A Weighted Difference of Anisotropic and Isotropic Total Variation for Relaxed Mumford-Shah Color and Multiphase Image Segmentation

In a class of piecewise-constant image segmentation models, we propose to incorporate a weighted difference of anisotropic and isotropic total variation (AITV) to regularize the partition boundaries in an image. To deal with the nonconvex AITV term, we apply the difference-of-convex algorithm (DCA), in which the subproblems can be minimized by the primal-dual hybrid gradient method with line search. We can prove that the DCA iterations converge to a limit point of the proposed model. We discuss the AITV extension of the Chan-Vese model and the fuzzy region competition model. A generalization to color image segmentation is also explained. In the numerical experiments, we compare our proposed models with the classic convex approaches and the two-stage segmentation methods (denoising and then thresholding) on various images, showing that our models are effective in image segmentation and robust with respect to impulsive noises.