Reformulating and Optimizing the Mumford-Shah Functional on a Graph - A Faster, Lower Energy Solution

Active contour formulations predominate current minimization of the Mumford-Shah functional (MSF) for image segmentation and filtering. Unfortunately, these formulations necessitate optimization of the contour by evolving via gradient descent, which is known for its sensitivity to initialization and the tendency to produce undesirable local minima. In order to reduce these problems, we reformulate the corresponding MSF on an arbitrary graph and apply combinatorial optimization to produce a fast, low-energy solution. The solution provided by this graph formulation is compared with the solution computed via traditional narrow-band level set methods. This comparison demonstrates that our graph formulation and optimization produces lower energy solutions than gradient descent based contour evolution methods in significantly less time. Finally, by avoiding evolution of the contour via gradient descent, we demonstrate that our optimization of the MSF is capable of evolving the contour with non-local movement.