Distance functions and geodesics on points clouds

Abstract

An algorithm for computing intrinsic distance functions and geodesics on sub-manifolds of Rd given by point clouds is introduced in this paper. The basic idea is that, as shown in this paper, intrinsic distance functions and geodesics on general co-dimension sub-manifolds of Rd can be accurately approximated by the extrinsic Euclidean ones computed in a thin offset band surrounding the manifold. This permits the use of computationally optimal algorithms for computing distance functions in Cartesian grids. We then use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on sub-manifolds of Rd, a computationally optimal approach. For point clouds, the offset band is constructed without the need to explicitly find the underlying manifold, thereby computing intrinsic distance functions and geodesics on point clouds while skipping the manifold reconstruction step. The case of point clouds representing noisy samples of a sub-manifold of Euclidean space is studied as well. All the underlying theoretical results are presented, together with experimental examples, and comparisons to graph-based distance algorithms.