Efficient tensor tomography in fan-beam coordinates

TL;DR

This paper introduces a comprehensive analysis and efficient inversion method for tensor tomography in fan-beam coordinates on the Euclidean disk, addressing both scalar and tensor cases with novel range characterizations and numerical validation.

Contribution

It presents a new range characterization for the Radon transform in fan-beam coordinates and an efficient reconstruction method for tensors, including non-solenoidal representatives.

Findings

Fast inversion of scalar Radon transform using moment conditions

Effective reconstruction of tensors with non-solenoidal representatives

Numerical examples demonstrating the method's accuracy and efficiency

Abstract

We propose a thorough analysis of the tensor tomography problem on the Euclidean unit disk parameterized in fan-beam coordinates. This includes, for the inversion of the Radon transform over functions, using another range characterization first appearing in [Pestov-Uhlmann, IMRN 2004] to enforce in a fast way classical moment conditions at all orders. When considering direction-dependent integrands (e.g., tensors), a problem where injectivity no longer holds, we propose a suitable representative (other than the traditionally sought-after solenoidal candidate) to be reconstructed, as well as an efficient procedure to do so. Numerical examples illustrating the method are provided at the end.