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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: [email protected]
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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 

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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