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Stereology for some classes of polyhedrons

Part of: General convexity Inference from stochastic processes Geometric 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 geometry 62M30: Spatial processes
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.

References

[1] Adrian, H. (1992) Statistical characteristics of selected polyhedra. Acta Stereol. 11, 143–155.Google Scholar
[2] Bach, G. (1965) Zufallsschnitte durch ein Haufwerk von Rotationsellipsoiden mit konstantem Achsenverhältnis. Z. angew. Math. Phys. 16, 224–232.CrossRefGoogle Scholar
[3] Coleman, R. (1981) Intercept length of random probes through boxes. J. Appl. Prob. 18, 276–282.CrossRefGoogle Scholar
[4] Cruz Orive, L.-M. (1976) Particle size-shape distributions: the general spheroid problem, I. Mathematical model. J. Microsc. 107, 135–176.Google Scholar
[5] Cruz Orive, L.-M. (1978) Particle size-shape distributions: the general spheroid problem, II. Stochastic model and practical guide. J. Microsc. 112, 235–253.Google Scholar
[6] Fullman, R. L. (1953) Measurement of particle sizes in opaque bodies. Trans. AIME 197, 447–452.Google Scholar
[7] Hartmann, S. F., Mücklich, F., Ohser, J., Dressler, W. and Petzow, G. (1992) Quantitative Charakterisierung von Si3N4-Gefügen durch räumliche Parameter. Sonderb. Prakt. Metallogr. 24, 263–274.Google Scholar
[8] Hull, F. C. and Houk, W. J. (1953) Statistical grain structure studies: Plane distribution curves of regular polyhedrons. Trans. AIME 197, 565–572.Google Scholar
[9] Itoh, H. (1970) An analytical expression of the intercept length distribution of cubic particles. Metallography 3, 407–417.CrossRefGoogle Scholar
[10] Kleinwächter, A. and Zähle, M. (1986) Size distribution stereology for quasiellipsoids in. Statistics 17, 453–454.CrossRefGoogle Scholar
[11] Kneser, H. (1963) Schnitte durch Tetraeder. Proc 1st Internat. Congr. Stereol., Wien, pp. 11/1–11/4.Google Scholar
[12] Little, R. J. A. and Rubin, D. B. (1987) Statistical Analysis with Missing Data. Wiley, New York.Google Scholar
[13] Lorz, U. and Hahn, U. (1993) Geometric characteristics of spatial Voronoi tessellations and planar sections. J. Visual Communication and Image Representation. Submitted.Google Scholar
[14] Lorz, U. and Krawietz, R. (1992) Random plane sections of needle-shaped particles. Metallurg. Found. Engineering 18, 495–511.Google Scholar
[15] Myers, J. (1963) Sectioning of polyhedrons. Proc. 1st Internat. Congr. Stereol., Wien, pp. 15/1–15/6.Google Scholar
[16] Myers, E. J. (1967) Size distributions of cubic particles. In Stereology, ed. Elias, H., pp. 187–188. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[17] Naumovich, N. V. and Kriskovets, T. I. (1982) Influence of the body shape on the profile of chord length distribution. Acta Stereol. 1, 51–59.Google Scholar
[18] Obenaus, P. and Herrmann, M. (1990) Methode zur Charakterisierung von Stengelkristalliten in Siliziumnitrid-Keramik. Prakt. Metallogr. 27, 503–513.CrossRefGoogle Scholar
[19] Paul, J. L. (1981) Distribution curves of sectional area through some families of convex particles. J. Microsc. 122, 165–172.CrossRefGoogle Scholar
[20] Saltykov, S. A. (1974) Stereometrische Metallographie. Deutscher Verlag für Grund-stoffindustrie, Leipzig.Google Scholar
[21] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
[22] Silverman, B. W., Jones, M. C., Nychka, D. W. and Wilson, J. D. (1990) A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography. J. R. Statist. Soc. B52, 271–324.Google Scholar
[23] Socha, P. (1988) Random plane section of a polyhedron. Acta Stereol. 7, 41–46.Google Scholar
[24] Stoyan, D. (1982) Stereological formulae for size distributions through marked point processes. Prob. Math. Statist. 2, 161–166.Google Scholar
[25] Sukiasian, G. S. (1982) On random section of polyhedra. Dokl. Akad. Nauk SSR 263, 809–812.Google Scholar
[26] Underwood, E. E. (1970) Quantitative Stereology. Addison-Wesley, Reading, MA.Google Scholar
[27] Vardi, Y., Shepp, L. A. and Kaufmann, L. (1985) A statistical model for positron emission tomography (with comments). J. Amer. Statist. Assoc. 80, 8–37.CrossRefGoogle Scholar
[28] Voss, K. (1978) Zur numerischen Auswertung von Schnittflächenverteilungen IV. Biom. J. 20, 425–434.CrossRefGoogle Scholar
[29] Voss, K. (1982) Frequencies of n-polygons in planar sections of polyhedra. J. Microsc. 128, 111–120.CrossRefGoogle Scholar
[30] Wasén, J. and Warren, R. (1991) A Catalogue of Stereological Characteristics of Selected Solid Bodies. Vol. 1, Polyhedrons. Chalmers University of Technology, Dept of Engineering Metals.Google Scholar