Distance Functions and Geodesics on Point Clouds

An new paradigm for computing intrinsic distance functions and geodesics on sub-manifolds of given by point clouds is introduced in this paper. The basic idea is that, as shown here, intrinsic distance functions and geodesics on general co-dimension sub-manifolds of can be accurately approximated by extrinsic Euclidean ones computed inside a thin offset band surrounding the manifold. This permits the use of computationally optimal algorithms for computing distance functions in Cartesian grids. We use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on sub-manifolds of , 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 submanifold of Euclidean space is studied as well. All the underlying theoretical results are presented along with experimental examples for diverse applications and comparisons to graph-based distance algorithms.