Higher Order Variationall Methods for Computer Vision

This research proposal 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 artefacts 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.

In this project, we investigated so-called higher order variational methods for computer vision. The basic idea of variational models is to represent a certain computer vision problem as an optimization problem i.e. one tries to find that solution that attains the minimal solution of a suitable cost functional. The key to the success of this approach is to design the cost functional - the variational model - in a way that it best reflects the physical properties of the 3D world. In order to achieve this, we developed variational models of higher order. Here, the order of the model refers to the degree of regularity of the solution.We basically investigated two different kinds of models:The first model is based on the so-called total generalized variation regulariszer that allows to reconstruct piecewise polynomial solutions while still being a convex cost functional. In course of the project we extended the model such that it can be applied for 3D reconstruction problems for car driver assistance and light field cameras. It turned out the variational models of higher lead to a significantly improved reconstruction quality compared to state-of-the-art first-order models.The second model is inspired by the human visual system. The basic idea is to represent a 2D image in a certain 3D space, where the third dimension corresponds to the orientation of the local image structure. This comes along with the advantage, that certain higher-order problems can be represented as first-order problem. For example, we were able to show that some difficult curvature minimization problems can be represented as convex energy functionals, which allows to efficiently compute an exact solution.In conclusion, we can say that we clearly achieved and even outreached our initial project goals. The most important results have already published at relevant conferences and journals.