Differential Geometric Aspects of Image Processing

Winter Term 2019

Differential Geometric Aspects of Image Processing

Lecturer: Dr. Marcelo Cárdenas
Office hour: Tuesday, 13:00 - 14:00.

Winter Term 2019

Lectures (3h) with exercises (1h), winter term 2019

Lectures: Tuesdays, 16-18, Building E1.3, Room HS 003,
Thursdays, 14-16, Building E1.3, Room HS 003

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 requirementsContents References

Undergraduate knowledge of mathematics. Students should be familiar with basic concepts of multivariate calculus and linear algebra as covered in introductory maths course. 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 of the following Thursday. 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.

  • Tuesday, February 25, 2020, Building E1.3, Lecture Hall 016, 14:00-16:00
  • Monday, March 16, 2020, Building E1.1, Lecture Hall 106, 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.

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

Slides of the lecture

Date Title (Last) Update
October 15 Introduction and Basic Concepts
October 17 Curves in the Euclidean Space
October 22 Curve Evolutions in the Plane
October 29 Evolution of Level Lines
October 31 Variational Approaches and Gradient Descent
November 5 Curvature Motion in Different Geometries
November 12 Curves and Surfaces
November 14 Regular Surfaces in 3D
November 19 Curvature of Surfaces
November 26 Surface Evolutions
November 28 Parallel Transport and Geodesics
December 3 Geodesics of Surfaces
December 10 Finding Shortest Paths
December 17 Hamilton Jacobi / Laplace Beltrami
January 7 Diffusion on Surfaces
January 14 Surfaces Diffusion
January 16 Introduction to Finite Elemets I
January 21 Introduction to Finite Elemets II
January 21 Surface Inpainting
February 4 Low Dim. Manifold Model
February 6 Beltrami Flow

Classroom Exercises

No. Solution Date of Discussion
C1 Solution to C1 Oct 24
C2 Solution to C2 Nov 7
C3 Solution to C3 Nov 21
C4 Solution to C4 Dec 5
C5 Solution to C5 Dec 12
C6 Solution to C6 Dec 19
C7 Solution to C7 Jan 9

Homework Exercises

No. Solution Deadline for Submission (Last) Update
H1 Solution to H1 Oct 31, lecture break
H2 Solution to H2 Nov 14, lecture break
H3 Solution to H3 Dec 12, lecture break
H4 Solution to H4 Feb 11, lecture break

  • 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.

References for topological and differential geometric foundations:

  • M. do Carmo, Differential geometry of curves and surfaces. Prentice Hall 1976.
  • M. do Carmo, Riemmanian Geometry. Birkhaeuser 1993.
  • H. W. Guggenheimer, Differential Geometry. McGrawHill 1963.
  • W. Klingenberg, Riemannian Geometry. de Gruyter 1982.

Further references will be given during the lecture.

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