Crystallographic and geodesic Radon transforms on SO(3): motivation, generalization, discretization

TL;DR

This paper explores the crystallographic and geodesic Radon transforms on SO(3), analyzing their properties, applications in texture analysis, and providing a discretization method for exact reconstruction of bandlimited functions.

Contribution

It develops a general framework for Radon transforms on compact Lie groups, analyzes the transforms on SO(3), and introduces a discretization approach for exact function reconstruction.

Findings

Both transforms are relevant in texture analysis.

The crystallographic Radon transform is non-invertible.

An exact reconstruction formula for bandlimited functions is provided.

Abstract

In this paper we consider the so-called crystallographic Radon transform (or crystallographic $X$-ray transform) and totally geodesic Radon transform on the group of rotations SO(3). As we show both of these transforms naturally appear in texture analysis, i.e. the analysis of preferred crystallographic orientation. Although we discuss only applications to texture analysis both transforms have other applications as well. In section 2 we start with motivations and applications. In sections 3 and 4 we develop a general framework on compact Lie groups. In section 5 we give a detailed analysis of the totally geodesic Radon transform on SO(3). In section \ref{relations} we compare crystallographic Radon transform on SO(3) and Funk transform on $S^{3}$. In section \ref{1} we show non-invertibility of the crystallographic transform. In section 8 we describe an exact reconstruction formula for bandlimited functions, which uses only a finite number of samples of their Radon transform. Some auxiliary results for this section are collected in Appendix.