The generalized totally geodesic Radon transform and its application to texture analysis

The generalized totally geodesic Radon transform associates the mean values over spherical tori to a function f defined on ??³??, where the elements of ??³ 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 into its involved geometry. The focus is on the correspondence of geometrical and group features rather than on the mapping of functions and their spaces. Copyright © 2008 John Wiley & Sons, Ltd.