Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-30T18:19:39.488Z Has data issue: false hasContentIssue false
  • English
  • Français

On the distribution of the spherical contact vector of stationary germ-grain models

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

G. Last  and
R. Schassberger
G. Last*
Affiliation:
Technical University of Braunschweig
R. Schassberger*
Affiliation:
Technical University of Braunschweig
*
∗ Postal address: Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstraße 14, Postfach 33,29, 38023 Braunschweig, Germany.
∗ Postal address: Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstraße 14, Postfach 33,29, 38023 Braunschweig, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝd to Ξ. For almost all points x ε ℝd there exists a unique point p(x) in the boundary of Ξ such that r(x) is the length of the vector x-p(x), which is called the spherical contact vector at x. In this paper we relate the distribution of the spherical contact vector to the times it takes a typical boundary point of Ξ to hit another grain if all grains start growing at the same time and at the same speed. The notion of a typical point is made precise by using the generalized curvature measures of Ξ. The result generalizes a well known formula for the Boolean model. Specific examples are discussed in detail.

Keywords

Spherical contact vectorspherical contact distributioncurvature measuresgerm-grain modelrandom settessellationrandom measurePalm probability

MSC classification

Primary: 60G57: Random measures
Secondary: 60D05: Geometric probability and stochastic geometry

Information

Type
Stochatic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 36 - 52
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Baddeley, A. and Gill, R.D. (1994). The empty space hazard of spatial pattern. Preprint 845. University Utrecht.Google Scholar
Baddeley, A. and Gill, R.D. (1997). Kaplan–Meier estimators of interpoint distributions for spatial point processes. Ann. Statist. 25, 263–292.CrossRefGoogle Scholar
Bonnesen, T. and Fenchel, W. (1934). Theorie der konvexen Körper. Springer, Berlin.Google Scholar
Federer, H. (1969). Geometric Measure Theory. Springer, Berlin.Google Scholar
Heinrich, L. (1998). Contact and chord length distribution of a stationary Voronoi tessellation. Adv. Appl. Prob. 30, to appear.CrossRefGoogle Scholar
Heinrich, L. and Molchanov, I.S. (1995). Central limit theorem for a class of random measures associated with germ-grain models. CWI report BS-R9518.Google Scholar
Last, G. and Schassberger, R. (1996a). A flow conservation law for surface processes. Adv. Appl. Prob. 28, 13–28.CrossRefGoogle Scholar
Last, G. and Schassberger, R. (1996b). On the spherical contact distribution of stationary random sets. Technical Report No 96/06. Technical University Braunschweig.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Mecke, J. and Muche, L. (1995). The Poisson Voronoi tessellation I. A basic identity. Math. Nachr. 176, 199–208.CrossRefGoogle Scholar
Möller (1989). Random tessellations in Rd . Adv. Appl. Prob. 24, 37–73.Google Scholar
Muche, L. and Stoyan, D. (1992). Contact and chord length distribution of the Poisson Voronoi tessellation. J. Appl. Prob. 29, 467–471.CrossRefGoogle Scholar
Okabe, A., Boots, B. and Sugihara, K. (1992). Spatial Tessellations. Concepts and Applications of Voronoi Diagrams. Wiley, Chichester.Google Scholar
Saxl, I. (1993). Contact distances and random free paths. J. Microsc. 170, 53–64.CrossRefGoogle Scholar
Schneider, R. (1980). Parallelmengen mit Vielfachheit und Steiner-Formeln. Geom. Dedicata 9, 111–127.CrossRefGoogle Scholar
Schneider, R. (1993). Convex surfaces, curvature and surface area measures. In Handbook of Convex Geometry. North-Holland, Amsterdam. pp. 273–299.CrossRefGoogle Scholar
Schneider, R. and Wieacker, J.A. (1993). Integral geometry. In Handbook of Convex Geometry. North-Holland, Amsterdam. pp. 1349–1390.CrossRefGoogle Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D., Kendall, W.S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. 2nd edn, Wiley, Chichester.Google Scholar
Weil, W. (1997). The mean normal distribution of stationary random sets and particle processes. In Advances in Theory and Applications of Random Sets, Proc. Int. Symposium, Fontainebleau, France, 9–11 Oct. 1996. ed. Jeulin, D.. World Scientific, Singapore. pp. 21–33.Google Scholar
Weil, W. and Wieacker, J.A. (1984). Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.CrossRefGoogle Scholar
Weil, W. and Wieacker, J.A. (1993). Stochastic geometry. In Handbook of Convex Geometry. North-Holland, Amsterdam. pp. 1391–1438.CrossRefGoogle Scholar
Zähle, M., (1986). Curvature measures and random sets. Prob. Theory Rel. Fields 71, 37–58.CrossRefGoogle Scholar