On the implementation of the multi-phase region segmentation, solving the hidden phase problem

We consider the Chan and Vese multiphase segmentation based on the partition of an image minimizing an energy involving a region-based data fidelity term and a regularization term. The common implementation of the continuous optimization of this segmentation framework, with multiple level set functions, raises some numerical issues which lead to poor performance of the method when handling more than two phases. We propose a general formulation of the multiphase model, and a permutation method, incorporated in the level-set based implementation of the multi-phase approach to handle in an original way the so-called hidden phase problem.