Distance Functions and Geodesics on Points Clouds
Abstract
An algorithm for computing intrinsic distance functions and geodesics on sub-manifolds vector r(sup d) 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 vector r(sup d) 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 vector r(sup d), 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 2005
- Accession Number
- ADA437158
Entities
People
- Facundo Mémoli
- Guillermo Sapiro
Organizations
- University of Minnesota