Exact and Minimal Sampling for Riemannian Curve Reconstruction

Develop an algorithm that, given a closed curve C on a Riemannian manifold M, constructs a sample set S ⊂ C that is minimal in cardinality while exactly satisfying the injectivity-radius–constrained ρ-sampling condition defined via the injective local feature size ilfs(p) = min(lfs(p), i_M(p)) and the injective reach of intervals on C, rather than relying on heuristic backtracking to extract a subset of samples meeting these conditions.

Background

In extending planar ρ-sampling to Riemannian manifolds, the paper introduces injective local feature size and injective reach to define a manifold-aware ρ-sampling scheme. For experiments, the authors approximate the medial axis using Voronoi diagrams and then employ a backtracking strategy to extract a subset of samples that satisfies their manifold sampling conditions.

The authors explicitly state that producing an exact sampling of minimal cardinality which meets these manifold sampling conditions is not currently achieved and remains open. Solving this would yield principled sample sets for curves on manifolds, improving both theoretical guarantees and practical efficiency for reconstruction pipelines that depend on such sampling.