Solving Minimal Surface Problems on Surfaces and Point Clouds

Minimal surface problems play an important role not only in physics or biology but also in mathematical signal and image processing. Although the computation of respective solutions is well-investigated in the setting of discrete images, only little attention has been payed to more complicated data, e.g., surfaces represented as meshes or point clouds. In this work we introduce a novel family of discrete total variation seminorms for weighted graphs based on the upwind gradient and incorporate them into an efficient minimization algorithm to perform total variation denoising on graphs. Furthermore, we demonstrate how to utilize the latter algorithm to uniquely solve minimal surface problems on graphs. To show the universal applicability of this approach, we illustrate results from filtering and segmentation of 3D point cloud data.