Mixed measures of convex cylinders and quermass densities of Boolean models
type: Item
instance of: scholarly article
title: Mixed measures of convex cylinders and quermass densities of Boolean models
zbMATH Open document ID: 1180.52011
author: Lars Michael Hoffmann
published in: Acta Applicandae Mathematicae
publication date: 2009-11-20T00:00:00Z
review text: A Boolean model of convex bodies is the best known example of a random closed set. It has been studied extensively in recent years and is among the most popular models from stochastic geometry. For the latter, quermass densities (also called densities of Minkowski functionals) are very interesting characteristics because a lot of algorithms for their estimation from real data are available. Whereas quermass densities can be defined for stationary random closed sets in the extended convex ring, for a stationary and isotropic Boolean model of convex bodies, iterations of the principal kinematic formula for quermass integrals can be used to express quermass densities in terms of the underlying Poisson particle process. To extend these results to stationary but not necessarily isotropic processes of convex particles and their Boolean models, translative integral formulas for curvature measures by R. Schneider and W. Weil were applied. For non-stationary Boolean models of convex bodies, quermass densities were introduced by \textit{H. Fallert} [Math. Nachr. 181, 165--184 (1996; Zbl 0858.60015)] using an iterated version of these translative integral formulas where the latter was obtained by W. Weil. Yet in some applications [\textit{K. Schladitz, S. Peters, D. Reinel-Bitzer, A. Wiegmann} and \textit{J. Ohser}, Comput. Mat. Sci. 38, 56--66 (2006)] it is necessary to work with Boolean models of convex cylinders, i.e. Minkowski sums of convex bodies and linear subspaces of \({\mathbb R}^d\) (= \(d\)-dimensional Euclidean space). In this case formulas for all quermass densities are only available for stationary and isotropic Boolean models for which they were proved by \textit{P. Davy} [Stereology -- a stationary viewpoint. Dissertation, Australian National University, Camberra (1978)]. Recently, \textit{M. Spiess} and \textit{E. Spodarev} obtained results for the specific volume and specific surface area of stationary Boolean models [``Anisotropic dilated Poisson \(k\)-flat…
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 5636289
zbMATH Keywords: curvature measure
zbMATH Keywords: fibre process
zbMATH Keywords: Minkowski functionals
zbMATH Keywords: particle processes
zbMATH Keywords: Poisson process
zbMATH Keywords: specific intrinsic volume
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
full work available at URL: s10440-008-9269-4
OpenAlex ID: W2077820865
cites work: Q4284039
cites work: Quermaßdichten für Punktprozesse konvexer Körper und Boolesche Modelle
cites work: Translative integral formulae for convex bodies
cites work: Q5287551
cites work: Translative and Kinematic Integral Formulae for Curvature Measures
cites work: Q4000464
cites work: Anisotropic Poisson processes of cylinders
cites work: Q4842952
cites work: Point processes of cylinders, particles and flats
cites work: Iterations of translative integral formulae and non-isotropic Poisson processes of particles
cites work: Mixed measures and functionals of translative integral geometry
DOI: 10.1007/S10440-008-9269-4
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description: scientific article; zbMATH DE number 5636289