Abstract

We propose a novel spatially continuous framework forconvex relaxations based on functional lifting. Our methodcan be interpreted as a sublabel–accurate solution to mul-tilabel problems. We show that previously proposed func-tional lifting methods optimize an energy which is linear between two labels and hence require (often infinitely) manylabels for a faithful approximation. In contrast, the proposed formulation is based on a piecewise convex approximation and therefore needs far fewer labels – see Fig. 1. In comparison to recent MRF-based approaches, our method is formulated in a spatially continuous setting and showsless grid bias. Moreover, in a local sense, our formulationis the tightest possible convex relaxation. It is easy to implement and allows an efficient primal-dual optimization on GPUs. We show the effectiveness of our approach on several computer vision problems.