On the Curve Reconstruction in Riemannian Manifolds

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 set of a curve, 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 applications of curve reconstruction to ordering motion frames.