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.