Research Projects (2010)

This research project is devoted to the study of higher order convex variational methods for problems in computer vision. First order methods, i.e. methods which take into account first order derivatives have shown a great success for a variety of inverse computer vision problems. This success is mostly due to the introduction of total variation methods by Rudin, Osher and Fatemi in 1992. Total variation methods exhibit the important property to preserve sharp discontinuities in the solution while the associated optimization problem is still convex. This leads to robust problem solutions, independent of any initialization. Besides this, total variation methods also exhibit some disadvantages. First, total variation methods favor piecewise constant solutions which leads to staircaising artifacts in image restoration problems and to the preference of fronto‐parallel structures in stereo problems. Second, total variation methods introduce a shrinking bias in shape optimization problems. The aim of this project is therefore to study higher order convex variational methods in order to improve the shortcomings of first order methods. We therefore propose to investigate two approaches. The first approach is based on the so‐called generalized total variation method, recently introduced by Bredies, Kunisch and Pock. It provides a framework to recover piecewise polynomial functions based on a convex functional. We expect that this method leads to significant improvements of stereo and motion estimation problems. The second approach is based on the so‐called roto‐translation space introduced by Citti and Sarti in 2006. It allows to rewrite functionals incorporating curvature regularity by means of a convex first order functional in higher dimensions. We expect that this approach will significantly improve the performance of various shape optimization problems.