The multiphase image segmentation is modeled as a minimization problem with characteristic functions defined by level set functions

which leads to solutions of some gradient descent equations with low computation efficiency. This is improved via the dual method or Split-Bregman method using binary labeling functions and convex relaxation

thresholding techniques. In this paper

we propose a fast direct dual method (DDM) without convex relaxation and thresholding techniques. First

we design the DDM for the two-phase Chan-Vese model

which results in a binary solution of the primal variable in analytical form and a simple iterative formulation of the dual variable by using KKT (Karush-Kuhn-Tucker) conditions. Then

it is extended to the Chan-Vese model for multiphase image segmentation. The experimental results demonstrate that the proposed method has a better performance

and is more efficient than the gradient descent method

the dual method

and the Split-Bregman method.