Line transects, covariance functions and set convergence

A.J. Cabo, A. J. Baddeley

Research output: Contribution to journalArticleScientific


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 P. Waksman [Adv. Appl. Math. 8, No. 1, 38-52 (1987; Zbl 0638.60016)] 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.
Original languageEnglish
Pages (from-to)585-605
JournalAdvances in Applied Probability
Issue number3
Publication statusPublished - 1995
Externally publishedYes


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