Discrete varifolds and surface approximation

In this chapter, we give a brief account of the notion of discrete varifolds, which are general and flexible tools to represent in a common framework regular surfaces and a large category of discrete representations of surfaces, eg point clouds, triangulated surfaces or volumetric representations. In this setting, a new notion of discrete mean curvature can be defined, relying only on the varifold structure and not on any specific feature of the underlying discretization type. This notion of discrete mean curvature is obtained thanks to a regularization of the so-called first variation of the varifold, it is easy to compute, and we prove that it has nice convergence properties. We illustrate this notion on 2D and 3D examples.