A Multiresolution Stochastic Level Set Method for Mumford–Shah Image Segmentation

The Mumford-Shah model is one of the most successful image segmentation models. However, existing algorithms for the model are often very sensitive to the choice of the initial guess. To make use of the model effectively, it is essential to develop an algorithm which can compute a global or near global optimal solution efficiently. While gradient descent based methods are well-known to find a local minimum only, even many stochastic methods do not provide a practical solution to this problem either. In this paper, we consider the computation of a global minimum of the multiphase piecewise constant Mumford-Shah model. We propose a hybrid approach which combines gradient based and stochastic optimization methods to resolve the problem of sensitivity to the initial guess. At the heart of our algorithm is a well-designed basin hopping scheme which uses global updates to escape from local traps in a way that is much more effective than standard stochastic methods. In our experiments, a very high-quality solution is obtained within a few stochastic hops whereas the solutions obtained with simulated annealing are incomparable even after thousands of steps. We also propose a multiresolution approach to reduce the computational cost and enhance the search for a global minimum. Furthermore, we derived a simple but useful theoretical result relating solutions at different spatial resolutions.