A discrete spherical x-ray transform of orientation distribution functions using bounding cubes

Abstract

We investigate a cubed sphere parametrization of orientation space with the aim of constructing a discrete voxelized version of the spherical x-ray transform. For tracing the propagation of a unit great circle through the partition subsets, the frustums of the cubed sphere, a fast procedure is proposed. The circle's parts in each frustum are gnomonically mapped into line segments inside the bounding cubes. The line segments constitute a convex polygon with vertexes indicating frustum exit–entry points. Thus the problem of system matrix calculation is reduced to the tracing of line segments within rectangular voxel arrays partitioning the bounding cubes. Hence algebraic reconstruction techniques can be used in a comprehensive way for orientation distribution function estimation from diffraction data.