Towards statistics of crystal orientations in quantitative texture anaylsis

Abstract

By the application of Rodrigues parameters, crystal orientations are represented as points on the unit semi-hypersphere $4+ c R 4 or equivalently on the projective hyperplane H3c R 4. For the statistical analysis of orientation data, probability models on $4+-H 3 are required. Among different hyperspherical analogs of the normal distribution in Euclidean space, corresponding to various of its characterizations, the Bingham model distribution is" characterized as the hyperspherical analog for statistical purposes. Maximum-likelihood estimates of its parameter matrices and single-sample significance tests of uniformity and rotational symmetry are presented. In terms of probability theory and stochastic processes, the hyperspherical Browniandiffusion distribution is the analog of the normal distribution in Euclidean space. An orientation distribution derived by Savelova lind. Lab. (USSR), (1984), 50, 468-474] and erroneously related to a central-limit-type theorem for rotations by Matthies, Muller & Vinel [Textures Mierostruct. (1988), 10, 77-96], with the Brownian-diffusion distribution on $4+ -H 3, recalls the fact that no simple analog of the central limit theorem in Euclidean space exists for arbitrary spaces and manifolds, e.g. for S p, SP+, SO(p), p > 2. Therefore, interpretations of preferred orientations in terms of mechanisms of texture development related to an imaginary simple analog of the central limit theorem would generally be misleading. Individual crystal-orientation measurements may initially require different nonparametric methods of analysis. As a counterpart of pole-to-orientation density inversion of diffraction data, hyperspherical kernel orientation density estimation is suggested for individual orientation measurements. With the application of non-negative kernels, the estimated orientation density is always non-negative. The critical smoothing parameter of this method is determined on information-theoretical grounds. For patterns of preferred orientation too complex to be sufficiently approximated by the parametric Bingham model distribution, a clustering of individual orientation 0021-8898/93/010112-10506.00 measurements into disjunct classes is suggested that relates similarity of orientations to multimodality of the estimated orientation density; each class of orientations may then be individually analyzed by parametric methods.

Orientation distribution analysis in deformed grains

“…The importance of developing statistical analyses of individual orientation data was initially recognized by Schaeben (1993). The ®rst step in this statistical analysis is usually to determine the average orientation; hence averaging methods for a set of near-orientations were proposed by Krieger Lassen et al (1994) and Humbert et al (1996).…”

geomstats: a Python Package for Riemannian Geometry in Machine Learning

“…In biology, the sphere S 2 is used in analysis of protein structures [23]. In physics, the semi-hypersphere S 4 + is used to encode the projective space P 4 for representing crystal orientations in applied crystallography [36].…”

Inferential statistics of electron backscatter diffraction data from within individual crystalline grains

“…with a random q 2 S 3 , with a symmetric ð4 Â 4Þ matrix A, and with the hypergeometric function 1 F 1 ð1=2; 2; Þ of matrix argument seems appropriate (Bingham, 1964(Bingham, , 1974Schaeben, 1993;Kunze & Schaeben, 2004, 2005. It is emphasized that the densities f ðAEq; AÞ and f ðAEq; A þ tIÞ, with t 2 R and the ð4 Â 4Þ identity matrix I, define the same distribution.…”