Welcome to the homepage of the lecture

Differential Equations in Image Processing and Computer Vision

Winter Term 2017 / 2018

Differential Equations in Image Processing and Computer Vision

Three Computer Science Teaching Awards (Summer Terms 2003 and 2006, Winter Term 2015)
One Mathematics Teaching Award (Summer Term 2009)

Lecturer: Prof. Joachim Weickert

Coordinator of tutorials: Aaron Wewior

Winter Term 2017 / 2018

Lectures (4h) with theoretical exercises (2h)
(9 ETCS points)

Tuesday, 8-10 a.m., Building E 1.3, Lecture Hall 001
Friday, 10-12 a.m., Building E 1.3, Lecture Hall 001

First lecture: Tuesday, October 17, 2017

Tutorials: 2 hours each week; see below.

NewsSynopsisPrerequisitesTutorialsRegistrationWritten ExamsContentsSelf TestMaterial for the Programming AssignmentsExample Solutions for the AssignmentsReferences

27.10. The lecture from Tuesday, October 31, will be moved to Monday, October 30, 6-8 p.m., Building E1.3, Lecture Hall 001.
The tutorial for group T1 from Tuesday, October 31, will be moved to Thursday, November 2, 4-6 p.m., Building E1.3, Seminar Room 016.
The tutorial for group T2 from Tuesday, October 31, will be moved to Thursday, November 2, 6-8 p.m., Building E1.3, Seminar Room 016.

20.10. Registration is closed.

20.10. The lectures on Tuesday start at 8:20 a.m.

17.10. Registration is open.

Many modern techniques in image processing and computer vision make use of methods based on partial differential equations (PDEs) and variational calculus. Moreover, many classical methods may be reinterpreted as approximations of PDE-based techniques. In this course we will get an in-depth insight into these methods. For each of these techniques, we will discuss the basic ideas as well as theoretical and algorithmic aspects. Examples from the fields of medical imaging and computer aided quality control illustrate the various application possibilities.

Since this class guides its participants to many research topics in our group, its attendance is required for everyone who wishes to pursue a master thesis in our group.

Equally suited for students of visual computing, mathematics and computer science. Requires undergraduate knowledge in mathematics (e.g. ''Mathematik für Informatiker I-III''). Knowledge in image processing or differential equations is useful, but not required. The lectures will be given in English.

A combination of classroom and homework assignments (including theoretical as well as programming problems) is offered. The classroom assignments are intended to be solved in the tutorials and are not graded. The homework assignments are intended to be solved at home and have to be submitted on Friday before the lecture. In order to qualify for the exam you must

  • attend 80% of the tutorials (we do check)
  • gain 50% of all points from the homework assignments

Working in groups of up to 3 people is permitted, but all persons must be in the same tutorial group. Both classroom as well as homework assignments will be discussed in the tutorials.

If you have questions concerning the assignments or tutorials, please do not hesitate to contact Aaron Wewior.

Two groups are scheduled:

  • Group T1:
    Tuesday, 10-12 a.m., Building E1.3, Seminar Room 016
    Tutor: Aaron Wewior
  • Group T2:
    Tuesday, 2-4 p.m., Building E1.3, Seminar Room 016
    Tutor: Tobias Alt

The tutorial group can be reached via the mail addresses:
dic-x -- at -- mia.uni-saarland.de
where x has to be replaced by the group name (t1 or t2).

Registration is now closed. You can still check in which group you are via here.

Please do not forget to register also in the HISPOS system.

The first written exam will take place on
Thursday, February 8, 2018 from 2:00 p.m. to 5:00 p.m.
in the Günther Hotz Lecture Hall.

The second written exam will take place on
Thursday, April 5, 2018 from 2:00 p.m. to 5:00 p.m.
in the Günther Hotz Lecture Hall.

In order to qualify for the exams you must

  • attend 80% of the tutorials (we do check)
  • gain 50% of all points from the homework assignments

In case of qualification, you are allowed to take part in both exams. The better grade counts.

