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1.
It is well known that the complexity of the Delaunay
triangulation of $n$ points in $\RR ^d$, i.e., the number of its
simplices, can be $\Omega (n^{\lceil {d}/{2}\rceil })$. In
particular, in $\RR ^3$, the number of tetrahedra can be quadratic.
Put another way, if the points are uniformly distributed in a cube or a
ball, the expected complexity of the Delaunay triangulation is only
linear. The case of points distributed on a surface is of great
practical importance in reverse engineering since most surface
reconstruction algorithms first construct the Delaunay triangulation
of a set of points measured on a surface.
In this paper we bound the complexity of the Delaunay triangulation
of points distributed on the boundary of a given polyhedron. Under a
mild uniform sampling condition, we provide deterministic asymptotic
bounds on the complexity of the three-dimensional Delaunay triangulation of the
points when the sampling density increases. More precisely, we show
that the complexity is $O(n^{1.8})$ for general polyhedral surfaces
and $O(n\sqrt{n})$ for convex polyhedral surfaces.
Our proof uses a geometric result of independent interest that states
that the medial axis of a surface is well approximated by a subset of
the Voronoi vertices of the sample points. 相似文献
2.
Triangulations in CGAL 总被引:7,自引:0,他引:7
Jean-Daniel Boissonnat Olivier Devillers Sylvain Pion Monique Teillaud Mariette Yvinec 《Computational Geometry》2002,22(1-3):5-19
This paper presents the main algorithmic and design choices that have been made to implement triangulations in the computational geometry algorithms library
. 相似文献
3.
4.
Jean-Daniel Boissonnat David Cohen-Steiner Gert Vegter 《Discrete and Computational Geometry》2008,39(1-3):138-157
This paper addresses the problem of piecewise linear approximation of implicit surfaces. We first give a criterion ensuring
that the zero-set of a smooth function and the one of a piecewise linear approximation of it are isotopic. Then, we deduce
from this criterion an implicit surface meshing algorithm certifying that the output mesh is isotopic to the actual implicit
surface. This is the first algorithm achieving this goal in a provably correct way. 相似文献
5.
Jean-Daniel Boissonnat Leonidas J. Guibas Steve Y. Oudot 《Discrete and Computational Geometry》2009,42(1):37-70
It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low
dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation
on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results
do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density. On the
positive side, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between
restricted Delaunay triangulation and witness complex hold on higher-dimensional manifolds as well. We derive from our structural
results an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales. The algorithm
combines a farthest-point refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not
require the input point sample to be sparse. Its running time is bounded by c(d)n
2, where n is the size of the input point cloud, and c(d) is a constant depending solely (yet exponentially) on the dimension d of the ambient space. Although this running time makes our reconstruction algorithm rather theoretical, recent work has shown
that a variant of our approach can be made tractable in arbitrary dimensions, by building upon the results of this paper.
This work was done while S.Y. Oudot was a post-doctoral fellow at Stanford University. His email there is no longer valid. 相似文献
6.
Dr. Camille Lecourt Sabrina Dhambri Khalil Yamani Dr. Guillaume Boissonnat Dr. Simon Specklin Dr. Etienne Fleury Karim Hammad Eric Auclair Serge Sablé Dr. Antonio Grondin Dr. Paola B. Arimondo Dr. François Sautel Dr. Georges Massiot Dr. Christophe Meyer Prof. Dr. Janine Cossy Dr. Geoffroy Sorin Dr. Marie-Isabelle Lannou Prof. Dr. Janick Ardisson 《Chemistry (Weinheim an der Bergstrasse, Germany)》2019,25(11):2745-2749
A strategy for the assembly of the entire carbon backbone of a stereoisomer of the antitumor marine natural product hemicalide has been investigated. The devised convergent approach relies on Horner–Wadsworth–Emmons and Julia–Kocienski olefination reactions for the construction of the C6=C7 and C34=C35 double bonds, respectively, an aldol reaction to create the C27−C28 bond, and a Suzuki–Miyaura cross-coupling as the endgame to form the C15−C16 bond. 相似文献
7.
J.-D. Boissonnat M. Sharir B. Tagansky M. Yvinec 《Discrete and Computational Geometry》1998,19(4):485-519
The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance
functions. Specifically, if S is a set of n points in general position in R
d
, the maximum complexity of its Voronoi diagram under the L
∞
metric, and also under a simplicial distance function, are both shown to be . The upper bound for the case of the L
∞
metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R
d
. This complexity is , for d ≥ 1 , and it improves to , for d ≥ 2 , if all the hypercubes have the same size. Under the L
1
metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R
3
is shown to be . We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases
significantly when the points are in degenerate configurations. (This increase does not occur with an appropriate modification
of the diagram definition.) Finally, on-line algorithms are proposed for computing the Voronoi diagram of n points in R
d under a simplicial or L
∞
distance function. Their expected randomized complexities are for simplicial diagrams and for L
∞
-diagrams.
Received July 31, 1995, and in revised form September 9, 1997. 相似文献
8.
We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in
\mathbbRd{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of
\mathbbR 3{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of
\mathbbR d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the
ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading
to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e2){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k). 相似文献
9.
Jean-Daniel Boissonnat Frank Nielsen Richard Nock 《Discrete and Computational Geometry》2010,44(2):281-307
The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into
regions, each region consisting of the points that are closer to a given object than to the others. We may define various
variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper,
we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman
divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence
measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi
diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman
diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi
diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. 相似文献
10.
Delaunay triangulations and Voronoi diagrams have found numerous
applications in surface modeling, surface mesh generation,
deformable surface modeling and surface reconstruction. Many
algorithms in these applications begin by constructing the
three-dimensional Delaunay triangulation of a finite set of points
scattered over a surface. Their running-time therefore depends on
the complexity of the Delaunay triangulation of such point sets.
Although the complexity of the Delaunay triangulation of points in
R3 may be quadratic in the worst case, we show in this paper
that it is only linear when the points are distributed on a fixed
set of well-sampled facets of R3 (e.g. the planar polygons in a
polyhedron). Our bound is deterministic and the constants are
explicitly given. 相似文献