Potts model, parametric maxflow and k-submodular functions

TL;DR

This paper introduces a faster algorithm for minimizing Potts energy functions in computer vision, reducing maxflow computations from linear to logarithmic in the number of labels, and connects it to k-submodular functions.

Contribution

It presents a novel method that reduces the number of maxflow computations for Potts model minimization from linear to logarithmic, improving efficiency.

Findings

Reduces maxflow computations from O(k) to O(log k)

Allows faster alpha expansion for unlabeled pixels

Establishes a connection to k-submodular functions

Abstract

The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19,20]. It identifies a part of an optimal solution by running $k$ maxflow computations, where $k$ is the number of labels. The number of "labeled" pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to $O(\log k)$ maxflow computations (or one {\em parametric maxflow} computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for {\em Tree Metrics}. We also show a connection to {\em $k$-submodular functions} from combinatorial optimization, and discuss {\em $k$-submodular relaxations} for general energy functions.