  The Radon transform Rf of functions f on SO(3) has recently been applied
extensively in texture analysis, i.e. the analysis of preferred
crystallographic orientation. In practice one has to determine the orientation
probability density function f \in L2(SO(3)) from Rf \in L2(S2\times S2) which
is known only on a discrete set of points. Since one has only partial
information about Rf the inversion of the Radon transform becomes an ill-posed
inverse problem. Motivated by this problem we define a new notion of the Radon
transform Rf of functions f on general compact Lie groups and introduce two
approximate inversion algorithms which utilize our previously developed
generalized variational splines on manifolds. Our new algorithms fit very well
to the application of Radon transform on SO(3) to texture analysis.
