Tight convex relaxations for vector-valued labeling problems

The multi-label problem is of fundamental importance to computer vision, yet finding global minima of the associated energies is very hard and usually impossible in practice. Recently, progress has been made using continuous formulations of the multi-label problem and solving a convex relaxation globally, thereby getting a solution with optimality bounds. In this work, we develop a novel framework for continuous convex relaxations, where the label space is a continuous product space. In this setting, we can combine the memory efficient product relaxation of [9] with the much tighter relaxation of [5], which leads to solutions closer to the global optimum. Furthermore, the new setting allows us to formulate more general continuous regularizers, which can be freely combined in the different label dimensions. We also improve upon the relaxation of the products in the data term of [9], which removes the need for artificial smoothing and allows the use of exact solvers.