On the Reconstruction of Geodesic Subspaces of ${\mathbb{R}}^N$

Abstract

We consider the topological and geometric reconstruction of a geodesic subspace of ${\mathbb{R}}^N$ both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for a successful reconstruction. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. For geodesic subspaces of ${\mathbb{R}}^2$, we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown shape of interest.

References in corpus (1)

• Metric thickenings of Euclidean submanifolds

Cited by in corpus (1)

• Recovering the homology of immersed manifolds