Steve Y. Oudot. On the Topology of the Restricted Delaunay
Triangulation and Witness Complex in Higher Dimensions. Technical
Report, Stanford University, November 2006. LANL arXiv:0803.1296v1
[cs.CG], http://arxiv.org/abs/0803.1296. Published
in the full version
of this
paper.
It is a well-known fact that, under mild sampling conditions, the
restricted Delaunay triangulation provides good topological
approximations of 1- and 2-manifolds. We show that this is not the
case for higher-dimensional manifolds, even under stronger sampling
conditions. Specifically, it is not true that, for any compact closed
submanifold M of R^n, and any sufficiently dense uniform
sampling L of M, the Delaunay triangulation of L
restricted to M is homeomorphic to M, or even homotopy
equivalent to it. Besides, it is not true either that, for any
sufficiently dense set W of witnesses, the witness complex
of L relative to W contains or is contained in the restricted Delaunay
triangulation of L.
@techreport{o-ntrdwchd-06 , author = "S. Y. Oudot" , title = "On the topology of the restricted {Delaunay} triangulation and witness complex in higher dimensions" , month = "November" , year = "2006" , institution = "Stanford University" , note = "LANL arXiv:0803.1296v1 [cs.CG], url{http://arxiv.org/abs/0803.1296}" } |