Generalized Splines for Radon Transform on Compact Lie Groups with Applications to Crystallography

Abstract

Abstract. 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 ∈ L 2 (SO(3)) from Rf ∈ L 2 (S 2 × S 2 ) 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.

Splines and Wavelets on Geophysically Relevant Manifolds

“…In section 6 results which are similar to the results of section 5 obtained for the group Radon transform on SO(3) (see [5], [6]). …”

“…In this situation the results of section 3 can be summarized in the following statement which was proved in [5]. …”

“…For the proof of the next theorem we refer to [5]. Moreover, it shows that f can be reconstructed as a limit (when m goes to infinity) of functions S 2 m+2 +t (f ) which were constructed by using only the values of the Radon transform Rf (x ν , y ν ).…”

“…The goal of the present study is to describe new constructions and applications of splines and bandlimited and localized frames in a space L 2 (M ), where M is a compact Riemannian manifold. Our article is a summary of some results for compact manifolds that were obtained in [5], [6], [24], [46]- [55]. To the best of our knowledge these are the papers which contain the most general results about splines, frames and Shannon sampling on compact Riemannian manifolds along with applications to Radon-type transforms on manifolds.…”

On Radon transforms on compact Lie groups

“…The Radon transform on SO(3) has been studied because of this application (see e.g. [11,2]). It turns out that SO(3) is the only connected, compact Lie group of rank one with injective Radon transform, whence it appears as a special case in the proof of our main result below.…”