Differential Geometric Aspects of Image Processing

Summer Term 2008

Differential Geometric Aspects of Image Processing

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

Summer 2008

Lectures (2h) with exercises (1h), summer term 2008

Lectures: Tuesdays 14–16 (2 p.m.–4 p.m.), Bld. E1.3, Lecture hall 001
Tutorials: Wednesdays 18–20 (6 p.m.–8 p.m.) every other week, Bld. E2.5, room H07 (»Zeichensaal«), starting April 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.

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Registration for the course: Online registration has been closed. Please e-mail Martin Welk if you have missed it.

HISPOS registration period: April 28–May 16,2008 (strict!)
In case your study programme is subject to HISPOS, you will need to register for this course also in the HISPOS system! Otherwise your exam may be invalid!
To my best knowledge (without guarantee) HISPOS affects all Computer Science programmes, bachelor/master studies in Mathematics, but not Visual Computing and diploma programmes in Mathematics.
Be aware that neither registration in HISPOS can replace registration with the lecturer nor vice versa.

Exam schedule: There will be two written exams

  • Thursday, July 24, 1400–1530h (2:00–3:30 pm), building E1.3, lecture hall 001,
  • Thursday, October 16, 1400–1530h (2:00–3:30 pm), building E1.3, lecture hall 001.

You can take one or both exams, the better grade counts.

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 exam at end of course. Two exams are offered:

  • Thursday, July 24, 1400–1530h (2:00–3:30 pm), building E1.3, lecture hall 001,
  • Thursday, October 16, 1400–1530h (2:00–3:30 pm), building E1.3, lecture hall 001.

You can take one or both exams, the better grade counts.

A semester apparatus (shared with the course Differential Equations in Image Processing and Computer Vision) is available in the Campus Library for Computer Science and Applied Mathematics.

  • F. Cao, Geometric Curve Evolution and Image Processing. 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 Last update
1 Introduction, Basic Concepts April 15 2008.04.16.1625
2 Curves and Curve Evolutions in Plane April 22 2008.04.29.1840
3 Level Sets April 29 2008.04.29.1838
4 Variational Approaches and Gradient Descents May 13 2008.05.13.1900
5 Curvature Motion in Different Geometries May 20 2008.05.21.1324
6 Curves and Surfaces in Euclidean Space May 27 2008.07.10.1436
7 Surface Evolution and Surface Diffusion June 3 2008.06.03.2004
8 Surface Smoothing June 10 2008.06.10.1913
9 Textured Surface Diffusion, Beltrami Flow, and Geodesic Active Contours June 17 2008.06.17.1836
10 Extensions to Geodesic Active Contours June 24 2008.06.24.2000
11 PDE Filters for Multi-Channel Images July 1 2008.07.01.1958
12 Advanced Differential Geometric Ideas in Multi-Channel Image Processing July 8 2008.07.08.1212
13 Applications of Differential Geometric Ideas July 15 2008.07.15.1054

Martin Welk / March 30–October 6, 2008

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