Optimization of Markov Random Fields in Computer Vision

A large variety of computer vision tasks can be formulated using Markov Random Fields (MRF). Except in certain special cases, optimizing an MRF is intractable, due to a large number of variables and complex dependencies between them. In this thesis, we present new algorithms to perform inference in MRFs, that are either more efficient (in terms of running time and/or memory usage) or more effective (in terms of solution quality), than the state-of-the-art methods. First, we introduce a memory efficient max-flow algorithm for multi-label submodular MRFs. In fact, such MRFs have been shown to be optimally solvable using max-flow based on an encoding of the labels proposed by Ishikawa, in which each variable $X_i$ is represented by $\ell$ nodes (where $\ell$ is the number of labels) arranged in a column. However, this method in general requires $2\,\ell^2$ edges for each pair of neighbouring variables. This makes it inapplicable to realistic problems with many variables and labels, due to excessive memory requirement. By contrast, our max-flow algorithm stores $2\,\ell$ values per variable pair, requiring much less storage. Consequently, our algorithm makes it possible to optimally solve multi-label submodular problems involving large numbers of variables and labels on a standard computer. Next, we present a move-making style algorithm for multi-label MRFs with robust non-convex priors. In particular, our algorithm iteratively approximates the original MRF energy with an appropriately weighted surrogate energy that is easier to minimize. Furthermore, it guarantees that the original energy decreases at each iteration. To this end, we consider the scenario where the weighted surrogate energy is multi-label submodular (i.e., it can be optimally minimized by max-flow), and show that our algorithm then lets us handle of a large variety of non-convex priors. Finally, we consider the fully connected Conditional Random Field (dense CRF) with Gaussian pairwise potentials that has proven popular and effective for multi-class semantic segmentation. While the energy of a dense CRF can be minimized accurately using a Linear Programming (LP) relaxation, the state-of-the-art algorithm is too slow to be useful in practice. To alleviate this deficiency, we introduce an efficient LP minimization algorithm for dense CRFs. To this end, we develop a proximal minimization framework, where the dual of each proximal problem is optimized via block-coordinate descent. We show that each block of variables can be optimized in a time linear in the number of pixels and labels. Consequently, our algorithm enables efficient and effective optimization of dense CRFs with Gaussian pairwise potentials. We evaluated all our algorithms on standard energy minimization datasets consisting of computer vision problems, such as stereo, inpainting and semantic segmentation. The experiments at the end of each chapter provide compelling evidence that all our approaches are either more efficient or more effective than all existing baselines

An ILP Solver for Multi-label MRFs with Connectivity Constraints

Integer Linear Programming (ILP) formulations of Markov random fields (MRFs) models with global connectivity priors were investigated previously in computer vision, e.g., \cite{globalinter,globalconn}. In these works, only Linear Programing (LP) relaxations \cite{globalinter,globalconn} or simplified versions \cite{graphcutbase} of the problem were solved. This paper investigates the ILP of multi-label MRF with exact connectivity priors via a branch-and-cut method, which provably finds globally optimal solutions. The method enforces connectivity priors iteratively by a cutting plane method, and provides feasible solutions with a guarantee on sub-optimality even if we terminate it earlier. The proposed ILP can be applied as a post-processing method on top of any existing multi-label segmentation approach. As it provides globally optimal solution, it can be used off-line to generate ground-truth labeling, which serves as quality check for any fast on-line algorithm. Furthermore, it can be used to generate ground-truth proposals for weakly supervised segmentation. We demonstrate the power and usefulness of our model by several experiments on the BSDS500 and PASCAL image dataset, as well as on medical images with trained probability maps.Comment: 19 page

Scalable Semidefinite Relaxation for Maximum A Posterior Estimation

Maximum a posteriori (MAP) inference over discrete Markov random fields is a fundamental task spanning a wide spectrum of real-world applications, which is known to be NP-hard for general graphs. In this paper, we propose a novel semidefinite relaxation formulation (referred to as SDR) to estimate the MAP assignment. Algorithmically, we develop an accelerated variant of the alternating direction method of multipliers (referred to as SDPAD-LR) that can effectively exploit the special structure of the new relaxation. Encouragingly, the proposed procedure allows solving SDR for large-scale problems, e.g., problems on a grid graph comprising hundreds of thousands of variables with multiple states per node. Compared with prior SDP solvers, SDPAD-LR is capable of attaining comparable accuracy while exhibiting remarkably improved scalability, in contrast to the commonly held belief that semidefinite relaxation can only been applied on small-scale MRF problems. We have evaluated the performance of SDR on various benchmark datasets including OPENGM2 and PIC in terms of both the quality of the solutions and computation time. Experimental results demonstrate that for a broad class of problems, SDPAD-LR outperforms state-of-the-art algorithms in producing better MAP assignment in an efficient manner.Comment: accepted to International Conference on Machine Learning (ICML 2014