Advances in Applied Probability
- Adv. in Appl. Probab.
- Volume 40, Number 1 (2008), 31-48.
Estimation of the mean normal measure from flat sections
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 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.
Adv. in Appl. Probab. Volume 40, Number 1 (2008), 31-48.
First available in Project Euclid: 16 April 2008
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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]
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. https://projecteuclid.org/euclid.aap/1208358885