Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T16:02:13.040Z Has data issue: false hasContentIssue false
  • English
  • Français

On second-order formulas in anisotropic stereology

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  01 July 2016

Viktor Beneš
Viktor Beneš*
Affiliation:
Czech Technical University
*
* Postal address: Department of Mathemaics, FSI, Czech Technical University, Karlovo Nám. 13, 12135 Prague 2, Czech Republic.
Get access Rights & Permissions [Opens in a new window]

Abstract

Formulas for anisotropic stereology of fibre and surface processes are presented. They concern the relation between second-order quantities of the original process and its projections and sections. Various mathematical tools for handling these formulas are presented, including stochastic optimization. Finally applications in stereology are discussed, relating to intensity estimators using anisotropic sampling designs. Variances of these estimators are expressed and evaluated for processes with the Poisson property.

Keywords

ANISOTROPIC SAMPLINGBUFFON TRANSFORMESTIMATION VARIANCELENGTH INTENSITYPAIR CORRELATION FUNCTIONPROJECTION MEASURE

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 326 - 343
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

Baddeley, A. (1985) An anisotropic sampling design. In Geobild'85, ed. Nagel, W., pp. 92–97. Friedrich-Schiller Universitat Jena.Google Scholar
Baddeley, A. and Cruz-Orive, L. M. (1995) The Rao-Blackwell theorem in stereology and some counterexamples. Adv. Appl. Prob. 27, 2–19.Google Scholar
Beneš, V. (1993) On anisotropic sampling in stereology. Proc. 6ECS Prague, Acta Stereol. 12, 185–190.Google Scholar
Beneš, V., Chadoeuf, J. and Ohser, J. (1994) On some characteristics of anisotropic fibre processes. Math. Nachr. 169, 5–17.Google Scholar
Chadoeuf, J. and Beneš, V. (1994) On some estimation variances in spatial statistics. Proc. 5th Prague Symposium on Asymptotic Statistics, Kybernetika 30, 245–262.Google Scholar
Cruz-Orive, L. M., Hoppeler, H., Mathieu, O. and Weibel, E. R. (1985) Stereological analysis of anisotropic structures using directional statistics. J. R. Statist. Soc. C 34, 14–32.Google Scholar
Hilliard, J. E. (1967) Determination of structural anisotropy. In Stereology, ed. Elias, H., pp. 219–227. Springer-Verlag, New York.CrossRefGoogle Scholar
Kemperman, J. H. B. (1968) The general moment problem, a geometric approach. Ann. Math. Statist. 39, 93–122.CrossRefGoogle Scholar
Lantuejoul, C. (1988) Some stereological and statistical consequences derived from Cartier's formula. J. Microscopy 151, 265–276.Google Scholar
Miles, R. (1974) Synopsis of ‘Poisson flats in Euclidean spaces’. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G., pp. 202–228. Wiley, New York.Google Scholar
Ohser, J. (1991) Variances of different estimators for the specific line length. Stoch. Geometry, Geom. Statistics, Stereology, Oberwolfach. Unpublished.Google Scholar
Pohlmann, S., Mecke, J. and Stoyan, D. (1981) Stereological formulas for stationary surface processes. Math. Operatforsch. Statist. 12, 429–440.Google Scholar
Rataj, J. (1994) Estimation of oriented direction distribution of a planar body. Friedrich-Schiller Univ. Jena, Preprint Math/94/1.Google Scholar
Sandau, K. and Hahn, U. (1994) Some remarks on the accuracy of surface area estimation using the spatial grid. J. Microscopy 173, 67–72.Google Scholar
Schwandtke, A. (1988) Second-order quantities for stationary weighted fibre processes. Math. Nachr. 139, 321–334.Google Scholar
Stoyan, D. and Ohser, J. (1984) Cross-correlation measure of weighted random measures and their estimation (in Russian). Teor. Verojatnost. Primen. 29, 338–347.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and Its Applications. Akademie-Verlag, Berlin.Google Scholar
Vedel Jensen, E. B. and Kieu, K. (1992) A note on recent research in second-order stereology. Acta Stereol. 11, 569–579.Google Scholar
Weil, W. (1993) The determination of shape and mean shape from sections and projections. Acta Stereol. 12(2), 73–84.Google Scholar
Zähle, M. (1990) A kinematic formula and moment measures of random sets. Math. Nachr. 149, 325–340.Google Scholar