The cone-beam transform and spherical convolution operators

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].

References in corpus (1)

• An SVD in Spherical Surface Wave Tomography

Cited by in corpus (5)

• Sliced Optimal Transport on the Sphere
• Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group
• A Frame Decomposition of the Funk-Radon Transform
• Commuting integral and differential operators and the master symmetries of the Korteweg-de Vries equation
• Sampling theorems for inverse problems on Riemannian manifolds