287 research outputs found
Weighted Poisson-Delaunay Mosaics
Slicing a Voronoi tessellation in with a -plane gives a
-dimensional weighted Voronoi tessellation, also known as power diagram or
Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay
mosaic to the radius of the smallest empty circumscribed sphere whose center
lies in the -plane gives a generalized discrete Morse function. Assuming the
Voronoi tessellation is generated by a Poisson point process in ,
we study the expected number of simplices in the -dimensional weighted
Delaunay mosaic as well as the expected number of intervals of the Morse
function, both as functions of a radius threshold. As a byproduct, we obtain a
new proof for the expected number of connected components (clumps) in a line
section of a circular Boolean model in $\mathbb{R}^n
Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics
Using the geodesic distance on the -dimensional sphere, we study the
expected radius function of the Delaunay mosaic of a random set of points.
Specifically, we consider the partition of the mosaic into intervals of the
radius function and determine the expected number of intervals whose radii are
less than or equal to a given threshold. Assuming the points are not contained
in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of
the convex hull in , so we also get the expected number of
faces of a random inscribed polytope. We find that the expectations are
essentially the same as for the Poisson-Delaunay mosaic in -dimensional
Euclidean space. As proved by Antonelli and collaborators, an orthant section
of the -sphere is isometric to the standard -simplex equipped with the
Fisher information metric. It follows that the latter space has similar
stochastic properties as the -dimensional Euclidean space. Our results are
therefore relevant in information geometry and in population genetics
Three-dimensional alpha shapes
Frequently, data in scientific computing is in its abstract form a finite
point set in space, and it is sometimes useful or required to compute what one
might call the ``shape'' of the set. For that purpose, this paper introduces
the formal notion of the family of -shapes of a finite point set in
\Real^3. Each shape is a well-defined polytope, derived from the Delaunay
triangulation of the point set, with a parameter \alpha \in \Real controlling
the desired level of detail. An algorithm is presented that constructs the
entire family of shapes for a given set of size in time , worst
case. A robust implementation of the algorithm is discussed and several
applications in the area of scientific computing are mentioned.Comment: 32 page
Topological Data Analysis with Bregman Divergences
Given a finite set in a metric space, the topological analysis generalizes
hierarchical clustering using a 1-parameter family of homology groups to
quantify connectivity in all dimensions. The connectivity is compactly
described by the persistence diagram. One limitation of the current framework
is the reliance on metric distances, whereas in many practical applications
objects are compared by non-metric dissimilarity measures. Examples are the
Kullback-Leibler divergence, which is commonly used for comparing text and
images, and the Itakura-Saito divergence, popular for speech and sound. These
are two members of the broad family of dissimilarities called Bregman
divergences.
We show that the framework of topological data analysis can be extended to
general Bregman divergences, widening the scope of possible applications. In
particular, we prove that appropriately generalized Cech and Delaunay (alpha)
complexes capture the correct homotopy type, namely that of the corresponding
union of Bregman balls. Consequently, their filtrations give the correct
persistence diagram, namely the one generated by the uniformly growing Bregman
balls. Moreover, we show that unlike the metric setting, the filtration of
Vietoris-Rips complexes may fail to approximate the persistence diagram. We
propose algorithms to compute the thus generalized Cech, Vietoris-Rips and
Delaunay complexes and experimentally test their efficiency. Lastly, we explain
their surprisingly good performance by making a connection with discrete Morse
theory
The Morse theory of \v{C}ech and Delaunay complexes
Given a finite set of points in and a radius parameter, we
study the \v{C}ech, Delaunay-\v{C}ech, Delaunay (or Alpha), and Wrap complexes
in the light of generalized discrete Morse theory. Establishing the \v{C}ech
and Delaunay complexes as sublevel sets of generalized discrete Morse
functions, we prove that the four complexes are simple-homotopy equivalent by a
sequence of simplicial collapses, which are explicitly described by a single
discrete gradient field.Comment: 21 pages, 2 figures, improved expositio
Quantifying Transversality by Measuring the Robustness of Intersections
By definition, transverse intersections are stable under infinitesimal
perturbations. Using persistent homology, we extend this notion to a measure.
Given a space of perturbations, we assign to each homology class of the
intersection its robustness, the magnitude of a perturbations in this space
necessary to kill it, and prove that robustness is stable. Among the
applications of this result is a stable notion of robustness for fixed points
of continuous mappings and a statement of stability for contours of smooth
mappings
Relaxed Disk Packing
Motivated by biological questions, we study configurations of equal-sized
disks in the Euclidean plane that neither pack nor cover. Measuring the quality
by the probability that a random point lies in exactly one disk, we show that
the regular hexagonal grid gives the maximum among lattice configurations.Comment: 8 pages => 5 pages of main text plus 3 pages in appendix. Submitted
to CCCG 201
On the Optimality of Functionals over Triangulations of Delaunay Sets
In this short paper, we consider the functional density on sets of uniformly
bounded triangulations with fixed sets of vertices. We prove that if a
functional attains its minimum on the Delaunay triangulation, for every finite
set in the plane, then for infinite sets the density of this functional attains
its minimum also on the Delaunay triangulations
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