Distance functions and geodesics on submanifolds of ℝ ^d and point clouds

A theoretical and computational framework for computing intrinsic distance functions and geodesics on submanifolds of ℝ ^d given by point clouds is introduced and developed in this paper. The basic idea is that, as shown here, intrinsic distance functions and geodesics on general co-dimension submanifolds of ℝ ^d 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 a computationally optimal approach also for the case of intrinsic distance functions on submanifolds of ℝ ^d. 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.