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≥2) 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 stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our two-stage method performs better than many standard two-phase or multiphase segmentation methods for very general images, including antimass, tubular, MRI, noisy, and blurry images.