Advances in Applied Probability

Estimation of the mean normal measure from flat sections

Markus Kiderlen

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We discuss the determination of the mean normal measure of a stationary random set ZRd 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 kk = 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.

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Adv. in Appl. Probab. Volume 40, Number 1 (2008), 31-48.

First available in Project Euclid: 16 April 2008

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

Random set anisotropy oriented mean normal measure rose of normal directions spherical projection vertical sections verification


Kiderlen, Markus. Estimation of the mean normal measure from flat sections. Adv. in Appl. Probab. 40 (2008), no. 1, 31--48. doi:10.1239/aap/1208358885.

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