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. In particular, we replace the total variation regularization in the Chan--Vese segmentation model and a fuzzy region competition model by the proposed AITV. To deal with the nonconvex nature of AITV, we apply the difference-of-convex algorithm (DCA), in which the subproblems can be minimized by the primal-dual hybrid gradient method with linesearch. The convergence of the DCA scheme is analyzed. In addition, a generalization to color image segmentation is discussed. In the numerical experiments, we compare the proposed models with the classic convex approaches and the two-stage segmentation methods (smoothing and then thresholding) on various images, showing that our models are effective in image segmentation and robust with respect to impulsive noises.