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Estimation of the mean normal measure from flat sections

Part of:Global differential geometryGeneral convexityGeometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Markus Kiderlen
Markus Kiderlen*
Affiliation:
University of Aarhus
*
∗ Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade Build. 1530, DK-8000 Aarhus C, Denmark. Email address: kiderlen@imf.au.dk
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Abstract

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We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝd by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.

Keywords

Random setanisotropyoriented mean normal measurerose of normal directionsspherical projectionvertical sectionsverification

MSC classification

Primary:60D05: Geometric probability and stochastic geometry
Secondary:53C65: Integral geometry52A22: Random convex sets and integral geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008, pp. 31 - 48
Copyright
Copyright © Applied Probability Trust 2008 

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