2,801 research outputs found
An obstruction to Delaunay triangulations in Riemannian manifolds
Delaunay has shown that the Delaunay complex of a finite set of points of
Euclidean space triangulates the convex hull of , provided
that satisfies a mild genericity property. Voronoi diagrams and Delaunay
complexes can be defined for arbitrary Riemannian manifolds. However,
Delaunay's genericity assumption no longer guarantees that the Delaunay complex
will yield a triangulation; stronger assumptions on are required. A natural
one is to assume that is sufficiently dense. Although results in this
direction have been claimed, we show that sample density alone is insufficient
to ensure that the Delaunay complex triangulates a manifold of dimension
greater than 2.Comment: This is a revision and extension of a note that appeared as an
appendix in the (otherwise unpublished) report arXiv:1303.649
Constructing Intrinsic Delaunay Triangulations of Submanifolds
We describe an algorithm to construct an intrinsic Delaunay triangulation of
a smooth closed submanifold of Euclidean space. Using results established in a
companion paper on the stability of Delaunay triangulations on -generic
point sets, we establish sampling criteria which ensure that the intrinsic
Delaunay complex coincides with the restricted Delaunay complex and also with
the recently introduced tangential Delaunay complex. The algorithm generates a
point set that meets the required criteria while the tangential complex is
being constructed. In this way the computation of geodesic distances is
avoided, the runtime is only linearly dependent on the ambient dimension, and
the Delaunay complexes are guaranteed to be triangulations of the manifold
A Cosmic Watershed: the WVF Void Detection Technique
On megaparsec scales the Universe is permeated by an intricate filigree of
clusters, filaments, sheets and voids, the Cosmic Web. For the understanding of
its dynamical and hierarchical history it is crucial to identify objectively
its complex morphological components. One of the most characteristic aspects is
that of the dominant underdense Voids, the product of a hierarchical process
driven by the collapse of minor voids in addition to the merging of large ones.
In this study we present an objective void finder technique which involves a
minimum of assumptions about the scale, structure and shape of voids. Our void
finding method, the Watershed Void Finder (WVF), is based upon the Watershed
Transform, a well-known technique for the segmentation of images. Importantly,
the technique has the potential to trace the existing manifestations of a void
hierarchy. The basic watershed transform is augmented by a variety of
correction procedures to remove spurious structure resulting from sampling
noise. This study contains a detailed description of the WVF. We demonstrate
how it is able to trace and identify, relatively parameter free, voids and
their surrounding (filamentary and planar) boundaries. We test the technique on
a set of Kinematic Voronoi models, heuristic spatial models for a cellular
distribution of matter. Comparison of the WVF segmentations of low noise and
high noise Voronoi models with the quantitatively known spatial characteristics
of the intrinsic Voronoi tessellation shows that the size and shape of the
voids are succesfully retrieved. WVF manages to even reproduce the full void
size distribution function.Comment: 24 pages, 15 figures, MNRAS accepted, for full resolution, see
http://www.astro.rug.nl/~weygaert/tim1publication/watershed.pd
Computing a Compact Spline Representation of the Medial Axis Transform of a 2D Shape
We present a full pipeline for computing the medial axis transform of an
arbitrary 2D shape. The instability of the medial axis transform is overcome by
a pruning algorithm guided by a user-defined Hausdorff distance threshold. The
stable medial axis transform is then approximated by spline curves in 3D to
produce a smooth and compact representation. These spline curves are computed
by minimizing the approximation error between the input shape and the shape
represented by the medial axis transform. Our results on various 2D shapes
suggest that our method is practical and effective, and yields faithful and
compact representations of medial axis transforms of 2D shapes.Comment: GMP14 (Geometric Modeling and Processing
The persistent cosmic web and its filamentary structure II: Illustrations
The recently introduced discrete persistent structure extractor (DisPerSE,
Soubie 2010, paper I) is implemented on realistic 3D cosmological simulations
and observed redshift catalogues (SDSS); it is found that DisPerSE traces
equally well the observed filaments, walls, and voids in both cases. In either
setting, filaments are shown to connect onto halos, outskirt walls, which
circumvent voids. Indeed this algorithm operates directly on the particles
without assuming anything about the distribution, and yields a natural
(topologically motivated) self-consistent criterion for selecting the
significance level of the identified structures. It is shown that this
extraction is possible even for very sparsely sampled point processes, as a
function of the persistence ratio. Hence astrophysicists should be in a
position to trace and measure precisely the filaments, walls and voids from
such samples and assess the confidence of the post-processed sets as a function
of this threshold, which can be expressed relative to the expected amplitude of
shot noise. In a cosmic framework, this criterion is comparable to friend of
friend for the identifications of peaks, while it also identifies the connected
filaments and walls, and quantitatively recovers the full set of topological
invariants (Betti numbers) {\sl directly from the particles} as a function of
the persistence threshold. This criterion is found to be sufficient even if one
particle out of two is noise, when the persistence ratio is set to 3-sigma or
more. The algorithm is also implemented on the SDSS catalogue and used to locat
interesting configurations of the filamentary structure. In this context we
carried the identification of an ``optically faint'' cluster at the
intersection of filaments through the recent observation of its X-ray
counterpart by SUZAKU. The corresponding filament catalogue will be made
available online.Comment: A higher resolution version is available at
http://www.iap.fr/users/sousbie together with complementary material (movie
and data). Submitted to MNRA
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
Only distances are required to reconstruct submanifolds
In this paper, we give the first algorithm that outputs a faithful
reconstruction of a submanifold of Euclidean space without maintaining or even
constructing complicated data structures such as Voronoi diagrams or Delaunay
complexes. Our algorithm uses the witness complex and relies on the stability
of power protection, a notion introduced in this paper. The complexity of the
algorithm depends exponentially on the intrinsic dimension of the manifold,
rather than the dimension of ambient space, and linearly on the dimension of
the ambient space. Another interesting feature of this work is that no explicit
coordinates of the points in the point sample is needed. The algorithm only
needs the distance matrix as input, i.e., only distance between points in the
point sample as input.Comment: Major revision, 16 figures, 47 page
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