Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T10:09:35.687Z Has data issue: false hasContentIssue false
  • English
  • Français

On estimation of the Euler number by projections of thin slabs

Part of: Topological manifolds Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

J. Rataj
J. Rataj*
Affiliation:
Charles University, Prague
*
∗ Postal address: Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. Email address: rataj@karlin.mff.cuni.cz
Get access Rights & Permissions [Opens in a new window]

Abstract

A stereological formula for the Euler number involving projections of the set in thin parallel slabs is considered. Sufficient conditions for the validity of this formula are derived.

Keywords

Euler-Poincaré characteristicmorphological closureconvex ringdeformation retract

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 57N15: Topology of $E^n$, $n$-manifolds ($4 less n less infty$)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 715 - 724
Copyright
Copyright © Applied Probability Trust 2004 

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

Supported by the Grant Agency of the Czech Republic, project 201/03/0946, and by MSM 113200007.

References

[1] Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, New York.Google Scholar
[2] Aubin, J.-P. and Frankowska, H. (1990). Set-Valued Analysis. Birkhäuser, Boston, MA.Google Scholar
[3] Dold, A. (1972). Lectures on Algebraic Topology. Springer, Berlin.CrossRefGoogle Scholar
[4] Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93, 418–491.CrossRefGoogle Scholar
[5] Fu, J. H. G. (1989). Curvature measures and generalized Morse theory. J. Differential Geometry 30, 619–642.Google Scholar
[6] Hadwiger, H. (1957). Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin.Google Scholar
[7] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
[8] Nagel, W., Ohser, J. and Pischang, K. (2000). An integral-geometric approach for the Euler–Poincaré characteristic of spatial images. J. Microscopy 198, 54–62.Google Scholar
[9] Ohser, J. and Nagel, W. (1996). The estimation of the Euler–Poincaré characteristic from observations on parallel sections. J. Microscopy 184, 117–126.CrossRefGoogle Scholar
[10] Rataj, J. (2002). Determination of spherical area measures by means of dilation volumes. Math. Nachr. 235, 143–162.Google Scholar
[11] Rataj, J. and Zähle, M. (2001). Curvatures and currents for unions of sets with positive reach. II. Ann. Global Anal. Geometry 20, 1–21.Google Scholar
[12] Rataj, J. and Zähle, M. (2003). Normal cycles of Lipschitz manifolds by approximation with parallel sets. Differential Geometry Appl. 19, 113–126.Google Scholar
[13] Sterio, D. G. (1984). The unbiased estimation of numbers and sizes of arbitrary particles using the disector. J. Microscopy 134, 127–136.Google Scholar
[14] Stoyan, D. (1990). Stereology and stochastic geometry. Internat. Statist. Rev. 58, 227–242.Google Scholar
[15] Worsley, K. J. (1995). Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. Appl. Prob. 27, 943–959.CrossRefGoogle Scholar
[16] Worsley, K. J. (1995). Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist. 23, 640–669.CrossRefGoogle Scholar