Combinatorial Continuous Maximal Flows

TL;DR

This paper introduces a combinatorial continuous max-flow method that avoids metrication artifacts in image processing tasks, providing a convergent, efficient algorithm with a clear dual formulation.

Contribution

It presents a novel combinatorial approach to continuous max-flow, ensuring convergence and eliminating metrication errors in applications like image segmentation.

Findings

The combinatorial continuous max-flow (CCMF) can be solved exactly with provable convergence.

CCMF produces solutions without metrication artifacts in image segmentation.

The dual problem clarifies the non-equivalence of max-flow and total variation in certain cases.

Abstract

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 metrication artefacts 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 metrication errors. 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 CCMF problem to find a null-divergence solution that exhibits no metrication artefacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.