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

TL;DR

This paper presents a method for topological and geometric reconstruction of geodesic subspaces in Euclidean space using filtrations, guaranteeing accurate homotopy and homology recovery under certain conditions.

Contribution

It introduces a novel reconstruction technique based on intrinsic length metrics and provides algorithms for homotopy equivalence with minimal Hausdorff distance in 2D.

Findings

Guarantees correct homotopy and homology groups reconstruction for certain geodesic subspaces.

Develops an algorithm for homotopy equivalent geometric complex with small Hausdorff distance in E2.

Utilizes Cech and Vietoris-Rips filtrations for shape reconstruction.

Abstract

We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the \v{C}ech 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.