Linkage between Piecewise Constant Mumford-Shah model and ROF model and its virtue in image segmentation

The piecewise constant Mumford-Shah (PCMS) model and the Rudin-Osher-Fatemi (ROF) model are two of the most famous variational models in image segmentation and image restoration, respectively. They have ubiquitous applications in image processing. In this paper, we explore the linkage between these two important models. We prove that for two-phase segmentation problem the optimal solution of the PCMS model can be obtained by thresholding the minimizer of the ROF model. This linkage is still valid for multiphase segmentation under mild assumptions. Thus it opens a new segmentation paradigm: image segmentation can be done via image restoration plus thresholding. This new paradigm, which circumvents the innate non-convex property of the PCMS model, therefore improves the segmentation performance in both efficiency (much faster than state-of-the-art methods based on PCMS model, particularly when the phase number is high) and effectiveness (producing segmentation results with better quality) due to the flexibility of the ROF model in tackling degraded images, such as noisy images, blurry images or images with information loss. As a by-product of the new paradigm, we derive a novel segmentation method, coined thresholded-ROF (T-ROF) method, to illustrate the virtue of manipulating image segmentation through image restoration techniques. The convergence of the T-ROF method under certain conditions is proved, and elaborate experimental results and comparisons are presented.