Mathematical Image Analysis Group, Saarland University

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.

News – Entrance requirements – Contents – Assessments / Exams – References – Download

News

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.

Entrance requirements

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.

Contents

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.

Assessments / Exams

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.

References

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.

Download

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