Convex relaxation for multilabel problems with product label spaces

Abstract

Convex relaxations for continuous multilabel problems have attracted a lot of interest recently [1,2,3,4,5]. Unfortunately, in previous methods, the runtime and memory requirements scale linearly in the total number of labels, making them very inefficient and often unapplicable for problems with higher dimensional label spaces. In this paper, we propose a reduction technique for the case that the label space is a product space, and introduce proper regularizers. The resulting convex relaxation requires orders of magnitude less memory and computation time than previously, which enables us to apply it to large-scale problems like optic flow, stereo with occlusion detection, and segmentation into a very large number of regions. Despite the drastic gain in performance, we do not arrive at less accurate solutions than the original relaxation. Using the novel method, we can for the first time efficiently compute solutions to the optic flow functional which are within provable bounds of typically 5% of the global optimum. © 2010 Springer-Verlag.