Multi-Scale Texture Modeling
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Multi-Scale Texture Modeling Ralf Hielscher · Helmut Schaeben
Received: 13 July 2006 / Accepted: 17 August 2007 / Published online: 15 December 2007 © International Association for Mathematical Geology 2007
Abstract Patterns of crystallographic preferred orientation are referred to as texture. The specific subject of texture analysis is the experimental determination and interpretation of the statistical distribution of orientations of crystals within a specimen of polycrystalline material, which could be metals or rocks. The objective is to relate an observed pattern of preferred orientation to its generating processes and vice versa. In geosciences, texture of minerals in rocks is used to infer constraints on their tectono-metamorphic history. Since most physical properties of crystals, such as elastic moduli, the coefficients of thermal expansion, or chemical resistance to etching depends on crystal symmetry and orientation, the presence of texture imparts directional properties to the polycrystalline material. A major issue of mathematical texture analysis is the resolution of the inverse problem to determine a reasonable orientation density function on SO(3) from measured pole intensities on S2 × S2 , which relates to the inverse of the totally geodesic Radon transform. This communication introduces a wavelet approach into mathematical texture analysis. Wavelets on the two-dimensional sphere S2 and on the rotational group SO(3) are discussed, and an algorithms for a wavelet decomposition on both domains following the ideas of Ta-Hsin Li is given. The relationship of these wavelets on both domains with respect to the totally geodesic Radon transform is investigated. In particular, it is shown that the Radon transform of these wavelets on SO(3) are again wavelets on S2 . A novel algorithm for the inversion of experimental pole intensities to an orientation density function based on this relationship is developed.
R. Hielscher () · H. Schaeben Geoscience Mathematics and Informatics, Freiberg University of Mining and Technology, 09596 Freiberg, Germany e-mail: [email protected] H. Schaeben e-mail: [email protected]
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Math Geosci (2008) 40: 63–82
Keywords Crystallographic preferred orientation · Texture · Orientation probability density function (ODF) · Pole probability density function (PDF) · Pole figure · Totally geodesic Radon transform · Spherical multi-scale analysis · Localization in space and frequency domain
1 Introduction A crystallographic orientation is a set of crystal symmetrically equivalent rotations between an individual crystal and the specimen. Patterns of crystallographic preferred orientations (‘texture’) of minerals in rocks are commonly used to infer constraints on their tectono-metamorphic history (Wenk 1985). They are usually presented as pole figures (fabric diagrams) P (h, r) of specific crystal lattice planes or directions as they are experimentally accessible by X-ray, neutron or synchrotron diffraction for some crystallographic forms. They represent the
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