An unconditionally stable hybrid method for image segmentation

In this paper, we propose a new unconditionally stable hybrid numerical method for minimizing the piecewise constant Mumford–Shah functional of image segmentation. The model is based on the Allen–Cahn equation and an operator splitting technique is used to solve the model numerically. We split the governing equation into two linear equations and one nonlinear equation. One of the linear equations and the nonlinear equation are solved analytically due to the availability of closed-form solutions. The other linear equation is discretized using an implicit scheme and the resulting discrete system of equations is solved by a fast numerical algorithm such as a multigrid method. We prove the unconditional stability of the proposed scheme. Since we incorporate closed-form solutions and an unconditionally stable scheme in the solution algorithm, our proposed scheme is accurate and robust. Various numerical results on real and synthetic images with noises are presented to demonstrate the efficiency, robustness, and accuracy of the proposed method.