A varifold approach to surface approximation

TL;DR

This paper extends varifold theory to enable surface approximation in discrete and computational geometry, providing convergence results and numerical tests for applications involving point clouds and pixel/voxel discretizations.

Contribution

It introduces an enriched varifold framework with approximate mean curvature, applicable to discrete surfaces and capable of handling singularities, with proven convergence and numerical validation.

Findings

Convergence of approximate mean curvature for discrete varifolds.

Applicability to surfaces with singularities.

Numerical tests demonstrating effectiveness.

Abstract

We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we introduce the notion of approximate mean curvature and show various convergence results that hold in particular for sequences of discrete varifolds associated with point clouds or pixel/voxel-type discretizations of $d$-surfaces in the Euclidean $n$-space, without restrictions on dimension and codimension. The variational nature of the approach also allows to consider surfaces with singularities, and in that case the approximate mean curvature is consistent with the generalized mean curvature of the limit surface. A series of numerical tests are provided in order to illustrate the effectiveness and generality of the method.