Generalized splines for Radon transform on compact Lie groups with applications to crystallography

TL;DR

This paper introduces generalized splines and approximate inversion algorithms for the Radon transform on compact Lie groups, particularly SO(3), enhancing texture analysis in crystallography by addressing partial and discrete data challenges.

Contribution

It defines a new Radon transform for functions on compact Lie groups and develops two spline-based algorithms for approximate inversion, tailored for texture analysis applications.

Findings

Algorithms effectively invert Radon transform with partial data

Spline methods improve stability and accuracy in texture analysis

Applicable to discrete and incomplete data scenarios

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 \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.