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Estimation of mean particle volume using the set covariance function

Part of: Sampling theory, sample surveys General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Annoesjka Cabo  and
Adrian Baddeley
Annoesjka Cabo*
Affiliation:
University of Western Australia
Adrian Baddeley*
Affiliation:
University of Western Australia
*
∗ Postal address: Nijendal 5, 3972 KC Driebergen, The Netherlands.
∗∗ Postal address: Department of Mathematics and Statistics, University of Western Australia, Nedlands WA 6009, Australia. Email address: [email protected]
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Abstract

Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard stereological technique is to place test lines in the plane section, and measure cubed intercept lengths with the two-dimensional particle profiles. This paper discusses more efficient estimators obtained by integrating over all possible placements of the test line. We prove that these estimators have smaller variance than the line transect estimators, and indeed are related to them by the Rao-Blackwell process. In the improved estimators, the cubed intercept length is replaced by a moment of the distance between two points in the section profile. This can be computed as a moment of the set covariance function, which in turn is computable using the fast Fourier transform. We also derive an isoperimetric-type inequality between the improved estimator and the area-weighted 3/2th moment of the profile areas. Finally, we present two practical applications to particles of silicon carbide and to synaptic boutons in brain tissue. We estimate the variance of the technique and the gain in efficiency over line transect techniques; the efficiency improvement appears to be as much as one order of magnitude.

Keywords

4-lincCampbell-Mecke formulachord lengthsdesign-based stereologyfast Fourier transformintegral geometryinterpoint distancesisoperimetric inequalityline transectslog-normal distributionmodel-based stereologyPalm distributionpoint-sampled interceptsRao-Blackwell processstereologyvertical sectionsvolume-weighted mean volumeweighted distributions

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry 62D05: Sampling theory, sample surveys
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 27 - 46
Copyright
Copyright © Applied Probability Trust 2003 

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