  Dense conditional random fields (CRF) with Gaussian pairwise potentials have
emerged as a popular framework for several computer vision applications such as
stereo correspondence and semantic segmentation. By modeling long-range
interactions, dense CRFs provide a more detailed labelling compared to their
sparse counterparts. Variational inference in these dense models is performed
using a filtering-based mean-field algorithm in order to obtain a
fully-factorized distribution minimising the Kullback-Leibler divergence to the
true distribution. In contrast to the continuous relaxation-based energy
minimisation algorithms used for sparse CRFs, the mean-field algorithm fails to
provide strong theoretical guarantees on the quality of its solutions. To
address this deficiency, we show that it is possible to use the same filtering
approach to speed-up the optimisation of several continuous relaxations.
Specifically, we solve a convex quadratic programming (QP) relaxation using the
efficient Frank-Wolfe algorithm. This also allows us to solve
difference-of-convex relaxations via the iterative concave-convex procedure
where each iteration requires solving a convex QP. Finally, we develop a novel
divide-and-conquer method to compute the subgradients of a linear programming
relaxation that provides the best theoretical bounds for energy minimisation.
We demonstrate the advantage of continuous relaxations over the widely used
mean-field algorithm on publicly available datasets.
