Integer Linear Programming (ILP) formulations of Markov random fields (MRFs) models with global connectivity priors were investigated previously in computer vision, e.g., globalinter,globalconn . In these works, only Linear Programming (LP) relaxations globalinter,globalconn or simplified versions 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 also be applied as a post-processing method on top of any existing multi-label segmentation approach. We demonstrate the power and usefulness of our model by several experiments on the BSDS500 image dataset, as well as on medical images with trained probability maps.
Graph cut is a popular technique for interactive image segmentation. However, it has certain shortcomings. In particular, graph cut has problems with segmenting thin elongated objects due to the ldquoshrinking biasrdquo. To overcome this problem, we propose to impose an additional connectivity prior, which is a very natural assumption about objects. We formulate several versions of the connectivity constraint and show that the corresponding optimization problems are all NP-hard. For some of these versions we propose two optimization algorithms: (i) a practical heuristic technique which we call DijkstraGC, and (ii) a slow method based on problem decomposition which provides a lower bound on the problem. We use the second technique to verify that for some practical examples DijkstraGC is able to find the global minimum. Minimum cut maximum flow algorithms on graphs have emerged as an increasingly useful tool for exactor approximate energy minimization in low-level vision. The combinatorial optimization literature provides many min-cut max-flow algorithms with different polynomial time complexity. Their practical efficiency, however, has to date been studied mainly outside the scope of computer vision. The goal of this paper is to provide an experimental comparison of the efficiency of min-cut max flow algorithms for applications in vision. We compare the running times of several standard algorithms, as well as a new algorithm that we have recently developed. The algorithms we study include both Goldberg-Tarjan style "push -relabel" methods and algorithms based on Ford-Fulkerson style "augmenting paths." We benchmark these algorithms on a number of typical graphs in the contexts of image restoration, stereo, and segmentation. In many cases, our new algorithm works several times faster than any of the other methods, making near real-time performance possible. An implementation of our max-flow min-cut algorithm is available upon request for research purposes.