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Densities of mixed volumes for Boolean models

Part of: Inference from stochastic processes General convexity Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Wolfgang Weil
Wolfgang Weil*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Mathematisches Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany. Email address: [email protected]
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Abstract

In generalization of the well-known formulae for quermass densities of stationary and isotropic Boolean models, we prove corresponding results for densities of mixed volumes in the stationary situation and show how they can be used to determine the intensity of non-isotropic Boolean models Z in d-dimensional space for d = 2, 3, 4. We then consider non-stationary Boolean models and extend results of Fallert on quermass densities to densities of mixed volumes. In particular, we present explicit formulae for a planar inhomogeneous Boolean model with circular grains.

Keywords

Boolean modelstationarymixed volumesPoisson processintensityinhomogeneous

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A39: Mixed volumes and related topics 60G55: Point processes 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 39 - 60
Copyright
Copyright © Applied Probability Trust 2001 

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References

[1] Davy, P. (1976). Projected thick sections through multidimensional particle aggregates. J. Appl. Prob. 13, 714–722 and 15, 456.Google Scholar
[2] Davy, P. (1978). Stereology—a statistical viewpoint. , Australian National University.CrossRefGoogle Scholar
[3] Fallert, H. (1996). Quermaßdichten für Punktprozesse konvexer Körper und Boolesche Modelle. Math. Nachr. 181, 165–184.Google Scholar
[4] Hahn, U., Micheletti, A., Pohlink, R., Stoyan, D. and Wendrock, H. (1999). Stereological analysis and modeling of gradient structures. J. Microscopy 195, 113–124.Google Scholar
[5] Kellerer, H. G. (1984). Minkowski functionals of Poisson processes. Z. Wahr-schein-lich-keits-th. 67, 63–84.Google Scholar
[6] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
[7] Mecke, K. (1994). Integralgeometrie in der Statistischen Physik. Harri Deutsch, Thun.Google Scholar
[8] Mecke, K. (1998). Integral geometry in statistical physics. Int. J. Modern Phys. B 12, 861–899.Google Scholar
[9] Micheletti, A. and Stoyan, D. (1998). Volume fraction and surface density for inhomogeneous random sets. Preprint 17/1998, Dipartimento di Matematica, Universitá degli Studi di Milano.Google Scholar
[10] Miles, R. E. (1976). Estimating aggregate and overall characteristics from thick sections by transmission microscopy. J. Microscopy 107, 227–233.CrossRefGoogle Scholar
[11] Molchanov, I. S. (1997). Statistics of the Boolean model for Practitioners and Mathematicians. John Wiley, New York.Google Scholar
[12] Quintanilla, J. and Torquato, S. (1997). Microstructure functions for a model of statistically inhomogeneous random media. Phys. Rev. E 55, 1558–1565.Google Scholar
[13] Quintanilla, J. and Torquato, S. (1997). Clustering in a continuum percolation model. Adv. Appl. Prob. 29, 327–336.Google Scholar
[14] Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press.Google Scholar
[15] Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.Google Scholar
[16] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, New York.Google Scholar
[17] Weil, W. (1988). Expectation formulas and isoperimetric properties for non-isotropic Boolean models. J. Microscopy 151, 235–245.Google Scholar
[18] Weil, W. (1990). Iterations of translative integral formulae and non-isotropic Poisson processes of particles. Math. Z. 205, 531–549.Google Scholar
[19] Weil, W. (1995). The estimation of mean shape and mean particle number in overlapping particle systems in the plane. Adv. Appl. Prob. 27, 102–119.CrossRefGoogle Scholar
[20] Weil, W. (1997). On the mean shape of particle processes. Adv. Appl. Prob. 29, 890–908.Google Scholar
[21] Weil, W. (1997). Mean bodies associated with random closed sets. Suppl. Rend. Circ. Mat. Palermo II 50, 387–412.Google Scholar
[22] Weil, W. (1999). Intensity analysis of Boolean models. Pattern Recognition 32, 1675–1684.Google Scholar
[23] Weil, W. (2001). Mixed measures and functionals of translative integral geometry. Math. Nachr. 223, 161–184.Google Scholar
[24] Weil, W. and Wieacker, J. A. (1984). Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.Google Scholar
[25] Zähle, M., (1986). Curvature mesures and random sets II. Z. Wahr-schein-lich-keits-th. 71, 37–58.Google Scholar