A Non-Heuristic Reduction Method For Graph Cut Optimization

Graph cuts optimization permits to minimize some Markov Random Fields (MRF) by computing a minimum cut (min-cut) in a relevant graph. Graph-cuts are very efficient and are now a well established field of research. However, due to the large amount of memory required for storing the graph, there application remains limited to the minimization of MRF involving a relatively small number of variables. An existing strategy to reduce the graph size restricts the graph construction to a subgraph, called reduced graph, whose nodes satisfy a predefined local condition. The test of the condition is evaluated on the fly during the graph construction. In this manner, the nodes of the reduced graph are typically located in a thin band surrounding the min-cut. In this paper, we propose a local test similar to the already existing tests for reducing large graphs. The advantage of the proposed condition is that we are able to provide theoretical guarantee that the min-cut in the reduced graph permits to construct a global minimizer of the MRF. This is a significant advantage over the already existing strategies. Once the theoretical guarantees are established, we present numerical experiments in the context of image segmentation. They confirm that the min-cut in the reduced graph provide a global minimum of the MRF and show empirically that the new test leads to memory gains similar to ones obtained with the existing heuristic test.