Narrow Results

We consider the topological and geometric reconstruction of a geodesic subspace of ${ℝ}^N$ 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 the reconstruction problem. 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. This technique provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and $\mu$–reach, and performs better under certain types of perturbations of the geodesic subspace. For geodesic subspaces of ${ℝ}^2$, we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown underlying space.

Given a compact geodesic space $X$, we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ to obtain what we call a persistence object. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and closed, Rips and Čech induced persistences. Amongst other results, we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence.