The Generalized Spherical Radon Transform and Its Application in Texture Analysis

The generalized spherical Radon transform associates the mean values over spherical tori to a function f defined on S ⊂ H, where the elements of S are considered as quaternions representing rotations. It is introduced into the analysis of crystallographic preferred orientation and identified with the probability density function corresponding to the angle distribution function W . Eventually, this communication suggests a new approach to recover an approximation of f from data sampling W . At the same time it provides additional clarification of a recently suggested method applying reproducing kernels and radial basis functions by instructive insight in its involved geometry. The focus is on the correspondence of geometrical and group features but not on the mapping of functions and their spaces. 1 Motivation and Introduction In texture analysis, i.e. the analysis of preferred crystallographic orientation, the orientation probability density function f representing the probability law of random orientations of crystal grains by volume is a major issue. In x–ray or neutron diffraction experiments spherical intensity distributions are measured which can be interpreted in terms of spherical probability distributions of distinghuished crystallographic axes. In texture analysis, they are referred to as pole probability density functions. Generally, if f is the orientation probability density function