The Spherical X-Ray Transform of Texture Goniometry

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In crystallography, a probability density function f(g) defined on the group SO(3) of rotations is referred to as orientation density function; the analysis of preferred crystallographic orientation is referred to as texture analysis. In nondestrcutive practical applications it is common practice to measure diffraction pole figures \( \tilde P(h,r) \) of a few crystallographic forms h with a texture goniometer. A pole figure is the result of sampling a spherical probability density function which is defined as tomographic projection of the orientation density function f. In terms of quaternions a pole density function is properly defined as average of f along circles and therefore referred to as the spherical X-ray transform of texture goniometry. Properties of this spherical X-ray transform are presented, including a range theorem, and open problems are addressed.