Discrete varifolds: A unified framework for discrete approximations of surfaces and mean curvature

Abstract

We propose a unified theory for the discretization ofmanifolds (triangulations, volumetric approximations, pixelization, point clouds etc.) which provides a common framework for describing discrete and continuous surfaces, allows a control of the weak regularity of the limit surface and provides a consistent notion of mean curvature. This is made possible by the theory of varifolds. Varifolds have been introduced more than 40 years ago as a generalized notion of k-surface, with a number of relevant applications (for example to the existence and regularity of soap bubble clusters and soap films). Our extension of the theory consists in a new discrete framework, including in particular a scale-dependent notion of mean curvature, by which one can approximate variational problems on (generalized) surfaces by minimizing suitable energies defined on discrete varifolds. As an example, we apply the theory of discrete varifolds to estimate the mean curvature of a point cloud and to approximate its evolution by mean curvature flow.