The Generalized Spherical Radon Transform and Its Application in Texture Analysis

TL;DR

This paper introduces a generalized spherical Radon transform for analyzing crystallographic orientations, providing a new method to recover functions from angle distribution data and clarifying related kernel-based techniques.

Contribution

It develops a novel generalized Radon transform on the 3-sphere, linking geometric and group features to improve texture analysis and function recovery methods.

Findings

Provides a new approach to recover functions from angle distribution data.

Clarifies the geometry behind kernel and radial basis function methods.

Enhances understanding of the relationship between spherical transforms and crystallographic orientation analysis.

Abstract

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.