Welcome to the homepage of the lecture

Differential Equations in
Image Processing and Computer Vision

Winter Term 2018 / 2019

Differential Equations in Image Processing and Computer Vision

Three Computer Science Teaching Awards (Summer Terms 2003 and 2006, Winter Term 2015)
One Mathematics Teaching Award (Summer Term 2009)

Lecturer: Dr. Pascal Peter

Coordinator of tutorials: Aaron Wewior

Winter Term 2018 / 2019

Lectures (4h) with theoretical exercises (2h)
(9 ETCS points)

Tuesday, 10-12 a.m., Building E 1.3, Lecture Hall 001
Friday, 2-4 p.m., Building E 1.3, Lecture Hall 001

First lecture: Tuesday, October 16

Tutorials: 2 hours each week; see below.

NewsSynopsisPrerequisitesTutorialsRegistrationWritten ExamsContentsSelf TestMaterial for the Programming AssignmentsExample Solutions for the AssignmentsReferences

05.04.19: The results for the second exam are online

15.02.19: The results for the first exam are online

04.02.19: The list of admitted students for the written exams is online

29.10.18: Tutorials for this week are shifted due to a public holiday.

23.10.18: Registration is closed.

16.10.18: Registration is open.

13.08.18: 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.

The tutorials include homework assignments (theory and programming) as well as classroom assignments. The programming assignments give an intuition about the way how image processing and computer vision algorithms work, while the theoretical assigments provide additional mathematical insights. Classroom assignments are supposed to be easier and should guide you gently to the main themes.

For the homework assignments you can obtain up to 24 points per week. Actively participating in the classroom assignments gives you 12 more points per week, regardless of the correctness of your solutions. To qualify for both exams you need 2/3 of all possible points. For 13 weeks, this comes down to 13 x 24 = 312 points. Working in groups of up to 3 people is permitted, but all persons must be in the same tutorial group.

If you miss a tutorial because you are sick, you can still get the points for participation, if you bring a doctor's certificate.

If you have questions concerning the tutorials, please do not hesitate to contact Aaron Wewior.

Two groups are scheduled:

  • Group T1:
    Thursday, 12-2 p.m., Building E1.3, Seminar Room 107
  • Group T2:
    Thursday, 4-6 p.m., Building E1.3, Seminar Room 107

The tutorial group can be reached via the mail addresses:
dic-x -- at -- mia.uni-saarland.de
where x has to be replaced by the group name (t1 or t2).

Due to a public holiday, the tutorials on November 1 have been moved to October 31. You can visit one of the two replacement tutorials:

  • 2-4 p.m., Building E1.1, Seminar Room 106
  • 4-6 p.m., Building E2.5, Seminar Room 2 (U.36)

You can attend any of the two tutorials and attendance is not mandatory.

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

There will be two written exams. The first one will be at the beginning and the second one at the end of the semester break.

The first written exam will take place on
Wednesday, February 13, 2019 from 2:00 p.m. to 5:00 p.m.
in in Building E 1.3, Lecture Hall 002.

The second written exam will take place on
Thursday, April 4, 2019 from 2:00 p.m. to 5:00 p.m.
in in Building E 1.3, Lecture Hall 002.

In order to qualify for the exams you need a total amount of 2/3 of all possible points from the homework and classroom 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 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 Aaron Wewior as soon as possible.

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.
  • 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.

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 4.10 in Bldg. E1.7 on Wednesday, February 20, 2018, from 11:00 a.m. to noon.

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 room 4.10 in Bldg. E1.7 on Tuesday, April 9, 2019, from 2:00 p.m. to 3:00 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.

16.10 Introduction, Overview
19.10 Linear Diffusion I: Basic Concepts
(contains classroom assignment C1 and homework H1)
23.10 Linear Diffusion II: Numerics, Limitations, Alternatives
26.10 Nonlinear Isotropic Diffusion I: Modelling and Continuous Theory
(contains classroom assignment C2 and homework H2)
30.10 Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory
02.11 Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
(contains classroom assignment C3 and homework H3)
06.11 Nonlinear Anisotropic Diffusion I: Modelling
09.11 Nonlinear Anisotropic Diffusion II: Continuous and Discrete Theory
(contains classroom assignment C4 and homework H4)
13.11 Nonlinear Anisotropic Diffusion III: Efficient Algorithms
16.11 Nonlinear Diffusion: Parameter Selection
(contains classroom assignment C5 and homework H5)
20.11 Variational Methods I: Basic Ideas
23.11 Variational Methods II: Discrete Aspects
(contains classroom assignment C6 and homework H6)
27.11 Variational Methods III: TV Regularisation and Primal-Dual Methods
30.11 Variational Methods IV: Functionals of Two Variables
(contains classroom assignment C7 and homework H7)
04.12 Vector- and Matrix-Valued Images
07.12 Unification of Denoising Methods
(contains classroom assignment C8 and homework H8)
11.12 Osmosis I: Continuous Theory and Modelling
14.12 Osmosis II: Discrete Theory and Efficient Algorithms
(contains classroom assignment C9 and homework H9)
18.12 Image Sequence Analysis I: Models for the Smoothness Term
21.12 Image Sequence Analysis II: Models for the Data Term
(contains classroom assignment C10 and homework H10)
08.01 Image Sequence Analysis III: Practical Aspects
11.01 Image Sequence Analysis IV: Numerical Methods
(contains classroom assignment C11 and homework H11)
15.01 Continuous-Scale Morphology I: Basic Ideas
18.01 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
(contains classroom assignment C12 and homework H12)
22.01 Curvature-Based Morphology I: Mean Curvature Motion
25.01 Curvature-Based Morphology II: Affine Morphological Scale-Space
(contains classroom assignment C13 and homework H13)
29.01 Self-Snakes and Active Contours
01.02 PDE-Based Image Compression I: Data Selection
(please take a look at the self-test problems)
05.02 PDE-Based Image Compression II: Optimised Encoding and Better PDEs
08.02 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.

06/02 Self Test Problem Sheet
10/02 Self Test Solution

Here you can download the material for the programming assignments:

Date Topic
19.10 H1 - Linear Diffusion, Gaussian Convolution
26.10 H2 - Linear Diffusion
02.11 H3 - Nonlinear Isotropic Diffusion
09.11 H4 - Anisotropic Diffusion
16.11 H5 - FED, Decorrelation
23.11 H6 - Diffusion-Reaction Methods
30.11 H7 - Primal-Dual Methods for TV Regularisation
07.12 H8 - Iterated Bilateral Filtering
14.12 H9 - Osmosis
21.12 H11 - Optic Flow
11.01 H12 - Morphology
18.01 H13 - Curvature-Based Morphology

Sample solutions are only available during the semester.

  • 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.
  • 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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