Inversion of the attenuated momenta ray transform of planar symmetric tensors

TL;DR

This paper introduces a stable reconstruction method for symmetric tensors in the plane from their attenuated momenta ray transform, extending Bukhgeim's $A$-analytic theory and demonstrating numerical results for the non-attenuated case.

Contribution

It develops a novel, constructive inversion technique for the attenuated momenta ray transform of symmetric tensors using an extension of $A$-analytic theory.

Findings

Reconstruction method is stable and effective for symmetric tensors.

Numerical implementation demonstrates the method for non-attenuated 1-tensors.

The approach extends existing $A$-analytic techniques to tensor tomography.

Abstract

We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim's $A$-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of 1-tensors. This new version now includes the results of the preprint arXiv: 2307.10758.