Cone fields and topological sampling in manifolds with bounded curvature

TL;DR

This paper establishes conditions under which offsets of noisy point clouds in manifolds with bounded curvature are homotopy equivalent to the original set, using critical point theory and stability results.

Contribution

It introduces a new reconstruction theorem linking Hausdorff distance bounds to homotopy equivalence via critical points in curved manifolds.

Findings

Reconstruction theorem for offsets in manifolds with bounded curvature.

Stability theorems for {3}-critical points in Riemannian manifolds.

Conditions ensuring offsets are homotopy equivalent to original sets.

Abstract

Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of {\mu}-critical points in an annular region. Since an offset of a set deformation retracts to the set itself provided that there are no critical points of the distance function nearby, we can use this theorem to show when the offset of a point cloud is homotopy equivalent to the set it is sampled from. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature. In the process, we prove stability theorems for {\mu}-critical points when the ambient space is a manifold.