Convex Formulation and Exact Global Solutions for Multi-phase Piecewise Constant Mumford-Shah Image Segmentation

摘要

Most variational models for multi-phase image segmentation are non-convex and possess multiple local minima, which makes solving for a global solution an extremely difficult task. In this work, we provide a method for computing a global solution for the (non-convex) multi-phase piecewise constant Mumford-Shah (spatially continuous Potts) image segmentation problem. Our approach is based on using a specific representation of the problem due to Lie et al. We then rewrite this representation using the dual formulation for total variation so that a variational convexification technique due to Pock et al. may be employed. Unlike some recent methods in this direction, our method can guarantee that a global solution is obtained. We believe our method to be the first in the literature that can make this claim. Once we have the convex optimization problem, we give an algorithm to compute a global solution. We demonstrate our algorithm on several multi-phase image segmentation examples, including a medical imaging application.