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Approximating the Riemannian Metric from Point Clouds via Manifold Moving Least Squares

by   Barak Sober, et al.

The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold ℳ of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution parameter h, state-of-the-art discrete methods yield O(h) provable approximations. In this paper, we investigate the convergence of such approximations made by Manifold Moving Least-Squares (Manifold-MLS), a method that constructs an approximating manifold ℳ^h using information from a given point cloud that was developed by Sober & Levin in 2019. In this paper, we show that provided that ℳ∈ C^k and closed (i.e. ℳ is a compact manifold without boundary) the Riemannian metric of ℳ^h approximates the Riemannian metric of ℳ,. Explicitly, given points p_1, p_2 ∈ℳ with geodesic distance ρ_ℳ(p_1, p_2), we show that their corresponding points p_1^h, p_2^h ∈ℳ^h have a geodesic distance of ρ_ℳ^h(p_1^h,p_2^h) = ρ_ℳ(p_1, p_2)(1 + O(h^k-1)) (i.e., the Manifold-MLS is nearly an isometry). We then use this result, as well as the fact that ℳ^h can be sampled with any desired resolution, to devise a naive algorithm that yields approximate geodesic distances with a rate of convergence O(h^k-1). We show the potential and the robustness to noise of the proposed method on some numerical simulations.


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