Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-20T03:33:34.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Stereological versions of integral geometric formulae for n-dimensional ellipsoids

Published online by Cambridge University Press:  24 August 2016

E. B. Jensen  and
J. Møller
E. B. Jensen*
Affiliation:
University of Aarhus
J. Møller*
Affiliation:
University of Aarhus
*
∗Postal address: Department of Theoretical Statistics, Institute of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗Postal address: Department of Theoretical Statistics, Institute of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark.
Get access Rights & Permissions [Opens in a new window]

Abstract

Recently, unbiased stereological estimators of moments of particle volume, based on measurements on lower-dimensional sections through the particles, have been developed by Jensen and Gundersen (1985). In this note, we derive an explicit form of these unbiased estimators valid for particles in Rn of ellipsoidal shape, and we establish a close relationship between the estimators and a known integral geometric formula for ellipsoids, due to Furstenberg and Tzkoni (1971). Furthermore, a stereological version of another integral geometric formula for n-dimensional ellipsoids, due to Guggenheimer (1973), is derived.

Keywords

GEOMETRIC PROBABILITYISOTROPIC SUBSPACESPARTICLE VOLUMEPOINT-SAMPLINGSTEREOLOGYUNIFORM RANDOM POINTS
Type
Short Communications
Information
Journal of Applied Probability , Volume 23 , Issue 4 , December 1986 , pp. 1031 - 1037
Copyright
Copyright © Applied Probability Trust 1986 

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.)

References

Blaschke, W. (1935a) Integralgeometrie 1. Ermittlung der Dichten für lineare Unterräume im En. Actualités Sci. Indust. 252, 1–22.Google Scholar
Blaschke, W. (1935b) Integralgeometrie 2. Zu Ergebnissen von M. W. Crofton. Bull. Math. Soc. Roumaine Sci. 37, 3–11.Google Scholar
Cruz-Orive, E. M. (1985) Estimating volumes from systematic hyperplane sections. I. Appl. Prob. 22, 518–530.Google Scholar
Furstenberg, H. and Tzkoni, I. (1971) Spherical functions and integral geometry. Israel I. Math. 10, 327–338.CrossRefGoogle Scholar
Guggenheimer, H. (1973) A formula of Furstenberg–Tzkoni type. Israel J. Math. 14, 281–282.CrossRefGoogle Scholar
Jensen, E. B. and Gundersen, H. J. G. (1985) The stereological estimation of moments of particle volume. J. Appl. Prob. 22, 82–98.Google Scholar
Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
Miles, R. E. (1973) A simple derivation of a formula of Furstenberg and Tzkoni. Israel J. Math. 14, 278–280.CrossRefGoogle Scholar
Miles, R. E. (1985) A comprehensive set of stereological formulae for embedded aggregates of not-necessarily-convex particles. I. Microsc. 138, 115–125.Google Scholar
Petkantschin, B. (1936) Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raum. Abh. Math. Sem. Univ. Hamburg 11, 249–310.CrossRefGoogle Scholar