Poisson Hyperplane Tessellations
Poisson Hyperplane Tessellations

Author: Daniel Hug

Publisher: Springer

Published: 2024-04-16

Total Pages: 0

ISBN-13: 9783031541032

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This book is the first comprehensive presentation of a central topic of stochastic geometry: random mosaics that are generated by Poisson processes of hyperplanes. It thus connects a basic notion from probability theory, Poisson processes, with a fundamental object of geometry. The independence properties of Poisson processes and the long-range influence of hyperplanes lead to a wide range of phenomena which are of interest from both a geometric and a probabilistic point of view. A Poisson hyperplane tessellation generates many random polytopes, also a much-studied object of stochastic geometry. The book offers a variety of different perspectives and covers in detail all aspects studied in the original literature. The work will be useful to graduate students (advanced students in a Master program, PhD students), and professional mathematicians. The book can also serve as a reference for researchers in fields of physics, computer science, economics or engineering.

Spatial Tessellations
Spatial Tessellations

Author: Atsuyuki Okabe

Publisher: John Wiley & Sons

Published: 2009-09-25

Total Pages: 696

ISBN-13: 047031785X

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Spatial data analysis is a fast growing area and Voronoi diagrams provide a means of naturally partitioning space into subregions to facilitate spatial data manipulation, modelling of spatial structures, pattern recognition and locational optimization. With such versatility, the Voronoi diagram and its relative, the Delaunay triangulation, provide valuable tools for the analysis of spatial data. This is a rapidly growing research area and in this fully updated second edition the authors provide an up-to-date and comprehensive unification of all the previous literature on the subject of Voronoi diagrams. Features: * Expands on the highly acclaimed first edition * Provides an up-to-date and comprehensive survey of the existing literature on Voronoi diagrams * Includes a useful compendium of applications * Contains an extensive bibliography A wide range of applications is discussed, enabling this book to serve as an important reference volume on this topic. The text will appeal to students and researchers studying spatial data in a number of areas, in particular, applied probability, computational geometry, and Geographic Information Science (GIS). This book will appeal equally to those whose interests in Voronoi diagrams are theoretical, practical or both.

Repulsion of Determinantal Point Processes and Stationary Poisson Tessellations in High Dimensions
Repulsion of Determinantal Point Processes and Stationary Poisson Tessellations in High Dimensions

Author: Elizabeth Watson O'Reilly

Publisher:

Published: 2019

Total Pages: 336

ISBN-13:

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In this dissertation, new results on stochastic geometric models in high dimensional space are presented. We first concentrate on a particular class of repulsive point processes called determinantal point processes (DPPs). We establish a coupling of a DPP and its reduced Palm version showing the repulsive effect of a point of the point process. This is used for discussing the degree of repulsiveness in DPPs, including Ginibre point processes and other specific parametric models for DPPs. We then study this repulsion for stationary DPPs in high dimensional Euclidean space. It is shown that for many families of DPPs, a typical point has no repulsive effect with high probability for large space dimension n. It is also proved that for some DPPs there exists an R* such that the repulsive effect occurs at a distance of [square root] nR* with high probability for large n. This R* is interpreted as the asymptotic reach of repulsion of the DPP. Examples of DPPs exhibiting this behavior are presented and an application to high dimensional Boolean models is given. The second half of this dissertation examines zero cells of stationary Poisson tessellations. First, a stationary stochastic geometric model is proposed for analyzing one-bit data compression. The data is assumed to be an unconstrained stationary set, and each data point is compressed using one bit with respect to each hyperplane in a stationary and isotropic Poisson hyperplane tessellation. Size metrics of the zero cell of the tessellation are studied to determine how the intensity of hyperplanes must scale with dimension to ensure sufficient separation of different data by the hyperplanes or sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and source coding. We then study the concentration of the norm of a random vector Y uniformly sampled in the centered zero cell of a stationary random tessellation in high dimensions. It is shown that for a stationary and isotropic Poisson-Voronoi tessellation, [mathematical equation] approaches one as the dimension approaches infinity. For a stationary and isotropic Poisson hyperplane tessellation, we prove that [mathematical equation] will be within a fixed range (R [subscript l], R [subscript u]) with probability approaching one as dimension n tends to infinity

Stochastic Geometry, Spatial Statistics and Random Fields
Stochastic Geometry, Spatial Statistics and Random Fields

Author: Volker Schmidt

Publisher: Springer

Published: 2014-10-24

Total Pages: 484

ISBN-13: 3319100645

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This volume is an attempt to provide a graduate level introduction to various aspects of stochastic geometry, spatial statistics and random fields, with special emphasis placed on fundamental classes of models and algorithms as well as on their applications, e.g. in materials science, biology and genetics. This book has a strong focus on simulations and includes extensive codes in Matlab and R which are widely used in the mathematical community. It can be seen as a continuation of the recent volume 2068 of Lecture Notes in Mathematics, where other issues of stochastic geometry, spatial statistics and random fields were considered with a focus on asymptotic methods.

Tensor Valuations and Their Applications in Stochastic Geometry and Imaging
Tensor Valuations and Their Applications in Stochastic Geometry and Imaging

Author: Eva B. Vedel Jensen

Publisher: Springer

Published: 2017-06-10

Total Pages: 469

ISBN-13: 3319519514

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The purpose of this volume is to give an up-to-date introduction to tensor valuations and their applications. Starting with classical results concerning scalar-valued valuations on the families of convex bodies and convex polytopes, it proceeds to the modern theory of tensor valuations. Product and Fourier-type transforms are introduced and various integral formulae are derived. New and well-known results are presented, together with generalizations in several directions, including extensions to the non-Euclidean setting and to non-convex sets. A variety of applications of tensor valuations to models in stochastic geometry, to local stereology and to imaging are also discussed.

Stochastic Analysis for Poisson Point Processes
Stochastic Analysis for Poisson Point Processes

Author: Giovanni Peccati

Publisher: Springer

Published: 2016-07-07

Total Pages: 359

ISBN-13: 3319052330

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Stochastic geometry is the branch of mathematics that studies geometric structures associated with random configurations, such as random graphs, tilings and mosaics. Due to its close ties with stereology and spatial statistics, the results in this area are relevant for a large number of important applications, e.g. to the mathematical modeling and statistical analysis of telecommunication networks, geostatistics and image analysis. In recent years – due mainly to the impetus of the authors and their collaborators – a powerful connection has been established between stochastic geometry and the Malliavin calculus of variations, which is a collection of probabilistic techniques based on the properties of infinite-dimensional differential operators. This has led in particular to the discovery of a large number of new quantitative limit theorems for high-dimensional geometric objects. This unique book presents an organic collection of authoritative surveys written by the principal actors in this rapidly evolving field, offering a rigorous yet lively presentation of its many facets.