Efficient Global Minimization for the Multiphase Chan-Vese Model of Image Segmentation

Abstract

The Mumford-Shah model is an important variational image segmentation model. A popular multiphase level set approach, the Chan-Vese model, was developed as a numerical realization by representing the phases by several overlapping level set functions. Recently, a variant representation of the Chan-Vese model with binary level set functions was proposed. In both approaches, the gradient descent equations had to be solved numerically, a procedure which is slow and has the potential of getting stuck in a local minima. In this work, we develop an efficient and global minimization method for a discrete version of the level set representation of the Chan-Vese model with 4 regions (phases), based on graph cuts. If the average intensity values of the different regions are sufficiently evenly distributed, the energy function is submodular. It is shown theoretically and experimentally that the condition is expected to hold for the most commonly used data terms. We have also developed a method for minimizing nonsubmodular functions, that can produce global solutions in practice should the condition not be satisfied, which may happen for the L1 data term.