Differential Geometric Aspects of Image Processing

Summer Term 2015

Differential Geometric Aspects of Image Processing

Lecturer: Martin Schmidt

Examiner: Prof. Dr. Joachim Weickert

Summer Term 2015

Lectures (2h) with exercises (1h), summer term 2015

Lectures: Monday 12.15-13-.5, Building E1.7, Room 4.12,
First Lecture: Tuesday, April 21, 2015

Tutorials: Friday 10-12, every other week, Building E1.3, Lecture Hall 003,
Next Tutorial: Friday, June 26, 2015

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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We moved the lectures to Monday, 12-14, E.17, 4.12

Please register for this course in HISPOS until May 27.

There is NO lecture on June 2.

Start of the lectures:
We decided to start the lectures at 8.25 am and include a 5min break.

Registration for the course:
The registration period is over.

Please remember that you have to register online for the exam in the HISPOS system of the Saarland University.

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 on manifolds
  • active contours and active regions.

The homework assignments are intended to be solved at home and have to be submitted in the lecture break, or earlier. In order to qualify for the exam you must obtain 50% of the possible points on average.
If you have qualified for the exam, you may participate in both exams. The better grade counts.

  • Thursday, August 6, 2015, Building E1.1, Lecture Hall 002, 14:00-16:00
  • Tuesday, October 13, 2015, Building E1.1, Lecture Hall 002, 14:00-16:00

Please do not forget to bring your student ID card with you.

These are the rules during the exams:

  • For the exams, you can use the course material (including lecture notes and example solutions from this web page) and hand-written notes, but neither books nor any other printed material.
  • Pocket calculators are not allowed.
  • Mobile phones, PDAs, laptops and other electronic devices have to be turned off.
  • Please keep the student ID card ready for an attendance check during the exam.
  • You are not allowed to take the exam sheets with you.
  • Solutions that are written with pencil will not be graded.

  • 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.

Further references will be given during the lecture.

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

Slides of the lecture

Date Title (Last) Update
April 21 Introduction and Basic Concepts
April 28 Curves and Curve Evolutions
May 5 Level Sets Update 1 (Slide 8): May 11
May 12 Variational Approaches and Gradient Descent
May 19 Curvature Motion in Different Geometries
May 26 Curves and Surfaces in Euclidean Space
June 2 NO lecture
June 9 Surface Evolution and Surface Diffusion
June 15 Surface Smoothing
June 21 Beltrami Framework, Geodesic Active Contours
June 21 Extensions to Geodesic Active Contours
July 6 PDE Filters for Multi-Channel Images
July 13 Advanced Differential Geometric Ideas in Multi-Channel Image Processing
July 20 Applications of Differential Geometric Ideas

Classroom Exercises

No. Solution Date of Discussion
C1 Solution to C1 April 24
C2 Solution to C2 May 8
C3 Solution to C3 May 22
C4 Solution to C4 June 12
C5 Solution to C5 June 26

Homework Exercises

No. Solution Deadline for Submission Date of Discussion (Last) Update
H1 Solution to H1 May 5, lecture break May 8
H2 Solution to H2 May 19, lecture break May 22 May 29. Update 1:
Minus sign
added in H2.1
H3 Solution to H3 June 9, lecture break June 12
H4 Solution to H4 June 22, lecture break June 26
H5 Solution to H5 July 6, lecture break July 10

Martin Schmidt / July 24, 2015

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