Combinatorial Optimization of the piecewise constant Mumford-Shah functional with application to scalar/vector valued and volumetric image segmentation

Abstract

Front propagation models represent an important category of image segmentation techniques in the current literature. These models are normally formulated in a continuous level sets framework and optimized using gradient descent methods. Such formulations result in very slow algorithms that get easily stuck in local solutions and are highly sensitive to initialization. In this paper, we reformulate one of the most influential front propagation models, the Chan–Vese model, in the discrete domain. The graph representability and submodularity of the discrete energy function is established and then max-flow/min-cut approach is applied to perform the optimization of the discrete energy function. Our results show that this formulation is much more robust than the level sets formulation. Our approach is not sensitive to initialization and provides much faster solutions than level sets. The results also depict that our segmentation approach is robust to topology changes, noise and ill-defined edges, i.e., it preserves all the advantages associated with level sets methods.