Lecture

Differential Equations in Image Processing and Computer Vision

Summer Term 2003

Differential Equations in Image Processing and Computer Vision

Lecturer: Prof. Dr. Joachim Weickert

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

The written exam took place on July 28. The results are displayed on floor 4, building 27.2, and can also be retrieved here. Opportunity for inspection of the graded exam sheets is given on August 21, from 4 p.m., in building 27.2, room 26 (basement, next to Mr. Hoffmann's flat).

Lectures: Tuesday, Friday 11-13 c.t., Building 45, Lecture Hall 3
First lecture: Friday, April 25, 2003

Exercises (please register!):
Tuesday 4-6 p.m. (group A)

• Theoretic exercises: lecture hall H07 (so-called Zeichensaal), basement of building 27.2;
  
• Programming exercises: room 012, building 45
  Wednesday 2-4 p.m. (group B)
  
• Theoretic exercises: room U12, building 36.1
  
• Programming exercises: room 012, building 45

Theoretic and programming exercises are held in weekly alternation.

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.

Date	Topic
25/4	Introduction, Overview
29/5	Mathematical Introduction
2/5	Linear Diffusion I: Basic Concepts
6/5	Linear Diffusion II: Numerics, Limitations, Alternatives
9/5	Nonlinear Isotropic Diffusion I: Modeling and Continuous Theory
13/5	Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory
16/5	Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
20/5	Nonlinear Anisotropic Diffusion I: Modeling
27/5	Nonlinear Anisotropic Diffusion II: Continuous and Discrete Aspects
30/5	Nonlinear Diffusion: Parameter Selection
3/6	Variational Methods I: Basic Ideas
6/6	Variational Methods II: Discrete Aspects
17/6	Variational Methods III: TV Denoising, Equivalence Results
20/6	Variational Methods IV: Mumford-Shah, Diffusion-Reaction
24/6	Vector- and Matrix-Valued Images
27/6	Image Sequence Analysis I: Global Methods
1/7	Image Sequence Analysis II: Local Methods
4/7	Image Sequence Analysis III: Combined Local-Global Methods
8/7	Image Sequence Analysis IV: Numerical Methods
11/7	Classical Continuous-Scale Morphology I: Basic Ideas
15/7	Classical Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
18/7	Curvature-Based Morphology I: Basic Ideas
22/7	Curvature-Based Morphology II: Applications
25/7	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. The exercises are supervised by Martin Welk and Dmitry Ovrutskiy.

Date	Topic
6/5	Assignment 1: Linear Diffusion Filtering
13/5	Assignment 3: Isotropic Nonlinear Diffusion Filtering
3/6	Assignment 5: Anisotropic Diffusion Filtering
17/6	Assignment 7: Variational Image Restoration
8/7	Assignment 9: Variational Optic Flow Computation
22/7	Assignment 11: Morphological Operations

• 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.

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