The exams will be closed book. These are the rules during the exams:

  • You are allowed and obliged to bring three things to your desk only: Your student ID card (Studierendenausweis), a ball-pen that has no function other than writing, and a so-called cheat sheet. This cheat sheet is a A4 page with formulas or important equations from the lecture. Please note that the cheat sheet has to be handwritten by yourself. It will be collected at the end of the exam, and you can get it back at the exam inspection.
  • In particular, electronic devices (including your cell phone), bags, jackets, briefcases, lecture notes, homework and classroom work solutions, additional handwritten notes, books, dictionaries, and paper are not allowed at your desk.
  • Please keep your student ID card ready for an attendance check during the exam.
  • Do not use pencils or pens that are erasable with a normal rubber.
  • You are not allowed to take anything with you that contains information about the exam. A violation of this rule means failing the DIC course.
  • You must stay until the exam is completely over.

Course material will be made available on this webpage in order to support the classroom teaching and the tutorials, not to replace them. Additional organisational information, examples and explanations that may be relevant for your understanding and the exam are provided in the lectures and tutorials. It is solely your responsibility - not ours - to make sure that you receive this infomation.

The following table shows a preliminary list of topics that will be covered during the semester.

17/10 Introduction, Overview
20/10 Linear Diffusion I: Basic Concepts
(contains classroom assignment C1 and homework H1)
24/10 Linear Diffusion II: Numerics, Limitations, Alternatives
27/10 Nonlinear Isotropic Diffusion I: Modelling and Continuous Theory
(contains classroom assignment C2 and homework H2)
30/10 Nonlinear Isotropic Diffusion II: Semidiscrete and Discrete Theory
This lecture has been moved! Time and place: 6 p.m., Building E 1.3, Lecture Hall 001
03/11 Nonlinear Isotropic Diffusion III: Efficient Sequential and Parallel Algorithms
(contains classroom assignment C3 and homework H3)
07/11 Nonlinear Anisotropic Diffusion I: Modelling
10/11 Nonlinear Anisotropic Diffusion II: Continuous and Discrete Theory
(contains classroom assignment C4 and homework H4)
14/11 Nonlinear Anisotropic Diffusion III: Efficient Algorithms
17/11 Nonlinear Diffusion: Parameter Selection
(contains classroom assignment C5 and homework H5)
21/11 Variational Methods I: Basic Ideas
24/11 Variational Methods II: Discrete Aspects
(contains classroom assignment C6 and homework H6)
28/11 Variational Methods III: TV Regularisation and Primal-Dual Methods
01/12 Variational Methods IV: Functionals of Two Variables
(contains classroom assignment C7 and homework H7)
05/12 Vector- and Matrix-Valued Images
08/12 Unification of Denoising Methods
(contains classroom assignment C8 and homework H8)
12/12 Osmosis I: Continuous Theory and Modelling
15/12 Osmosis II: Discrete Theory and Efficient Algorithms
(contains classroom assignment C9 and homework H9)
19/12 Image Sequence Analysis I: Models for the Smoothness Term
22/12 Image Sequence Analysis II: Models for the Data Term
(contains classroom assignment C10 and homework H10)
02/01 Image Sequence Analysis III: Practical Aspects
05/01 Image Sequence Analysis IV: Numerical Methods
(contains classroom assignment C11 and homework H11)
09/01 Continuous-Scale Morphology I: Basic Ideas
12/01 Continuous-Scale Morphology II: Shock Filters and Nonflat Morphology
(contains classroom assignment C12 and homework H12)
16/01 Curvature-Based Morphology I: Mean Curvature Motion
19/01 Curvature-Based Morphology II: Affine Morphological Scale-Space
(contains classroom assignment C13 and homework H13)
23/01 Self-Snakes and Active Contours
26/01 PDE-Based Image Compression I: Data Selection
(please take a look at the self-test problems)
30/01 PDE-Based Image Compression II: Optimised Encoding and Better PDEs
02/02 Summary and Outlook

At the end of the semester, there will be a self-test problem sheet that contains 6 problems, which are intended to be similar in style and difficulty to a 180-minutes written exam.

