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.

Announcements – Description – Prerequisites – Tutorials – Registration – Oral Exam – Contents – Assignments – Material for the Programming Assignments – Literature

• 2018-02-01: Example solution to assignment 07 is online.
  Lecture notes updated; corrected some typos and clarified some notation and naming.
• 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
Assignment 07	Duality, Monotone Operators	01/22	[download]	01/29

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]
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. 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.

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