Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-29T19:35:24.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogeneous line-segment processes

Published online by Cambridge University Press:  24 October 2008

Richard Cowan
Richard Cowan
Affiliation:
CSIRO, P.O. Box 218, Lindfield 2070, N.S.W., Australia
Get access Rights & Permissions [Opens in a new window]

Summary

A number of formulae of an integral geometric character are presented for the most general planar, homogeneous line-segment process, one in which a given segment length and orientation may depend upon (a) the point process of segment midpoints and (b) the lengths and orientations of other segments. The sense in which these formulae have a probabilistic/statistical interpretation is made precise. For the general process, two interpretations are given; one requires the theory of Palm distributions whilst the other depends upon ergodic results. When additional structure for the process is assumed, the integral geometric formulae lead to interesting, non-intuitive sampling formulae.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 3 , November 1979 , pp. 481 - 489
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Ambartsumian, R. V.Random fields of segments and random mosaics on a plane. Proc. Sixth Symp. Math. Statist. Prob. 3 (1970), 369–381.Google Scholar
(2)Ambartsumian, R. V. Convex polygons and random tesselations. Stochastic geometry, ed. Harding, and Kendall, (London, Wiley, 1974).Google Scholar
(3)Coleman, R.Sampling procedures for the lengths of random straight lines. Biometrika 59 (1972), 415–426.CrossRefGoogle Scholar
(4)Coleman, R. The distance from a given point to the nearest end of one member of a random process of linear segments. Stochastic geometry, ed. Harding, and Kendall, (London, Wiley, 1974).Google Scholar
(5)Cowan, R. Properties of ergodic random mosaic processes. (To appear Mathematische Nachrichten.)Google Scholar
(6)Cowan, R.The use of the ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10 (1978), 47–57.CrossRefGoogle Scholar
(7)Halmos, P.Measure theory (Princeton, N.J., Van Nostrand, 1956).Google Scholar
(8)Leadbetter, M. R.On basic results of point process theory. Proc. Sixth Symp. Math. Statist. Prob. 3 (1970), 449–462.Google Scholar
(9)Parker, P. and Cowan, R.ome properties of line-segment processes. J. Appl. Prob. 13 (1976), 96–107.CrossRefGoogle Scholar
(10)Santaló, L. A.Random processes of linear segments and graphs. Proc. Buffon. bi-centenary conf. Paris. (1977).Google Scholar