Stability of Curvature Measures

TL;DR

This paper establishes a stability framework for estimating curvature measures of compact sets from sampled data, ensuring robustness and accuracy in practical computations.

Contribution

It introduces a stability result for curvature measures of offsets of compact sets with positive μ-reach, enabling their estimation from finite samples.

Findings

Curvature measures can be computed for finite unions of balls.

Estimates of curvature measures are stable under Hausdorff perturbations.

Framework supports robust curvature estimation from point-cloud data.

Abstract

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive $\mu$-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive $\mu$-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature.