Welcome to the homepage of the lecture

Differential Equations in Image Processing and Computer Vision

Winter Term 2014 / 2015

Differential Equations in Image Processing and Computer Vision

Two Computer Science Teaching Awards (Summer Terms 2003 and 2006)
One Mathematics Teaching Award (Summer Term 2009)

Lecturer: Prof. Joachim Weickert

Coordinator of tutorials: Sarah Schäffer

Winter Term 2014 / 2015

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

Lectures: Tuesday, Friday 10-12 am c.t., Building E 1.3, Lecture Hall 003

First lecture: Tuesday, October 21, 2014

Tutorials: 2 hours each week; see below.

NEWS: The results of the first written exam are now online.

NEWS: The results of the second written exam are now online.

NEWS: The certificates are ready and can be fetched in room 4.22, building E1.7 (Ellen Wintringer, opening hours: Mon, Wed, Thu 08.30 - 12.15, Tue 13.30 - 16.15, Fri 10.00 - 12.30).



SynopsisPrerequisitesAssignments and TutorialsWritten ExamsContentsSelf TestMaterial for the Programming AssignmentsExample Solutions for the AssignmentsReferences



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.


A combination of classroom and homework assignments (including theoretical as well as programming problems) is offered. The classroom assignments are intended to be solved in the tutorials and are not graded. The homework assignments are intended to be solved at home and have to be submitted on Friday, at 10.00 am before the lecture. In order to qualify for the exam you must

  • attend 80% of the tutorials (we do check)
  • gain 50% of all points from the homework assignments

Working in groups of up to 3 people is permitted, but all persons must be in the same tutorial group. Both classroom as well as homework assignments will be discussed in the tutorials.

If you have questions concerning the assignments or tutorials, please do not hesitate to contact Sarah Schäffer.

Three groups are scheduled:

  • Group T1:
    Tuesday, 2-4 pm, Bldg. E1.3, Seminar Room 016
  • Group T2:
    Tuesday, 4-6 pm, Bldg. E1.3, Seminar Room 016
  • Group W1:
    Wednesday, 2-4 pm, Bldg. E2.5, Seminar Room 4 (Room U16)

The tutors 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, t2, or w1).

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


There will be two written exams:

The first written exam will take place on
Wednesday, February 18, 2015 from 2:00 pm to 5:00 pm
in Building E 1.3, Lecture Hall 002.

The second written exam will take place on
Tuesday, April 14, 2015 from 2:00 to 5:00 pm.
in Building E 1.3, Lecture Hall 002.

In order to qualify for the exams you must

  • attend 80% of the tutorials (we do check)
  • gain 50% of all points from the homework assignments

In case of qualification, you are allowed to take part in both exams. The better grade counts.

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 Sarah Schäffer as soon as possible.

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

These are the rules during the exams:

  • For the exams, you can use the course material (including lecture notes and example solutions from this web page) and hand-written notes, but neither books nor any other printed material.
  • Pocket calculators are not allowed.
  • Mobile phones, PDAs, laptops and other electronic devices have to be turned off.
  • Please keep the student ID card ready for an attendance check during the exam.
  • You are not allowed to take the exam sheets with you.
  • Solutions that are written with pencil will not be graded.

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

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 first written exam has the opportunity to inspect his/her graded solutions in room 4.10 in Bldg. E1.7 on Wednesday, February 25th, 2015, from 2:15 pm to 3:45 pm.

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 21st, 2015, from 4:15 pm to 5:45 pm.


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.

DateTopic
21/10 Introduction, Overview
24/10 Linear Diffusion I: Basic Concepts
(contains classroom assignment C1 and homework H1)
28/10 Linear Diffusion II: Numerics, Limitations, Alternatives
31/10 Nonlinear Isotropic Diffusion I: Modelling and Continuous Theory
(contains classroom assignment C2 and homework H2)
04/11 Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory
07/11 Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
(contains classroom assignment C3 and homework H3)
11/11 Nonlinear Anisotropic Diffusion I: Modelling
14/11 Nonlinear Anisotropic Diffusion II: Continuous and Discrete Theory
(contains classroom assignment C4 and homework H4)
18/11 Nonlinear Anisotropic Diffusion III: Efficient Algorithms
21/11 Nonlinear Diffusion: Parameter Selection
(contains classroom assignment C5 and homework H5)
25/11 Variational Methods I: Basic Ideas
28/11 Variational Methods II: Discrete Aspects
(contains classroom assignment C6 and homework H6)
02/12 Variational Methods III: TV Regularisation and Primal-Dual Methods
05/12 Variational Methods IV: Functionals of Two Variables
(contains classroom assignment C7 and homework H7)
09/12 Vector- and Matrix-Valued Images
12/12 Unification of Denoising Methods
(contains classroom assignment C8 and homework H8)
16/12 Osmosis I: Continuous Theory and Modelling
19/12 Osmosis II: Discrete Theory and Efficient Algorithms
(contains classroom assignment C9 and homework H9)
06/01 Image Sequence Analysis I: Models for the Smoothness Term
09/01 Image Sequence Analysis II: Models for the Data Term
(contains classroom assignment C10 and homework H10)
13/01 Image Sequence Analysis III: Practical Aspects
16/01 Image Sequence Analysis IV: Numerical Methods
(contains classroom assignment C11 and homework H11)
20/01 Continuous-Scale Morphology I: Basic Ideas
23/01 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
(contains classroom assignment C12 and homework H12)
27/01 Curvature-Based Morphology I: Mean Curvature Motion
30/01 Curvature-Based Morphology II: Affine Morphological Scale-Space
(contains classroom assignment C13 and homework H13)
03/02 Self-Snakes and Active Contours
06/02 PDE-Based Image Compression I: Data Selection
(please take a look at the self-test problems)
10/02 PDE-Based Image Compression II: Optimised Encoding and Better PDEs
13/02 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.

DateTopic
19/07 Self Test Problem Sheet


Here you can download the material for the programming assignments:

Date Topic
24/10 H1 - Linear Diffusion, Gaussian Convolution
31/10 H2 - Linear Diffusion
07/11 H3 - Nonlinear Isotropic Diffusion
14/11 H4 - Anisotropic Diffusion
21/11 H5 - FED, Decorrelation
28/11 H6 - Diffusion-Reaction Methods
05/12 H7 - Primal-Dual Methods for TV Regularisation
12/12 H8 - Iterated Bilateral Filtering
19/12 H9 - Osmosis
16/01 H11 - Optic Flow
23/01 H12 - Morphology
30/01 H13 - Curvature-Based Morphology


  • 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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