The cone-beam transform and spherical convolution operators

TL;DR

This paper provides a singular value decomposition of the generalized Funk-Radon transform and the cone-beam transform with spherical sources, advancing mathematical understanding of these integral operators in 3D imaging.

Contribution

It introduces a singular value decomposition for the generalized Funk-Radon transform and the cone-beam transform with sources on the sphere, generalizing previous results.

Findings

Singular value decomposition of the generalized Funk-Radon transform.

Singular value decomposition of the cone-beam transform with spherical sources.

Extension of Kazantsev's results to more general settings.

Abstract

The cone-beam transform consists of integrating a function defined on the three-dimensional space along every ray that starts on a certain scanning set. Based on Grangeat's formula, Louis [2016, Inverse Problems 32 115005] states reconstruction formulas based on a new generalized Funk-Radon transform on the sphere. In this article, we give a singular value decomposition of this generalized Funk-Radon transform. We use this result to derive a singular value decomposition of the cone-beam transform with sources on the sphere thus generalizing a result of Kazantsev [2015, J. Inverse Ill-Posed Probl. 23(2):173-185].