Stability of boundary measures

Frédéric Chazal (1) , David Cohen-Steiner (1) , Quentin Mérigot (1)

1 GEOMETRICA - Geometric computing

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.

Keywords: dimension detection; point clouds; curvature measures; volume of tubes; convex functions; nearest neighbor

Domains: Computational Geometry [cs.CG]; Classical Analysis and ODEs [math.CA]; Metric Geometry [math.MG]

Frédéric Chazal, David Cohen-Steiner, Quentin Mérigot. Stability of boundary measures. [Research Report] 2007, pp.20. ⟨inria-00154798v1⟩