Abstract
Spherical geometry of quaternions is employed to characterize the Bingham distribution on the 3-dimensional sphere \({\mathbb{S}}^3 \subset {\mathbb{R}}^4 \) as being uniquely composed of a bipolar, a circular and a spherical component. A new parametrization of its dispersion parameters provides a classification of patterns of crystallographic preferred orientations (CPO, or textures). It is shown that the Bingham distribution can represent most types of ideal CPO patterns; in particular single component, fiber and surface textures are represented by rotationally invariant bipolar, circular and spherical distributions, respectively. Pole figures of given crystal directions are derived by the spherical Radon transform of the Bingham probability density function of rotations, which are displayed for general and special cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Altmann, S. L., 1986, Rotations, quaternions, and double groups: Oxford University Press, Oxford, 317 p.
Bingham, C., 1964, Distributions on the sphere and on the projective plane: Unpublished PhD Thesis, Yale University, 80 p.
Bingham, C., 1974, An antipodally symmetric distribution on the sphere: Ann. Statist., v. 2, no. 6, p. 1201–1225.
Boogaart, K. G. v. d., 2002, Statistics for individual crystallographic orientation measurements: Dissertation TU Freiberg, Shaker-Verlag, Aachen, 159 p.
Boogaart, K. G. v. d., and Schaeben, H., 2002a, Kriging of regionalized directions, axes, and orienta-tions: I. Directions and axes: Math. Geol., v. 34, no. 5, p. 479–503.
Boogaart, K. G. v. d., and Schaeben, H., 2002b, Kriging of regionalized directions, axes, and orienta-tions: II. Orientations: Math. Geol., v. 34, no. 6, p. 671–677.
Bunge, H. J., 1982, Texture analysis in materials science: Mathematical methods: Butterworths, London, p. 193.
Cerejeiras, P., Schaeben, H., and Sommen, F., 2002, The spherical X-ray transform: Math. Methods Appl. Sci., v. 25, no. 16-18, p. 1493–1507.
Eschner, T., 1994, Generalized model function for quantitative texture analysis, in Bunge, H. J., Siegesmund, S., Skrotzki, W., and Weber, K., eds., Textures of geological materials: DGM Infor-mationsgesellschaft, Oberursel, p. 15–28.
Gürlebeck, K., and Sprößig, W., 1997, Quaternionic and Clifford calculus for physicists and engineers: Wiley, Chichester, 371 p.
Grewen, J., and Wassermann, G., 1955, Über die idealen Orientierungen einer Walztextur: Acta Metall., v. 3, no. 4, p. 354–360.
Helgason, S., 1959, Differential operators on homogeneous spaces: Acta Math., v. 102, no. 3-4, p. 239–299.
Helgason, S., 1999, The Radon transform: Birkhäuser, Boston, 188 p.
Kuipers, J. B., 1999, Quaternions and rotation sequences: A primer with applications to orbits, aerospace, and virtual reality: Princeton University Press, Princeton, 371 p.
Kunze, K., 1991, Zur quantitativen Texturanalyse von Gesteinen: Bestimmung, Interpretation und Simulation von Quarzteilgefügen: Dissertation RWTH Aachen, Veröff. Zentralinstitut Physik der Erde Nr. 117, Potsdam, 136 p.
Matthies, S., 1980, Standard functions in the texture analysis: Phys. Status Solidi B, v. 101, no. 2, p. K111–K115.
Matthies, S., Vinel, G. W., and Helming, K., 1987, Standard distributions in texture analysis, vol. I: Akademie Verlag, Berlin, 442 p.
Matthies, S., Helming, K., Steinkopff, T., and Kunze, K., 1988, Standard distributions for the case of fibre textures: Phys. Status Solidi B, v. 150, no. 1, p. K1–K5.
Meister, L., and Schaeben, H., 2004, A concise quaternion geometry of rotations: Math. Methods Appl. Sci. (in press).
Onstott, T. C., 1980, Application of the Bingham distribution function to paleomagnetic studies: J. Geo-phys. Res., v. 85, no. B3, p. 1500–1510.
Prentice, M. J., 1980, Orientation statistics without parametric assumptions: J. R. Stat. Soc., v. B48, no. 2, p. 214–222.
Schaeben, H., 1981, Mathematische Methoden der Analyse von Richtungsgefügen: Unpublished Dis-sertation, RWTH Aachen, 252 p.
Schaeben, H., 1996, Texture approximation or texture modelling with components represented by the von Mises-Fisher matrix distribution on SO(3) and the Bingham distribution on S 4+ : J. Appl. Cryst., v. 29, no. 5, p. 516–525.
Watson, G. S., 1983, Statistics on spheres: Wiley, New York, 238 p.
Wenk, H.-R. (ed.), 1985, Preferred orientation in deformed metals and rocks: An introduction to modern texture analysis: Academic Press, Orlando, 610 p.
Wood, A. T. A., 1988, Some notes on the Fisher-Bingham family on the sphere: Comm. Stat. Theory Methods, v. 17, no. 11, p. 3881–3897.
Wood, A. T. A., 1993, Estimation of the concentration parameters of the Fisher matrix distribution on SO(3) and the Bingham distribution on S q ,q≥2: Austral. J. Statist., v. 35, no. 1, p. 69–79.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kunze, K., Schaeben, H. The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis. Mathematical Geology 36, 917–943 (2004). https://doi.org/10.1023/B:MATG.0000048799.56445.59
Issue Date:
DOI: https://doi.org/10.1023/B:MATG.0000048799.56445.59