  This paper proposes an Allen-Cahn Chan-Vese model to settle the multi-phase
image segmentation. We first integrate the Allen--Cahn term and the Chan--Vese
fitting energy term to establish an energy functional, whose minimum locates
the segmentation contour. The subsequent minimization process can be attributed
to variational calculation on fitting intensities and the solution
approximation of several Allen-Cahn equations, wherein $n$ Allen-Cahn equations
are enough to partition $m = 2^n$ segments. The derived Allen-Cahn equations
are solved by efficient numerical solvers with exponential time integrations
and finite difference space discretization. The discrete maximum bound
principle and energy stability of the proposed numerical schemes are proved.
Finally, the capability of our segmentation method is verified in various
experiments for different types of images.
