We present a multiscale approach for minimizing the energy associated with Markov Random Fields (MRFs) with energy functions that include arbitrary pairwise potentials. The MRF is represented on a hierarchy of successively coarser scales, where the problem on each scale is itself an MRF with suitably defined potentials. These representations are used to construct an efficient multiscale algorithm that seeks a minimal-energy solution to the original problem. The algorithm is iterative and features a bidirectional crosstalk between fine and coarse representations. We use consistency criteria to guarantee that the energy is nonincreasing throughout the iterative process. The algorithm is evaluated on real-world datasets, achieving competitive performance in relatively short run-times.

We consider the problem of finding a geometrically consistent set of point matches between two images. We assume that local descriptors have provided a set of candidate matches, which may include many outliers. We then seek the largest subset of these correspondences that can be aligned perfectly using a nonrigid deformation that exerts a bounded distortion. We formulate this as a constrained optimization problem and solve it using a constrained, iterative reweighted least-squares algorithm. In each iteration of this algorithm we solve a convex quadratic program obtaining a globally optimal match over a subset of the bounded distortion transformations. We further prove that a sequence of such iterations converges monotonically to a critical point of our objective function. We show experimentally that this algorithm produces excellent results on a number of test sets, in comparison to several state-of-the-art approaches.