Differential Geometric Aspects of Image Processing

Summer Term 2006

Differential Geometric Aspects of Image Processing

Dr. Martin Welk (bld. E11, room 3.10.1, phone 0681-302-57343)

Summer 2006


Lectures (2h) with exercises (1h), summer term 2006

Lectures: Wednesdays 14–16 (2 p.m.–4 p.m.), Bld. E13, Lecture hall 001
Tutorials: Mondays 16–18 (4 p.m.–6 p.m.) every other week, Bld. E13, seminar room no. 16


Specialised course in mathematical image analysis, suitable for students in mathematics and computer science programs.
Participants learn how concepts of differential geometry can be applied in image processing.


Entrance requirementsContentsAssessments / ExamsReferencesDownload



Undergraduate knowledge of mathematics. For computer science students, this requirement is met by having completed the Mathematics for Computer Scientists lecture cycle.

Mathematical prerequisites which exceed the basic mathematics courses are provided within the lecture. Previous knowledge in either digital image processing or differential geometry is therefore helpful but not required.


The course is concerned with modern methods of digital image processing which rely on the differential geometry of curves and surfaces. This includes methods of image enhancement (like smoothing procedures) as well as feature extraction and segmentation (like locating contours using active contour models).

The lecture aims at combining theoretical foundation directly with a variety of applications from the above-mentioned fields; the range of topics extends up to recent research problems.

An introduction to the relevant concepts and results from differential geometry will be included in the course.

Topics include:

  • curves and surfaces in Euclidean space
  • level sets
  • curve and surface evolutions
  • variational formulations and gradient descents
  • diffusion of scalar and non-scalar data
  • diffusion on manifolds
  • active contours and active regions.


Written or oral exam at end of course


  • F. Cao, Geometric Curve Evolution and Image Processing. (see also the PDF script) Lecture Notes in Mathematics, vol. 1805, Springer, Berlin 2003.
  • R. Kimmel, Numerical Geometry of Images. Springer, Berlin 2004.
  • S. Osher, N. Paragios, eds., Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, Berlin 2003.
  • G. Sapiro, Geometric Partial Differential Equations and Image Analysis. Cambridge University Press 2001.
  • Articles from journals and conferences.

Participants of the course can download the lecture materials here (access password-protected):

No. Title Date
1 Introduction and Basic Concepts April 19
2 Curve and Curve Evolutions in Plane April 24
  Classroom Exercise 1 April 26
3 Level Sets May 3
4 Variational Approaches and Gradient Descents May 17
  Classroom Exercise 2
(includes solutions of Exercise 1)
May 22
5 Curvature Motion in Different Geometries May 24
  Classroom Exercise 3
(includes solutions of Exercise 2)
May 29
6 Curves and Surfaces in Euclidean Space May 31
7 Surface Evolutions and Surface Diffusion June 7
  Classroom Exercise 4
(includes solutions of Exercise 3)
June 12
8 Surface Smoothing, Beltrami Framework June 14
9 Geodesic Active Contours and Related Models June 21
  Classroom Exercise 5
(includes solutions of Exercise 4)
June 26
10 Geodesic Active Regions; PDE Filters for Multi-Channel Images June 28
11 Extensions on Multi-Channel Images July 5
  Classroom Exercise 6 July 10
12 Applications of Differential Geometric Ideas July 12
  Solutions of Classroom Exercises 5, 6 July 18
13 Colour Editing; Summary and Outlook July 19




Martin Welk / August 10, 2005–July 19, 2006

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