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, 1214 c.t., Building E1.3, Lecture Hall 001
Thursday, 810 c.t., Building E1.3, Lecture Hall 001
First lecture: Thursday, October 19, 2017
First Tutorial: Thursday, November 09, 2017; see also below.
Announcements –
Description –
Prerequisites –
Tutorials –
Registration –
Oral Exam –
Contents –
Assignments –
Material for the Programming Assignments –
Literature
 20180201: Example solution to assignment 07 is online.
Lecture notes updated; corrected some typos and clarified some notation
and naming.
 20171222: 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.
 20171030: 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
(multidimensional) calculus and linear algebra as covered in introductory
maths course (such as Mathematik für Informatiker IIII). 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
 20171109,
 20171123,
 20171207,
 20171221,
 20180104,
 20180118, and
 20180201.
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
(20171106).
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 contentrelated 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, HeavyBall, 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, LegendreFenchel Transform 
01/08 
[download] 
01/15 
Assignment 07 
Duality, Monotone Operators 
01/22 
[download] 
01/29 
No. 
Title 
Date 

Submit until 
Assignment 03 
Gradient Descent, HeavyBall, 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, HeavyBall, and Accelerated Gradient Descent 
06/01 
[download] 
Assignment 03 
Sample Programme: Gradient Descent, HeavyBall, 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, LegendreFenchel Transform 
01/18 
[download] 
Assignment 07 
Duality, Monotone Operators 
02/01 
[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. HiriartUrruty and C. Lemaréchal, Springer, 1993.
 Fundamentals of Convex Analysis
J.B. HiriartUrruty and C. Lemaréchal, Springer, 2001.
(Abridged version of the previous twovolume 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.
