An Unconstrained Multiphase Thresholding Approach for Image Segmentation

In this paper we provide a method to find global minimizers of certain non-convex 2-phase image segmentation problems. This is achieved by formulating a convex minimization problem whose minimizers are also minimizers of the initial non-convex segmentation problem, similar to the approach proposed by Nikolova, Esedoḡlu and Chan. The key difference to the latter model is that the new model does not involve any constraint in the convex formulation that needs to be respected when minimizing the convex functional, neither explicitly nor by an artificial penalty term. This approach is related to recent results by Chambolle. Eliminating the constraint considerably simplifies the computational difficulties, and even a straightforward gradient descent scheme leads to a reliable computation of the global minimizer. Furthermore, the model is extended to multiphase segmentation along the lines of Vese and Chan. Numerical results of the model applied to the classical piecewise constant Mumford-Shah functional for two, four and eight phase segmentation are shown.