Welcome to the homepage of the lecture

Convex Analysis for Image Processing

Winter Term 2017 / 2018

Convex Analysis for Image Processing

Dr. Matthias Augustin
Office hour: Wednesday, 14:30 - 15:30.

Winter Term 2017 / 2018

Lectures (3h) with exercises (1h);
(6 ETCS points)

Lectures:
Monday, 12-14 c.t., Building E1.3, Lecture Hall 001
Thursday, 8-10 c.t., Building E1.3, Lecture Hall 001

First lecture: Thursday, October 19, 2017

First Tutorial: Thursday, November 09, 2017; see also below.



AnnouncementsDescriptionPrerequisitesTutorialsRegistrationOral ExamContents AssignmentsMaterial for the Programming Assignments Literature



  • 2018-01-18: Example solution to assignment 06 is online. Corrected a typo in assignment 06.
  • 2018-01-11: Lecture notes updated; corrected some typos and clarified some arguments in chapter 9.
  • 2018-01-08: Assignment 06 is online.
    You can submit this assignment until 2018-01-15 before the lecture.
  • 2018-01-04: Example solution to assignment 05 is online, including programming parts.
  • 2017-12-22: Lecture notes updated; significant changes in Chapter 11 and added an appendix on the relation between subdifferentials and directional derivatives, including a proof of convex functions being differential at points where the subdifferential contains only one subgradient, and further calculus rules for the subdifferential.
  • 2017-10-30: Example solution to assignment 01 is online.
    Lecture notes updated; corrected some typos and clarified some arguments, e.g., why norms are convex but not strictly convex.
    Registration is now closed as the first tutorial was today.

Many problems in image processing or computer vision can be modeled as convex minimization problems. The restored image is thus the minimizer of a suitable convex energy functional and a numerical solution strategy can rely on fast optimization algorithms. Many such algorithms rely on knowledge of convex analysis. Therefore, this course wants to introduce students to some basic concepts in this rich field including applications in image processing.


This course is suitable for students of visual computing, mathematics, and computer science.
Students attending this course should be familiar with basic concepts of (multi-dimensional) calculus and linear algebra as covered in introductory maths course (such as Mathematik für Informatiker I-III). Mathematical prerequisites which exceed the basic mathematics courses are provided within the lecture. Lectures and tutorials will be in English. Knowledge from image processing may be helpful, but is not required.


There will be a total of 7 tutorials which will take place instead of regular lectures on

  • 2017-11-09,
  • 2017-11-23,
  • 2017-12-07,
  • 2017-12-21,
  • 2018-01-04,
  • 2018-01-18, and
  • 2018-02-01.

The tutorials include homework assignments which have to be submitted during the lecture break, or earlier and which will be graded. Working together in groups of up to 3 students is permitted and encouraged.

In order to qualify for the final exam, it is necessary to achieve 50% of the points of all assignment sheets in total. There will be either oral exams or a written exam, depending on the number of participants.


Registration is closed since the submitting date of the first assignment passed (2017-11-06).


In order to qualify for the final exam, it is necessary to achieve 50% of the points of all assignment sheets in total.

First exam: February 09, 2017
Second exam: April 06, 2017

  • You can attend both exams.
  • Each exam counts as one try.
  • Second exam can be taken to improve the grade.


Course material is available on this homepage in order to support the classroom teaching and the tutorials, not to replace them. Additional organizational information, examples and explanations that may be relevant for your understanding and the exam are provided in the lectures and tutorials.

Lecture notes with proofs

Lecture notes without proofs

First lecture organisational slides

First lecture content-related slides

No. Title Date Submit until
Assignment 01 Convex Sets and Convex Functions 10/30 [download] 11/06
Assignment 02 Lipschitz Continuous Gradients, Strong Convexity 11/13 [download] 11/20
Assignment 03 Gradient Descent, Heavy-Ball, and Accelerated Gradient Descent 11/27 [download] 12/04
Assignment 04 Projection, Subdifferentials 12/11 [download] 18/12
Assignment 05 Subdifferentials, Proximal Mapping, Proximal Gradient Method 12/18 [download] 01/02
Assignment 06 Conjugates, Legendre-Fenchel Transform 01/08 [download] 01/15


No. Title Date Submit until
Assignment 03 Gradient Descent, Heavy-Ball, and Accelerated Gradient Descent 11/27 [download] 12/04
Assignment 05 Proximal Gradient -- ISTA, FISTA, Tikhonov 12/18 [download] 01/02


No. Title Date
Assignment 01 Convex Sets and Convex Functions 11/07 [download]
Assignment 02 Lipschitz Continuous Gradients, Strong Convexity 11/23 [download]
Assignment 03 Gradient Descent, Heavy-Ball, and Accelerated Gradient Descent 06/01 [download]
Assignment 03 Sample Programme: Gradient Descent, Heavy-Ball, and Accelerated Gradient Descent for a Quadratic Function in 2d 06/01 [download]
Assignment 04 Projection, Subdifferentials 12/21 [download]
Assignment 05 Subdifferentials, Proximal Mapping, Proximal Gradient Method 01/04 [download]
Assignment 05 Sample Programme: Proximal Gradient -- ISTA, FISTA, Tikhonov 01/04 [download]
Assignment 06 Conjugates, Legendre-Fenchel Transform 01/18 [download]


There is no specific text book for this class, but here is a selection of some books covering many of the topics in this course, giving background material and providing further reading:

  • Convex Analysis
    R. T. Rockafellar, Princeton University Press, 1997.
  • Convex Analysis and Minimization Algorithms I+II
    J.-B. Hiriart-Urruty and C. Lemaréchal, Springer, 1993.
  • Fundamentals of Convex Analysis
    J.-B. Hiriart-Urruty and C. Lemaréchal, Springer, 2001.
    (Abridged version of the previous two-volume entry.)
  • Convex Optimization
    S. Boyd and L. Vandenberghe, Cambridge University Press, 2004.
    (Also available to download at the authors' homepages.)
  • Introductory Lectures on Convex Optimization - A Basic Course.
    Y. Nesterov, Kluwer Academic Publishers, 2004.
  • Convex Analysis and Optimization.
    D. P. Bertsekas, Athena Scientific, 2003.
  • Convex Optimization Algorithms.
    D. P. Bertsekas, Athena Scientific, 2015.
  • Convex Analysis and Monotone Operator Theory in Hilbert Spaces.
    H. H. Bauschke and P. L. Combettes, Springer, 2011.
  • Variational Analysis
    R. T. Rockafellar and R. J.-B. Wets, Springer, 1998.

Most of these and further books can be found in the mathematics and computer science library.

Further references will be provided during the lecture.


MIA Group
©2001-2016
The author is not
responsible for
the content of
external pages.