Bottleneck Potentials in Markov Random Fields

Ahmed Abbas, Paul Swoboda; Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2019, pp. 3175-3184

We consider general discrete Markov Random Fields(MRFs) with additional bottleneck potentials which penalize the maximum (instead of the sum) over local potential value taken by the MRF-assignment. Bottleneck potentials or analogous constructions have been considered in (i) combinatorial optimization (e.g. bottleneck shortest path problem, the minimum bottleneck spanning tree problem, bottleneck function minimization in greedoids), (ii) inverse problems with L_ infinity -norm regularization and (iii) valued constraint satisfaction on the (min,max)-pre-semirings. Bottleneck potentials for general discrete MRFs are a natural generalization of the above direction of modeling work to Maximum-A-Posteriori (MAP) inference in MRFs. To this end we propose MRFs whose objective consists of two parts: terms that factorize according to (i) (min,+), i.e. potentials as in plain MRFs, and (ii) (min,max), i.e. bottleneck potentials. To solve the ensuing inference problem, we propose high-quality relaxations and efficient algorithms for solving them. We empirically show efficacy of our approach on large scale seismic horizon tracking problems.