A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford-Shah Model and Thresholding

The Mumford--Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a two-stage segmentation method based on the Mumford--Shah model. The first stage of our method is to find a smooth solution $g$ to a convex variant of the Mumford--Shah model. Once $g$ is obtained, then in the second stage the segmentation is done by thresholding $g$ into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, $g$ can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle--Pock method. We prove that our method is convergent and that the solution $g$ is always unique. In our method, there is no need to specify the number of segments $K$ ($K\geq2$) before finding $g$. We can obtain any $K$-phase segmentations by choosing $(K-1)$ thresholds after $g$ is found in the first stage, and in the second...