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Differential Equations in Image Processing and Computer Vision

Summer Term 2006

Differential Equations in Image Processing and Computer Vision

Prof. Joachim Weickert

Summer Term 2006

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

Lectures: Tuesday, Friday 11-13 c.t., Building E13, Lecture Hall 1

First lecture: Friday, April 21, 2006

Tutorials: 2 hours each week; see below.

PrerequisitesSynopsisPlanned ContentsAssignmentsReferences

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.

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.

21/4 Introduction, Overview
25/4 Linear Diffusion I: Basic Concepts
28/4 Linear Diffusion II: Numerics, Limitations, Alternatives
2/5 Nonlinear Isotropic Diffusion I: Modeling and Continuous Theory
5/5 Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory
16/5 Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
19/5 Nonlinear Anisotropic Diffusion I: Modelling
23/5 Nonlinear Anisotropic Diffusion II: Theoretical and Numerical Aspects
26/5 Nonlinear Diffusion: Parameter Selection
30/5 Variational Methods I: Basic Ideas
2/6 Variational Methods II: Discrete Aspects
6/6 Variational Methods III: TV Denoising, Equivalence Results
9/6 Variational Methods IV: Mumford-Shah, Diffusion-Reaction
13/6 Vector- and Matrix-Valued Images
16/6 Image Interpolation
20/6 Image Sequence Analysis I: Models for the Smoothness Term
23/6 Image Sequence Analysis II: Models for the Data Term
27/6 Image Sequence Analysis III: Large Displacements and High Accuracy Methods
30/6 Image Sequence Analysis IV: Numerical Methods
4/7 Continuous-Scale Morphology I: Basic Ideas
7/7 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
11/7 Curvature-Based Morphology I: Mean Curvature Motion
14/7 Curvature-Based Morphology II: Affine Morphological Scale-Space
18/7 Self-Snakes and Active Contours
21/7 Summary and Outlook

A combination 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 Andres Bruhn and Dr. Michael Breuss.

2/5 P1 - Linear diffusion, convolution, finite differences
23/5 P2 - Nonlinear isotropic diffusion
6/6 P3 - Nonlinear anisotropic diffusion, diffusion-reaction filtering
20/6 P4 - Inpainting
4/7 P5 - Optic flow
18/7 P6 - Morphology, mean curvature motion

3 groups are scheduled for Tuesday and Wednesday,
the tutorials of group T2 will be held in German.

  • Group T1: Tue, 14-16,   Bldg. E13, room SR 15 (theory) and CIP 105 (programming)
  • Group T2: Tue, 16-18,   Bldg. E13, room SR 15 (theory) and CIP 105 (programming) - in German
  • Group W1: Wed, 14-16,   Bldg. E11, room U 12 (theory) and E13, room CIP 012 (programming)

You could enroll for a tutorial from Fri, Apr. 21, 2006, 14:00h
to Tue, Apr. 25, 2006, 16:00h.

The first written exam took place on July 26 from 2 to 5 PM in Building E13, Lecture Hall 002.

The second written exam took place on October 5 from 2 to 5 PM in
Building E2.5 (formerly 27.2), Lecture Hall I (Math Building)

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

The following thresholds were applied in determining the grades:

  • grade 1.0 : 55 - 62 points
  • grade 1.3 : 52 - 54 points
  • grade 1.7 : 50 - 51 points
  • grade 2.0 : 47 - 49 points
  • grade 2.3 : 44 - 46 points
  • grade 2.7 : 41 - 43 points
  • grade 3.0 : 38 - 40 points
  • grade 3.3 : 35 - 37 points
  • grade 3.7 : 33 - 34 points
  • grade 4.0 : 30 - 32 points
  • grade 5.0 : 0 - 29 points

The best results of both exams can be queried via our online query form.

Exam sheets of the second written exam can be inspected from October 16 to October 17 (monday to tuesday) after a personal appointment (via mail).

  • 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.

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