Mathematical Image Analysis Group, Saarland University

Welcome to the homepage of the lecture

Differential Equations in Image Processing and Computer Vision

Summer Term 2008

Differential Equations in Image Processing and Computer Vision

Lecturer: Prof. Dr. Joachim Weickert
Office hours: Friday, 14:15 - 15:15.

Coordinator of tutorials: Dr. Stephan Didas
Office hours: Wednesday, 14:00 - 15:00.

Summer Term 2008

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

Lectures: Tuesday, Friday 10-12 c.t., Building E13, Lecture Hall 1

First lecture: Tuesday, April 15, 2008

Tutorials: 2 hours each week; see below.

News:
The results of the second written exam are online, together with the statistics.
The inspection takes place on upcoming Wednesday, October 29, from 2-3 p.m. in room 3.06, building E1 1

Still, the results of the first written exam can be found here. The statistics of the results can be found here.

Prerequisites – Synopsis – Planned Contents – Assignments –
Tutorials – Written Exams – References

Prerequisites

Equally suited for students of 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.

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 master thesis in our group.

Contents

Date Topic
15/4 Introduction, Overview

18/4 Linear Diffusion I: Basic Concepts
(contains theoretical homework T1)

22/4 Linear Diffusion II: Numerics, Limitations, Alternatives
(contains classroom assignment C1)

25/4 Nonlinear Isotropic Diffusion I: Modeling and Continuous Theory
(contains programming assignment P1)

29/4 Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory

2/5 Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
(contains theoretical homework T2)

6/5 Nonlinear Anisotropic Diffusion I: Modelling
(contains classroom assignment C2)

9/5 Nonlinear Anisotropic Diffusion II: Theoretical and Numerical Aspects
(contains programming assignment P2)

13/5 Nonlinear Diffusion: Parameter Selection

16/5 Variational Methods I: Basic Ideas
(contains theoretical homework T3)

20/5 Variational Methods II: Discrete Aspects
(contains classroom assignment C3)

23/5 Variational Methods III: TV Denoising, Equivalence Results
(contains programming assignment P3)

27/5 Variational Methods IV: Functionals of Two Variables

30/5 Vector- and Matrix-Valued Images
(contains theoretical homework T4)

3/6 PDE-Based Image Interpolation
(contains classroom assignment C4)

6/6 Image Sequence Analysis I: Models for the Smoothness Term
(contains programming assignment P4)

10/6 Image Sequence Analysis II: Models for the Data Term

13/6 Image Sequence Analysis III: Large Displacements, High Accuracy Methods, Illumination Changes
(contains theoretical homework T5)

17/6 Image Sequence Analysis IV: Numerical Methods
(contains classroom assignment C5)

20/6 Continuous-Scale Morphology I: Basic Ideas
(contains programming assignment P5)

24/6 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology

27/6 Curvature-Based Morphology I: Mean Curvature Motion
(contains theoretical homework T6)

1/7 Curvature-Based Morphology II: Affine Morphological Scale-Space
(contains classroom assignment C6)

4/7 Self-Snakes and Active Contours
(contains programming assignment P6)

8/7 Unification of Denoising Methods

11/7 Summary and Outlook

Assignments

A combination of theoretical, programming and classroom assignments is offered. Previous experiences have shown that they are very helpful for understanding the methods.
Here you can download the material for the programming assignments:

Date Topic

25/4 P1 - Linear Diffusion, Gaussian Convolution

9/5 P2 - Nonlinear Isotropic Diffusion

23/5 P3 - Anisotropic Diffusion and Variational Methods

6/6 P4 - EED-Based Inpainting

20/6 P5 - Optic Flow

04/07 P6 - (Curvature-Based) Morphology

Registration for Tutorials

Three groups are scheduled for Tuesday and Thursday:

Group 1 (Luis Pizarro):
Tue, 16-18, Bldg. E2.4, seminar room 3 (theory) and bldg. E1 3 CIP pool room 104 (programming)
Group 2 (Markus Mainberger):
Thu, 12-14, Bldg. E1.3, seminar room 016 (theory) and bldg. E1 3 CIP pool room 104 (programming)
Group 3 (Markus Mainberger):
Thu, 16-18, Bldg. E2.4, seminar room 3 (theory) and bldg. E1 3 CIP pool room 104 (programming)

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 could register for the lecture and enroll for a tutorial from Tue, Apr. 15, 2008, 14:00h to Fri, Apr. 18, 2008, 16:00h.

Written Exams:

The first written exam has taken place on July 22 from 2 to 5 PM
in building E2 5, lecture hall I.

The second written exam will take place on October 14 from 2 to 5 PM
in building E2 5, lecture hall I.

You can bring your lecture notes and notes from the tutorials. You have to pass one exam, and the better grade counts.

In order to qualify for the exam you must

attend 80% of the programming and the theoretical tutorials (we do check)
solve 50% of all assignments (theory and programming) correctly. Working in groups of up to 3 people is permitted, but all persons must be in the same tutorial group.

References

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. Springer, New York, 2002.
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.