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.

中文翻译：

---

图上的 Mumford-Shah 泛函及其渐近线

我们考虑 Mumford-Shah 泛函对图的适应。这些基于对 Mumford-Shah 泛函的非局部近似的离散化。受机器学习应用的启发，我们研究与度量的随机样本相关的随机几何图。我们建立了图结构的条件，在该条件下，图 Mumford-Shah 泛函的极小值收敛到连续统 Mumford-Shah 泛函的极小值。此外，我们明确地确定了限制功能。此外，我们描述了一种用于计算图 Mumford-Shah 函数的近似最小值的有效算法。