On the Statistical Interpretation of the Piecewise Smooth Mumford-Shah Functional

Abstract: In region-based image segmentation, two models dominate the field: the Mumford-Shah functional and statistical approaches based on Bayesian inference. Whereas the latter allow for numerous ways to describe the statistics of intensities in regions, the first includes spatially smooth approximations. In this paper, we show that the piecewise smooth Mumford-Shah functional is a first order approximation of Bayesian a-posteriori maximization where region statistics are computed in local windows. This equivalence not only allows for a statistical interpretation of the full Mumford-Shah functional. Inspired by the Bayesian model, it also offers to formulate an extended Mumford-Shah functional that takes the variance of the data into account.