Saarland University | Mathematical Image Analysis Group | Teaching

Summer Term 2004

Differential Equations in Image Processing and Computer Vision

Dr. Martin Welk

Lectures (4h) with theoretical and programming exercises (2h)
(9 credit points)

Monday 14-16 c.t., Building 45, Lecture Hall 3;
Thursday 14-16 c.t., Building 45, Lecture Hall 2

First lecture: Monday, April 26, 2004

(please register – to register, send an e-mail to Martin Welk)

Theoretic and programming exercises are held in weekly alternation.
The exercises are supervised by Dr. Bernhard Burgeth.

In the first week (April 27/28), a mathematical introduction was given, comprising multidimensional differential calculus and basic facts about differential equations.


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


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 diploma or master thesis in our group.


No. Date Topic Assignments
1 26/4 Introduction, Overview
2 29/4 Linear Diffusion Filtering I: Basic Concepts T1
3 3/5 Linear Diffusion Filtering II: Numerical Aspects, Limitations, Alternatives P1
4 6/5 Nonlinear Isotropic Diffusion Filtering I: Modeling and Continuous Theory
5 10/5 Nonlinear Isotropic Diffusion Filtering II: Semidiscrete and Discrete Theory
6 13/5 Nonlinear Isotropic Diffusion Filtering III: Efficient Sequential and Parallel Algorithms
7 17/5 Nonlinear Anisotropic Diffusion I: Modeling P2, T2
8 24/5 Nonlinear Anisotropic Diffusion II: Theoretical and Discrete Aspects P3
9 27/5 Diffusion Filtering: Parameter Selection T3
10 3/6 Variational Methods I: Basic Ideas
11 7/6 Variational Methods II: Discrete Aspects T4
12 14/6 Variational Methods III: TV Denoising, Equivalence Results P4
13 17/6 Variational Methods IV: Mumford-Shah, Diffusion-Reaction
14 21/6 Vector- and Matrix-Valued Images
15 24/6 Image Sequence Analysis I: Global Methods T5
16 28/6 Image Sequence Analysis II: Local Methods P5
17 1/7 Image Sequence Analysis III: Combined Local-Global Methods
18 5/7 Image Sequence Analysis IV: Numerical Methods for the Variational Approaches
19 8/7 Continuous-Scale Morphology I: Basic Ideas, PDE Formulation, Shock Filters T6
20 12/7 Continuous-Scale Morphology II – Curvature-Based Morphology I P6
21 15/7 Curvature-Based Morphology II: Affine-Invariant Evolution, Extensions, Applications
22 19/7 Image Segmentation by Active Contour Methods
23 22/7 Summary and Outlook


A balance of theoretical and programming assignments will be offered. Previous experiences have shown that they are very helpful for understanding the methods that are presented in the lectures. Exercises are supervised by Bernhard Burgeth and Mostafa Khabouze.

Programming Assignments

No. Topic Tutorials Problem sheet Program and image files
P1 Linear Diffusion Filtering 4–5/5 Lecture 3 (3/5) Download tar file
P2 Isotropic Nonlinear Diffusion 18–19/5 Lecture 7 (17/5) Download tar file
P3 Anisotropic Nonlinear Diffusion 25–26/5 Lecture 8 (24/5) Download tar file
P4 Diffusion--Reaction Filtering 15–16/6 Lecture 12 (14/6) Download tar file
P5 Variational Optic Flow Computation 29–30/6 Lecture 16 (28/6) Download tar file
P6 Classical and Curvature-Based Morphology 13–14/7 Lecture 20 (12/7) Download tar file

Theory Assignments

No. Topic(s) Tutorials Problem sheet Deadline
T1 Linear diffusion, convolution, finite differences 11–12/5 Lecture 2 (29/4) 6/5
T2 Isotropic nonlinear diffusion, numerical schemes 1–2/6 Lecture 7 (17/5) 27/5
T3 Anisotropic diffusion, stopping criteria 8–9/6 Lecture 9 (27/5) 3/6
T4 Euler-Lagrange equations; diffusion-reaction discretisations 22–23/6 Lecture 11 (7/6) 17/6
T5 Energy functionals; wavelet shrinkage; optic flow 6–7/7 Lecture 15 (24/6) 1/7
T6 Half-quadratic regularisation, iterative solvers, morphology 20–21/7 Lecture 19 (8/7) 15/7


Martin Welk / July 22, 2004 / back to MIA home.
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