Mumford–Shah functionals on graphs and their asymptotics

We consider adaptations of the Mumford–Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford–Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford–Shah functionals converge to a minimizer of a continuum Mumford–Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford–Shah functional.