A wide range of low level vision problems have been formulated in terms of finding the most probable assignment of a Markov Random Field (or equivalently the lowest energy configuration). Perhaps the most successful example is stereo vision. For the stereo problem, it has been shown that finding the global optimum is NP hard but good results have been obtained using a number of approximate optimization algorithms.

In this paper we show that for standard benchmark stereo pairs, the global optimum can be found in about 30 minutes using a variant of the belief propagation (BP) algorithm. We extend previous theoretical results on reweighted belief propagation to account for possible ties in the beliefs and using these results we obtain easily checkable conditions that guarantee that the BP disparities are the global optima. We verify experimentally that these conditions are typically met for the standard benchmark stereo pairs and discuss the implications of our results for further progress in stereo.