Curve Reconstruction in Riemannian Manifolds: Ordering Motion Frames

TL;DR

This paper extends geometric curve reconstruction methods to Riemannian manifolds, proving that minimal spanning trees can accurately reconstruct curves and demonstrating applications in ordering motion frames.

Contribution

It introduces a method for curve reconstruction in Riemannian manifolds using minimal spanning trees, with proofs and practical examples.

Findings

Minimal spanning trees correctly reconstruct smooth and closed curves in Riemannian manifolds.

The approach accounts for local topological changes via the injectivity radius.

Successful reconstruction demonstrated on practical examples, including motion frame ordering.

Abstract

In this article we extend the computational geometric curve reconstruction approach to curves in Riemannian manifolds. We prove that the minimal spanning tree, given a sufficiently dense sample, correctly reconstructs the smooth arcs and further closed and simple curves in Riemannian manifolds. The proof is based on the behaviour of the curve segment inside the tubular neighbourhood of the curve. To take care of the local topological changes of the manifold, the tubular neighbourhood is constructed in consideration with the injectivity radius of the underlying Riemannian manifold. We also present examples of successfully reconstructed curves and show an applications of curve reconstruction to ordering motion frames.