Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T17:34:30.152Z Has data issue: false hasContentIssue false

Second-order stereology for planar fibre processes

Published online by Cambridge University Press:  01 July 2016

V. Weiss
Affiliation:
Friedrich-Schiller-Universität, Jena
W. Nagel*
Affiliation:
Friedrich-Schiller-Universität, Jena
*
* Postal address for both authors: Friedrich-Schiller-Universität, Fakultät für Mathematik und Informatik, 07740 Jena, Germany.

Abstract

Three different stereological methods for the determination of second-order quantities of planar fibre processes which have been suggested in the literature are considered. Proofs of the formulae are given (also by using a new integral geometric formula), relations between the methods are derived and the prerequisites are discussed. Furthermore, edge-corrected unbiased estimators for the second-order quantities are given.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1994 

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

[1] Ambartzumian, R. V. (1973) On random fields of segments and random mosaics in the plane. Teor. Veroyatn. Prim. 18, 515526; corrections 19, 600.Google Scholar
[2] Ambartzumian, R. V. (1981) Stereology of random planar segment processes. Rend. Sem. Mat. Torino 39, 147159.Google Scholar
[3] Ambartzumian, R. V. (1982) Combinatorial Integral Geometry. Wiley, Chichester.Google Scholar
[4] Ambartzumian, R. V. and Oganian, V. K. (1975) Homogeneous and isotropic fibre fields in the plane. Izv. AN Armen. SSR Ser. Math. 10, 509528.Google Scholar
[5] Hanisch, K.-H. (1985) On the second-order analysis of stationary and isotropic planar fibre processes by a line intercept method. In Geobild '85, Wiss. Beiträge der FSU Jena, 141146.Google Scholar
[6] Jensen, E. B., KiêU, K. and Gundersen, H. J. G. (1990) Second-order stereology. Acta Stereol. 9, 1535.Google Scholar
[7] Mecke, J. (1981) Formulas for stationary planar fibre processes III–Intersections with fibre systems. Math. Operationsf. Statist., Ser. Statist. 12, 201210.Google Scholar
[8] Mecke, J. and Stoyan, D. (1980) Formulas for stationary planar fibre processes–General theory. Math. Operationsf. Statist., Ser. Statist. 11, 267279.Google Scholar
[9] Nagel, W. (1987) An application of Crofton's formula to moment measures of random curved measures. Forschungserg. der FSU Jena N/87/12.Google Scholar
[10] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
[11] Schwandtke, A. (1988) Second-order quantities for stationary weighted fibre processes. Math. Nachr. 139, 321334.CrossRefGoogle Scholar
[12] Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Akademie-Verlag, Berlin.Google Scholar
[13] Zähle, M. (1982) Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, 4972.CrossRefGoogle Scholar
[14] Zähle, M. (1990) A kinematic formula and moment measures of random sets. Math. Nachr. 149, 325340.CrossRefGoogle Scholar