Computational Information Geometry Wonderland

Given a point set, one would like to find out its features without having to first reconstruct the underlying geometric structure: the mesh, a connected point set with its overall global topology. A feature can be defined as a point with high curvature, for example. One way to proceed is to compute geometric boundary measures. The idea dates back from 1959 in Federer's article: H. Federer: Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418?. 491.

The basic idea is to expand by a ball of radius r all points of the input set, and project for all points of the surface of that Minkowski union to the closest point (wrt to L2 norm distance). You'll end up with a geometric probabily measure on the discrete point set (or a continous pm. if you started with a manifold).

Now the catch is that if you pertubate the input point cloud (or object), how much will it influend on the pm. you'll get? A nice result of Chazal, Cohen-Steiner and Merigot (Stability of boundary measures) proves that the distance between original/perturbated input object (wrt. to Haussdorff) and the distance between the corresponding geometric source/perturbated pms. (wrt to Wasserstein) is 1/2 Holder. Informally, stable, or smooth, thus making it valuable for analysis.

More in Stability of boundary measures (INRIA RR 6219)