Convex Optimization for Computer Vision

Being able to determine the argument that minimizes a (possibly nonsmooth) convex cost function effeciently is of great practical relevance. For example, convex variational methods are one of the most powerful techniques for many computer vision and image processing problems, e.g. denoising, deblurring, inpainting, stereo matching, optical flow computation, segmentation, or super resolution. In this lecture we will discuss first order convex optimization methods to implement and solve the aforementioned problems efficiently. Particular attention will be paid to problems including constraints and non-differentiable terms, giving rise to methods that exploit the concept of duality such as the primal-dual hybrid gradient method or the alternating directions methods of multipliers. This lecture will cover the mathematical background for proving why the investigated methods converge as well as their efficient practical implementation.

We will cover the following topics:

Mathematical background

• Convex sets and functions
• Existence and uniqueness of minimizers
• Subdifferentials
• Convex conjugates
• Saddle point problems and duality

Numerical methods

• (Sub-)Gradient descent schemes
• Proximal point algorithm
• Primal-dual hybrid gradient method
• Augmented Lagrangian methods
• Acceleration schemes, adaptive step sizes, and heavy ball methods

Example applications in computer vision and signal processing problems, including

• Image denoising, deblurring, inpainting, segmentation
• Implementation in MATLAB

Location:  Monday in room H-C 7326, Wednesday in room H-A 7106, Hölderlinstraße 3

Time and Date: Monday 12:15 - 14:00, Wednesday 10:15 - 12:00

Start: April 18th, 2017, 12:15

The lecture is held in English. 

Location: Room H-A 7116 ,  Hölderlinstraße 3

Time and Date: Wednesday 12:15 - 13:45

Start: April 26th, 2017
Webpage:  Exercise and Materials

The exercise sheets consist of two parts, theoretical and programming exercises. The exercise sheets will be passed out in the lecture on Monday and you have one week to solve them. The solutions will be discussed in the exercises on Wednesday two days later. 

During the course of the lecture, we will pose a challenge to solve an optimization problem as quickly as possible. The challenge ends on Monday 17.07 23:59 . The best solution will receive a prize. The challenges will be a good preparation for the final exam!

Submission instructions: The source code should be sent via e-mail to michael.moeller@uni-siegen.de
 

Challenge: To be announced

Leaderboard

The exam will be oral.