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Stereology for some classes of polyhedrons

Part of:Inference from stochastic processesGeneral convexityGeometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

J. Ohser and
F. Mücklich
J. Ohser*
Affiliation:
Freiberg University of Mining and Technology
F. Mücklich*
Affiliation:
Max-Planck Institute for Metals Research, Stuttgart
*
* Postal address: Freiberg University of Mining and Technology, Institute of Ferrous Metallurgy, Leipziger Str. 34, D-09596 Freiberg, Germany.
** Postal address: Max-Planck-Institute for Metals Research, Institute for Materials Science, Powder Metallurgical Laboratory, Heisenbergstr. 5, D-70569 Stuttgart, Germany.
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Abstract

A general method for solving stereological problems for particle systems is applied to polyhedron structures. We suggested computing the kernel function of the respective stereological integral equation by means of computer simulation. Two models of random polyhedrons are investigated. First, regular prisms are considered which are described by their size and shape. The size-shape distribution of a stationary and isotropic spatial ensemble of regular prisms can be estimated from the size-shape distribution of the polygons observed in a section plane. Secondly, random polyhedrons are constructed as the convex hull of points which are uniformly distributed on surfaces of spheres. It is assumed that the size of the polyhedrons and the number of points (i.e. the number of vertices) are random variables. Then the distribution of a spatially distributed ensemble of polyhedrons is determined by its size-number distribution. The corresponding numerical density of this bivariate size-number distribution can be stereologically determined from the estimated numerical density of the bivariate size-number distribution of the intersection profiles.

Keywords

ILL-POSED PROBLEMSMAXIMUM LIKELIHOODCOMPUTER SIMULATION

MSC classification

Primary:60D05: Geometric probability and stochastic geometry
Secondary:52A22: Random convex sets and integral geometry62M30: Spatial processes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995, pp. 384 - 396
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Menendez Pelayo, Valencia, Spain, on 21–24 September 1993.

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