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Differential Equations in Image Processing and Computer Vision

Summer Term 2007

Differential Equations in Image Processing and Computer Vision

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

Coordinator of tutorials: Dr. Bernhard Burgeth
Office hours: Tuesday, 15:00 - 16:00.

Summer Term 2007

NEWS: Results of the second written exam are available now, see below.

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

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

Tutorials: 2 hours each week; see below.

PrerequisitesSynopsisPlanned ContentsAssignmentsWritten ExamsReferences

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 master thesis in our group.

17/4 Introduction, Overview
20/4 Linear Diffusion I: Basic Concepts
(contains theoretical assignment T1)
24/4 Linear Diffusion II: Numerics, Limitations, Alternatives
(contains classroom assignment C1)
27/4 Nonlinear Isotropic Diffusion I: Modeling and Continuous Theory
(contains programming assignment P1)
4/5 Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory
(contains theoretical assignment T2)
8/5 Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
(contains classroom assignment C2)
11/5 Nonlinear Anisotropic Diffusion I: Modelling
(contains programming assignment P2)
15/5 Nonlinear Anisotropic Diffusion II: Theoretical and Numerical Aspects
18/5 Nonlinear Diffusion: Parameter Selection (Corrected)
(contains theoretical assignment T3)
22/5 Variational Methods I: Basic Ideas
(contains classroom assignment C3)
25/5 Variational Methods II: Discrete Aspects
(contains programming assignment P3)
29/5 Variational Methods III: TV Denoising, Equivalence Results
1/6 Variational Methods IV: Functionals of Two Variables
(contains theoretical assignment T4)
5/6 Vector- and Matrix-Valued Images
(contains classroom assignment C4)
8/6 PDE-Based Image Interpolation
(contains programming assignment P4)
12/6 Image Sequence Analysis I: Models for the Smoothness Term
15/6 Image Sequence Analysis II: Models for the Data Term
(contains theoretical assignment T5)
19/6 Image Sequence Analysis III: Large Displacements and High Accuracy Methods
(contains classroom assignment C5)
22/6 Image Sequence Analysis IV: Numerical Methods
(contains programming assignment P5)
26/6 Continuous-Scale Morphology I: Basic Ideas
29/6 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
(contains theoretical assignment T6)
3/7 Curvature-Based Morphology I: Mean Curvature Motion
(contains classroom assignment C6)
6/7 Curvature-Based Morphology II: Affine Morphological Scale-Space
(contains programming assignment P6)
10/7 Self-Snakes and Active Contours
(contains important announcements)
13/7 Unification of Denoising Methods
17/7 Summary and Outlook

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:

27/4 P1 - Linear Diffusion, Gaussian Convolution
11/5 P2 - Isotropic Nonlinear Diffusion
25/5 P3 - Anisotropic Nonlinear Diffusion
8/6 P4 - EED-Based Inpainting
22/6 P5 - Optic Flow
6/7 P6 - Morphology

Three groups are scheduled for Tuesday and Wednesday:

  • Group T1 (Sebastiano Barbieri):
    Tue, 16-18, Bldg. E1.3, room SR 15 (theory) and CIP pool room 105 (programming)

  • Group W1 (Luis Pizarro):
    Wed, 14-16, Bldg. E2.5, lecture hall 3 (theory) and CIP pool room 105, Bldg. E1.3 (programming)

  • Group W2 (Luis Pizarro):
    Wed, 16-18, Bldg. E2.5, lecture hall 3 (theory) and CIP pool room 105, Bldg. E1.3 (programming)

You could enroll for a tutorial from Tue, Apr. 17, 2007, 14:00h
to Fri, Apr. 20, 2007, 16:00h.

The first written exam has taken place on July 24 from 2 to 5 PM
in Building E13, Lecture Halls 002 and 003.

The second written exam has taken place on October 11 from 2 to 5 PM
in Building E13, Lecture Hall 002.

The results of the second written exam can be found here.

The following thresholds were applied in determining the grades in the second exam:

  • grade 1.0 : 37 - 39 points (2)
  • grade 1.3 : 35 - 36 points (2)
  • grade 1.7 : 33 - 34 points (0)
  • grade 2.0 : 31 - 32 points (2)
  • grade 2.3 : 29 - 30 points (0)
  • grade 2.7 : 27 - 28 points (1)
  • grade 3.0 : 25 - 26 points (0)
  • grade 3.3 : 23 - 24 points (2)
  • grade 3.7 : 21 - 22 points (1)
  • grade 4.0 : 19 - 20 points (4)
  • grade 5.0 : 00 - 18 points (2)

  • 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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