  The generalized spherical Radon transform associates the mean values over
spherical tori to a function $f$ defined on $\mathbb{S}^3 \subset \mathbb{H}$,
where the elements of $\mathbb{S}^3$ 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 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.
