Welcome to the homepage of the lecture

Differential Equations in Image Processing and Computer Vision

Summer Term 2010

Differential Equations in Image Processing and Computer Vision

Two Computer Science Teaching Awards (summer term 2003 and 2006)
One Mathematics Teaching Award (summer term 2009)

Lecturer: Dr. Andrés Bruhn
Office hours: Friday, 14:15 - 15:15.

Coordinator of tutorials: Markus Mainberger

Summer Term 2010

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

Lectures: Tuesday, Friday 10-12 am c.t., Building E 1 3, Lecture Hall 001

First lecture: Tuesday, April 13, 2010

Tutorials: 2 hours each week; see below.


NEWS: The certificates are ready and can be fetched in room 111, building E2.4 (Geschaeftszimmer Mathematik, Frau Voss, opening hours for certificates: Mon-Thu 9.00-11.30 am).



SynopsisPrerequisitesAssignments and TutorialsWritten ExamsContents Material for the Programming Assignments References



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 usually to be submitted Fridays, 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 Markus Mainberger. The tutorials are conducted by Pascal Peter and Christian Schmaltz.

Three groups are scheduled Wednesdays and Thursdays:

  • Group 1 (Christian Schmaltz):
    Wed, 10-12 am, Bldg. E1.3, seminar room 015
  • Group 2 (Pascal Peter):
    Thu, 2-4 pm, Bldg. E2.5, seminar room 2 (H04)
  • Group 3 (Pascal Peter):
    Thu, 4-6 pm, Bldg. E2.5, seminar room 2 (H04)

The tutors can be reached via the mail addresses:
dic-g# -- at -- mia.uni-saarland.de
where # has to be replaced by the group number.

You had to register for the lecture and enroll for a tutorial on this web page from Tue, Apr. 13, 2010, 2.00 pm to Fri, Apr. 16, 2010, 4.00 pm. Still you can check in which group you are via web form.


There will be two written exams:

The first written exam has taken place on
Tuesday, July 27, 2010 from 2:00 pm to 5:00 pm,
in Building E 2.5, Lecture Hall I

The second written exam will take place on
Tuesday, October 12, 2010 from 2:00 to 5:00 pm,
in Building E 2.2, AudioMO.

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.

Here you can download the admission list.

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.
  • 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.

Each student who has participated in the first written exam had the opportunity to inspect his/her graded solutions in room 3.06 in Bldg. E1.1 on Friday, August 6th, 2010, from 2:00 pm to 3:30 pm.


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 had the opportunity to inspect his/her graded solutions in room 3.06 in Bldg. E1.1 on Monday, October 18th, 2010, from 3:30 pm to 4:30 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
13/04 Introduction, Overview
16/04 Linear Diffusion I: Basic Concepts
(contains classroom assignment C1 and homework H1)
20/04 Linear Diffusion II: Numerics, Limitations, Alternatives
23/04 Linear Diffusion III: The Scale Invariant Feature Transform (SIFT)
(contains classroom assignment C2 and homework H2)
27/04 Nonlinear Isotropic Diffusion I: Modeling and Continuous Theory
30/04 Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory
(contains classroom assignment C3 and homework H3)
04/05 Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
07/05 Nonlinear Anisotropic Diffusion I: Modelling
(contains classroom assignment C4 and homework H4)
11/05 Nonlinear Anisotropic Diffusion II: Theoretical and Numerical Aspects
14/05 Nonlinear Diffusion: Parameter Selection
(contains classroom assignment C5 and homework H5)
18/05 Variational Methods I: Basic Ideas
21/05 Variational Methods II: Discrete Aspects
(contains classroom assignment C6 and homework H6)
25/05 Variational Methods III: TV Denoising, Equivalence Results
28/05 Variational Methods IV: Functionals of Two Variables
(contains classroom assignment C7 and homework H7)
01/06 Vector- and Matrix-Valued Images
04/06 Unification of Denoising Methods
(contains classroom assignment C8 and homework H8)
08/06 Image Sequence Analysis I: Models for the Smoothness Term
11/06 Image Sequence Analysis II: Models for the Data Term
(contains classroom assignment C9 and homework H9)
15/06 Image Sequence Analysis III: Large Displacements, High Accuracy Methods, Confidence Measures
18/06 Image Sequence Analysis IV: Numerical Methods
(contains classroom assignment C10 and homework H10)
22/06 Continuous-Scale Morphology I: Basic Ideas
25/06 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
(contains classroom assignment C11 and homework H11)
29/06 Curvature-Based Morphology I: Mean Curvature Motion
02/07 Curvature-Based Morphology II: Affine Morphological Scale-Space
(contains classroom assignment C12 and homework H12)
06/07 Self-Snakes and Active Contours
09/07 PDE-Based Image Interpolation I: Basic Ideas
(please take a look at the self-test problem)
13/07 PDE-Based Image Interpolation II: Image Compression
16/07 Current Research in the MIA Group
20/07 Summary and Outlook


Here you can download the material for the programming assignments:

Date Topic
16/04 H1 - Linear Diffusion, Gaussian Convolution
23/04 H2 - Linear Diffusion
07/05 H4 - Nonlinear Isotropic Diffusion
14/05 H5 - Nonlinear Anisotropic Diffusion
21/05 H6 - Selection of the Stopping Time
28/05 H7 - Diffusion-Reaction Methods
04/06 H8 - Iterated Bilateral Filtering
18/06 H10 - Optic Flow
25/06 H11 - Morphology
02/07 H12 - 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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