Hostname: page-component-6d856f89d9-gndc8 Total loading time: 0 Render date: 2024-07-16T08:06:50.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Unbiased Stereological Estimation of d-Dimensional Volume in ℝn from an Isotropic Random Slice Through a Fixed Point

Part of: Global differential geometry Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

E. B. Vedel Jensen  and
K. Kiêu
E. B. Vedel Jensen*
Affiliation:
University of Aarhus
K. Kiêu*
Affiliation:
Institut National de la Recherche Agronomique
*
* Postal address: Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
** Postal address: Laboratoire de Biométrie, Institut National de la Recherche Agronomique, Route de Saint-Cyr, F-78000 Versailles, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

Unbiased stereological estimators of d-dimensional volume in ℝn are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝn through O hits a fixed point in ℝn is given.

Keywords

CROFTON'S FORMULAINTEGRAL GEOMETRYISOTROPIC SUBSPACESSLICESSTEREOLOGY

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 53C65: Integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 26 , Issue 1 , March 1994 , pp. 1 - 12
Copyright
Copyright © Applied Probability Trust 1994 

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

Evans, S. M. and Gundersen, H. J. G. (1989) Estimation of spatial distributions using the nucleator. Acta Stereol. 8, 395–400.Google Scholar
Gundersen, H. J. G. (1988) The nucleator. J. Microsc. 151, 3–21.Google Scholar
Gundersen, H. J. G., Bagger, P., Bendtsen, T. F., Evans, S. M., Korbo, L., Marcussen, N., Møller, A., Nielsen, K., Nyengaard, J. R., Pakkenberg, B., Sørensen, F. B., Vesterby, A. and West, M. J. (1988) The new stereological tool: Disector, fractionator, nucleator and point sampled intercepts and their use in pathological research and diagnosis. APMIS 96, 857–881.Google Scholar
Jensen, E. B. and Gundersen, H. J. G. (1989) Fundamental stereological formulae based on isotropically orientated probes through fixed points with applications to particle analysis. J. Microsc. 153, 249–267.Google Scholar
Jensen, E. B. V. and Kieu, K. (1992) Unbiased stereological estimation of d-dimensional volume in n from an isotropic random slice through a fixed point. Research Report No. 257, Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, Denmark.Google Scholar
Jensen, E. B., Kieu, K. and Gundersen, H. J. G. (1990) Second-order stereology. Acta Stereol. 9, 15–35.Google Scholar
Kieu, K. and Jensen, E. B. V. (1993) Stereological estimation based on isotropic slices through fixed points. J. Microsc. 170, 45–51.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its Applications, Volume 1. Addison-Wesley, Reading, MA.Google Scholar
Zähle, M. (1982) Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, 49–72.Google Scholar