  The piecewise constant Mumford-Shah (PCMS) model and the Rudin-Osher-Fatemi
(ROF) model are two important variational models in image segmentation and
image restoration, respectively. In this paper, we explore a linkage between
these models. We prove that for the two-phase segmentation problem a partial
minimizer of the PCMS model can be obtained by thresholding the minimizer of
the ROF model. A similar linkage is still valid for multiphase segmentation
under specific 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, called
thresholded-ROF (T-ROF) method, to illustrate the virtue of managing image
segmentation through image restoration techniques. The convergence of the T-ROF
method is proved, and elaborate experimental results and comparisons are
presented.
