Previous |  Up |  Next

Article

Keywords:
planar anisotropy; planar fibre system; Steiner compact analysis
Summary:
The paper concerns estimation of anisotropy of planar fibre systems using the relation between the rose of directions and the rose of intersections. The discussion about the properties of the Steiner compact estimator is based on both theoretical and simulation results. The approach based on the distribution of the Prokhorov distance between the estimated and true rose of directions is developed. Finally the curved test systems are investigated in both Fourier and Steiner compact analysis of anisotropy.
References:
[1] Baddeley A.: An anisotropic sampling design. In: Geobild’85 (W. Nagel, ed.). FSU Jena 1985, pp. 92–97
[2] Digabel H.: Determination practique de la rose des directions. In: 15 fascicules de morphologie mathematique appliquee (6), Fontainebleau 1975
[3] Heyer H.: Probability Measures on Locally Compact Groups. Springer, Berlin 1977 MR 0501241 | Zbl 0528.60010
[4] Hilliard J. E.: Specification and measurement of microstructural anisotropy. Trans. Metall. Soc. AIME 224 (1962), 1201–1211
[5] Kanatani K. I.: Stereological determination of structural anisotropy. Internat. J. Engrg. Sci. 22 (1984), 531–546 DOI 10.1016/0020-7225(84)90055-7 | MR 0750003 | Zbl 0564.73014
[6] Kufner A., Kadlec J.: Fourier Series. Academia, Praha 1971 MR 0393989 | Zbl 0215.17901
[7] Matheron G.: Random Sets and Integral Geometry. Wiley, New York 1975 MR 0385969 | Zbl 0321.60009
[8] Mecke J.: Formulas for stationary planar fibre processes III-intersections with fibre systems. Math. Oper. Statist., Ser. Statist. 12 (1981), 201–210 MR 0618605 | Zbl 0472.60016
[9] Philofski E. M., Hilliard J. E.: On the measurement of the orientation distribution of lineal and areal arrays. Trans. ASM 27 (1967), 1, 79–86
[10] Rachev S. T.: Probability Metrics. Wiley, New York 1991 MR 1105086 | Zbl 1178.91046
[11] Rataj J., Saxl I.: Analysis of planar anisotropy by means of the Steiner compact. J. Appl. Probab. 26 (1989), 490–502 DOI 10.2307/3214407 | MR 1010938 | Zbl 0694.60010
[12] Rychlik T.: Order statistics of variables with given marginal distributions. In: Distributions with Fixed Marginals and Related Topics (L. Rüschendorf, B. Schweizer and M. D. Taylor, eds.), IMS Lecture Notes – Monograph Series 28 (1996), pp. 297–306 MR 1485539
[13] Schneider R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia Math. Appl. 44 (1993) MR 1216521 | Zbl 0798.52001
[14] Stoyan S., Kendall W. S., Mecke J.: Stochastic Geometry and Its Applications. Second edition. Wiley, New York, Chichester 1995 Zbl 1155.60001
Partner of
EuDML logo