Papers › On the Reconstruction of Geodesic Subspaces of ℝᴺ
On the Reconstruction of Geodesic Subspaces of ℝᴺ
Brittany Terese Fasy, Rafal Komendarczyk, Sushovan Majhi, Carola Wenk
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We consider the topological and geometric reconstruction of a geodesic subspace of ℝᴺ 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 ℝ², 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.
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