A convex relaxation approach for computing minimal partitions

In this work we propose a convex relaxation approach for computing minimal partitions. Our approach is based on rewriting the minimal partition problem (also known as Potts model) in terms of a primal dual Total Variation functional. We show that the Potts prior can be incorporated by means of convex constraints on the dual variables. For minimization we propose an efficient primal dual projected gradient algorithm which also allows a fast implementation on parallel hardware. Although our approach does not guarantee to find global minimizers of the Potts model we can give a tight bound on the energy between the computed solution and the true minimizer. Furthermore we show that our relaxation approach dominates recently proposed relaxations. As a consequence, our approach allows to compute solutions closer to the true minimizer. For many practical problems we even find the global minimizer. We demonstrate the excellent performance of our approach on several multi-label image segmentation and stereo problems.