
J.D. Boissonnat, S. Y. Oudot. Provably Good Sampling and
Meshing of Lipschitz Surfaces. Proc. 22nd Annual ACM
Sympos. Comput. Geom., pages 337346, 2006.
Abstract:
In the last decade, a great deal of work has been devoted to the
elaboration of a sampling theory for smooth surfaces. The goal was to
ensure a good reconstruction of a given
surface S from a finite subset E of S.
The sampling conditions proposed so far offer guarantees
provided that E is sufficiently dense with respect to the local
feature size of S, which can be true only if S is smooth since the
local feature size vanishes at singular points.
In this paper, we
introduce a new measurable quantity, called the Lipschitz radius,
which plays a role similar to that of the local feature size in the
smooth setting, but which turns out to be welldefined and positive on
a much larger class of shapes. Specifically, it characterizes the
class of Lipschitz surfaces, which includes in particular all
piecewise smooth surfaces such that the normal variation around
singular points is not too large.
Our main result is that, if S is a Lipschitz surface and E is a sample
of S such that any point p of S is at distance less than a fraction of
the Lipschitz radius of S, then we obtain similar guarantees as in the
smooth setting. More precisely, we show that the Delaunay
triangulation of E restricted to S is a 2manifold isotopic to S lying
at bounded Hausdorff distance from S, provided that its facets are not
too skinny.
We further extend this result to the case of loose samples. As a
straightforward application, the Delaunay refinement algorithm we
proved correct for smooth surfaces works fine and offers the same
topological and geometric guarantees for Lipschitz surfaces.
Bibtex:
@inproceedings{bopgsmls06
, author = {J.D. Boissonnat and S. Oudot}
, title = {Provably Good Sampling and Meshing of {Lipschitz} Surfaces}
, year = {2006}
, pages = {337346}
, booktitle = {Proc. 22nd Annual ACM Sympos. Comput. Geom.}
}

