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Line transects, covariance functions and set convergence

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

A. J. Cabo  and
A. J. Baddeley
A. J. Cabo*
Affiliation:
CWI, Amsterdam
A. J. Baddeley*
Affiliation:
University of Western Australia and University of Leiden
*
* Postal address: Lindenheuvel 16, 1217 JX Hilversum, The Netherlands.
** Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
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Abstract

We define the ‘linear scan transform' G of a set in ℝd using information observable on its one-dimensional linear transects. This transform determines the set covariance function, interpoint distance distribution, and (for convex sets) the chord length distribution. Many basic integral-geometric formulae used in stereology can be expressed as identities for G. We modify a construction of Waksman (1987) to construct a metric η for ‘regular' subsets of ℝd defined as the L1 distance between their linear scan transforms. For convex sets only, η is topologically equivalent to the Hausdorff metric. The set covariance function (of a generally non-convex set) depends continuously on its set argument, with respect to η and the uniform metric on covariance functions.

Keywords

LINEAR SCAN TRANSFORMCHORD LENGTH DISTRIBUTION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 585 - 605
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Menendez Pelayo, Valencia, Spain, on 21–24 September 1993.

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