Here you can download the material for the programming assignments:

Date Topic
20/10 H1 - Linear Diffusion, Gaussian Convolution
27/10 H2 - Linear Diffusion
03/11 H3 - Nonlinear Isotropic Diffusion
10/11 H4 - Anisotropic Diffusion
17/11 H5 - FED, Decorrelation
24/11 H6 - Diffusion-Reaction Methods
01/12 H7 - Primal-Dual Methods for TV Regularisation
08/12 H8 - Iterated Bilateral Filtering
15/12 H9 - Osmosis
05/01 H11 - Optic Flow
12/01 H12 - Morphology
19/01 H13 - Curvature-Based Morphology

Here you can download example solutions for the assignments:

Date Assignment
27/10 Assignment C1: Partial Derivatives, Otsu's Axiomatic Derivation of Gaussian Scale-Space
27/10 Assignment H1: Derivatives, Gaussian Convolution and Linear Diffusion
03/11 Assignment C2: 1-D Linear Diffusion
03/11 Assignment H2: Finite Difference Approximation, Diffusivities, Linear Diffusion
10/11 Assignment C3: Stencils and Discrete Diffusion Processes
10/11 Assignment H3: Maximum-Minimum Principle and Average Grey Level Invariance, Discrete Scale-Space Properties of the Semi-Implicit Scheme, Consistency of the AOS Scheme, Nonlinear Isotropic Diffusion
17/11 Assignment C4: Diffusion Tensor of EED
17/11 Assignment H4: Discretisation of Anisotropic Nonlinear Diffusion, Integration Model for Anisotropic Diffusion, Anisotropic Diffusion
24/11 Assignment C5: Fast Explicit Diffusion
24/11 Assignment H5: Filter Factorisations into Explicit Diffusion Steps, Parameter Adaption under Rescaling
01/12 Assignment C6: Continuous Variational Regularisation, Convex Functionals and Forward Diffusion
01/12 Assignment H6: Gradient Domain Methods, Discrete Energy Minimisation, Stability of Diffusion-Reaction Discretisations, Diffusion-Reaction Methods
08/12 Assignment C7: Forward-Backward Splitting
08/12 Assignment H7: Proximal Problems and Energy Minimisation, Half-Quadratic Regularisation, Primal-Dual Methods for TV Regularisation
15/12 Assignment C8: Multiple Choice, Windowed Smoothness Terms
15/12 Assignment H8: Matrix-Valued Diffusion Filtering, Mean and Median Filtering, Iterated Bilateral Filtering
22/12 Assignment C9: Transition Matrix for 1-D Osmosis
22/12 Assignment H9: Average Grey Value Invariance of Osmosis, Analysis of the Explicit Osmosis Scheme, Osmosis Filtering
05/01 Assignment C10: Eigenvalues and Eigenvectors of Nagel's Method, Photometric Invariants
05/01 Assignment H10: Optic Flow Regularisation, Design of Global Optic Flow Methods, Motion Tensors
12/01 Assignment C11: Data Terms for Large Displacements
12/01 Assignment H11: JIF Regularisation, Parabolic and Elliptic Problem, Flow-Driven Isotropic Optic Flow
19/01 Assignment C12: Discretisation of 1-D Erosion
19/01 Assignment H12: Slope Transform, Dilation and Erosion

  • J. Weickert: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.
  • G. Aubert and P. Kornprobst: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Second Edition, Springer, New York, 2006.
  • T. F. Chan and J. Shen: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia, 2005.
  • F. Cao: Geometric Curve Evolutions and Image Processing. Lecture Notes in Mathematics, Vol. 1805, Springer, Berlin, 2003.
  • R. Kimmel: The Numerical Geometry of Images. Springer, New York, 2004.
  • G. Sapiro: Geometric Partial Differential Equations in Image Analysis. Cambridge University Press, 2001.
  • Articles from journals and conferences.

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