A crucial part of solving a statistical inverse problem with Bayesian approach is constructing prior probability distributions that depict our a priori beliefs. In this talk we introduce novel ideas how to construct prior distributions that prefer sharp edges. This is achieved by showing that estimating posterior distribution produces Mumford-Shah -type segmentations. The key aspects to this research are to show that the limiting infinite-dimensional probability distributions are well-defined and to prove so-called discretization invariance phenomenon, i.e. nothing bad happens when discretization is refined. This talk is based on the doctoral thesis work of the author.