Aarhus University Seal

Estimation of the mean normal measure from flat sections

by Markus Kiderlen
Research Reports Number 497 (August 2007)
We discuss the determination of the mean normal measure of a stationary random set $Z\subset \mathbb{R}^d$ by measurements taken in 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 $k\ge 3$ or $k=2$ 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 by mean normal measures of intersections with almost all $m$-tuples of planes, when $m\ge \lfloor d/k\rfloor+1$. A consistent estimator for the mean normal measure of $Z$, based on stereological measurements in vertical sections, is also presented.
Format available: PDF (486 KB)
Published in Adv. Appl. Prob. 40, 31-48 (2008).
This primarily serves as Thiele Research Reports number 10-2007, but was also published in Research Reports