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)

Lectures:
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

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

• Tuesday 2-4 p.m.
  Theoretic exercises: building 27.1, seminar room no. 7, Programming exercises: building 45, room 105
• Wednesday 9-11 a.m.
  Theoretic exercises: building 45, seminar room no. 15, Programming exercises: building 45, room 105
• Wednesday 4-6 p.m.
  Theoretic exercises: building 45, seminar room no. 16, Programming exercises: building 45, room 105

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.

Prerequisites

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.

Synopsis

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.

Contents

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	–

Assignments

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

References

• J. Weickert: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.

• G. Sapiro: Geometric Partial Differential Equations in Image Analysis. Cambridge University Press, 2001.

• G. Aubert and P. Kornprobst: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, New York, 2002.

• Articles from journals and conferences.