Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T15:50:47.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Rotational versions of the Crofton formula

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Eva B. Vedel Jensen
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
*
* Postal address: Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark.
Get access Rights & Permissions [Opens in a new window]

Abstract

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the ‘opposite’ problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.

Keywords

MINKOWSKI FUNCTIONALSHYPERGEOMETRIC FUNCTIONSLOCAL STEREOLOGY

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 87 - 96
Copyright
Copyright © Applied Probability Trust 1995 

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.)

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

Bonnesen, T. and Fenchel, W. (1934) Theorie der konvexen Körper. Springer-Verlag, Berlin.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1965) Table of Integrals, Series, and Products. Academic Press, New York.Google Scholar
Gundersen, H. J. G. (1988) The nucleator. J. Microsc. 151, 3–21.CrossRefGoogle ScholarPubMed
Jensen, E. B. V. (1994) Rotational versions of the Crofton formula. Research Report 285, Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus.Google Scholar
Jensen, E. B. and Gundersen, H. J. G. (1987) Stereological estimation of surface area of arbitrary particles. In Proceedings of the 7th International Congress for Stereology, Caen 1987. Acta Stereol. 6/III, 25–30.Google Scholar
Jensen, E. B. and Gundersen, H. J. G. (1989) Fundamental stereological formulae based on isotropically oriented probes through fixed points with applications to particle analysis. J. Microsc. 153, 249–267.CrossRefGoogle Scholar
Jensen, E. B. V. and Kieu, K. (1992) A new integral geometric formula of the Blaschke–Petkantschin type. Math. Nachr. 156, 57–74.CrossRefGoogle Scholar
Jensen, E. B. V. and Kieu, K. (1994) Unbiased stereological estimation of d-dimensional volume in ℝn from an isotropic random slice through a fixed point. Adv. Appl. Prob. 26, 1–12.Google Scholar
Jensen, E. B., Kiêu, K. and Gundersen, H. J. G. (1990) Second-order stereology. In Proceedings of the 5th European Congress for Stereology, Freiburg 1989. Acta Stereol. 9/I, 15–35.Google Scholar
Miles, R. E. (1979) Some new integral geometric formulae, with stochastic applications. J. Appl. Prob. 16, 592–606.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its Applications , Vol. 1. Addison-Wesley, Reading, MA.Google Scholar
Schneider, R. (1981) Crofton's formula generalized to projected thick sections. Rend. Circ. Mat. Palermo 30, 157–160.Google Scholar
Schneider, R. (1993) Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications , Vol. 44, Cambridge University Press.Google Scholar
Zähle, M. (1982) Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, 49–72.Google Scholar
Zähle, M. (1990) A kinematic formula and moment measures of random sets. Math. Nachr. 149, 325–340.CrossRefGoogle Scholar