term
(contains preparatory assignment P8 and homework H8)
15/12 Image Sequence Analysis III: Practical Aspects
17/12 Image Sequence Analysis IV: Numerical Methods
(contains preparatory assignment P9 and homework H9)
05/01 Osmosis I: Continuous Theory and Modelling
07/01 Osmosis II: Discrete Theory and Efficient Algorithms
(contains preparatory assignment P10 and homework H10)
12/01 Continuous-Scale Morphology I: Basic Ideas
14/01 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
(contains preparatory assignment P11 and homework H11)
19/01 Curvature-Based Morphology I: Mean Curvature Motion
21/01 Curvature-Based Morphology II: Affine Morphological Scale-Space
(contains preparatory assignment P12 and homework H12)
26/01 Self-Snakes and Active Contours
28/01 Backward Parabolic PDEs and M-smoothers
(contains preparatory assignment P13 and homework H13)
02/02 PDE-Based Image Compression I: Data Selection
04/02 PDE-Based Image Compression II: Optimised Encoding and Better PDEs
(please take a look at the self-test problems)
09/02 PDEs and Learning
11/02 Summary and Outlook

Here you can download a self-test problem sheet, that contains 6 problems, which are intended to be similar in style and difficulty to a 180-minutes written exam.


Here you can download the material for the programming assignments:

Date Topic

Here you can download example solutions for the assignments:

Date Assignment

  • J. Weickert: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.
  • G. Aubert and P. Kornprobst: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Second Edition, Springer, New York, 2006.
  • T. F. Chan and J. Shen: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia, 2005.
  • K. Bredies, D. Lorenz: Mathematical Image Processing. Birkhäuser, basel, 2018.
  • F. Cao: Geometric Curve Evolutions and Image Processing. Lecture Notes in Mathematics, Vol. 1805, Springer, Berlin, 2003.
  • R. Kimmel: The Numerical Geometry of Images. Springer, New York, 2004.
  • G. Sapiro: Geometric Partial Differential Equations in Image Analysis. Cambridge University Press, 2001.
  • Articles from journals and conferences.

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