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Dense conditional random fields (CRFs) with Gaussian pairwise potentials have become a popular framework for modelling several problems in computer vision such as stereo correspondence and multi-class semantic segmentation. By modelling long-range interactions, dense CRFs provide a more detailed labelling compared to their sparse counterparts. Currently the state-of-the-art algorithm performs mean-field inference using a filter-based method to obtain accurate segmentations, but fails to provide strong theoretical guarantees on the quality of the solution. Whilst the underlying model of a dense CRF provides enough information to yield well defined segmentations, it lacks the richness introduced via higher order potentials. The mean-field inference strategy was also extended to incorporate higher order potentials, but again failed to obtain a bound on the quality of the solution. To this extent, we show that a dense CRF can be aggregated with sparse higher order potentials in a way that is amenable to continuous relaxations. We will then show that, by using a filter-based method, these continuous relaxations can be optimised efficiently using state-of-the-art algorithms. Specifically we will solve a quadratic programming (QP) relaxation using the Frank-Wolfe algorithm and a linear programming (LP) relaxation by developing a proximal minimisation framework. By exploiting labelling consistency in the higher order potentials and utilising the filter-based method, we are able to formulate the above algorithms such that each iteration has a complexity linear in the number of classes and random variables. The experiments are performed on the standard publicly available MSRC data set and demonstrate the low energies achieved from the minimisation and the accuracy of the resulting segmentations.

Efficient Relaxations for Dense CRFs with Sparse Higher-Order Potentials