178 research outputs found
Poisson–Delaunay Mosaics of Order k
The order-k Voronoi tessellation of a locally finite set ⊆ℝ decomposes ℝ into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold
On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane
For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles in 1970
On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane
For a locally finite set in , the order- Brillouin
tessellations form an infinite sequence of convex face-to-face tilings of the
plane. If the set is coarsely dense and generic, then the corresponding
infinite sequences of minimum and maximum angles are both monotonic in . As
an example, a stationary Poisson point process in is locally
finite, coarsely dense, and generic with probability one. For such a set, the
distribution of angles in the Voronoi tessellations, Delaunay mosaics, and
Brillouin tessellations are independent of the order and can be derived from
the formula for angles in order- Delaunay mosaics given by Miles in 1970
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
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
Image Sampling with Quasicrystals
We investigate the use of quasicrystals in image sampling. Quasicrystals
produce space-filling, non-periodic point sets that are uniformly discrete and
relatively dense, thereby ensuring the sample sites are evenly spread out
throughout the sampled image. Their self-similar structure can be attractive
for creating sampling patterns endowed with a decorative symmetry. We present a
brief general overview of the algebraic theory of cut-and-project quasicrystals
based on the geometry of the golden ratio. To assess the practical utility of
quasicrystal sampling, we evaluate the visual effects of a variety of
non-adaptive image sampling strategies on photorealistic image reconstruction
and non-photorealistic image rendering used in multiresolution image
representations. For computer visualization of point sets used in image
sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary
materials, please visit at
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IST Austria Thesis
The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's
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