Combinatorial Continuous Maximal Flows

Camille Couprie (1) , Leo Grady (2) , Hugues Talbot (1) , Laurent Najman (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 Siemens Corporate Research

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a graph are known to exhibit grid (metrication) bias in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual min-cut problem) in these same applications to produce solutions with no grid bias. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow problem to find a solution that exhibits no grid bias and may be solved exactly by a fast, efficient algorithm with provable convergence.

Mots clés: Image Segmentation; Convex optimization; Discrete Calculus; Total Variation minimization

Domaines: Traitement des images [eess.IV]; Vision par ordinateur et reconnaissance de formes [cs.CV]; Optimisation et contrôle [math.OC]

Camille Couprie, Leo Grady, Hugues Talbot, Laurent Najman. Combinatorial Continuous Maximal Flows. TBA, 2010, pp.1-21. ⟨hal-00525822v1⟩