Stability of boundary measures

TL;DR

This paper introduces boundary measures at scale r for compact sets in Euclidean space, providing a stability theorem that enables their reliable computation from point clouds, with implications for feature detection and curvature measures.

Contribution

It presents a new boundary measure, proves its stability, and applies it to approximate Federer's curvature measures from point-cloud data.

Findings

Boundary measures can be computed for point clouds.

A quantitative stability theorem for boundary measures is established.

Stability results extend to Federer's curvature measures.

Abstract

We introduce the boundary measure at scale r of a compact subset of the n-dimensional Euclidean space. We show how it can be computed for point clouds and suggest these measures can be used for feature detection. The main contribution of this work is the proof a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer's curvature measures of a compact, allowing to compute them from point-cloud approximations of the compact.