On the $X$-ray transform of symmetric higher order tensors

TL;DR

This paper characterizes the range of the X-ray transform for symmetric tensor fields in the Euclidean plane, using a Hilbert-transform linked to A-analytic maps, advancing understanding of tensor tomography.

Contribution

It provides a new range characterization for the X-ray transform of symmetric tensors employing A-analytic maps and Hilbert-transforms, extending previous results in tensor tomography.

Findings

Range characterized via Hilbert-transform and A-analytic maps

Applicable to both attenuated and non-attenuated transforms

Advances theoretical understanding of tensor field reconstruction

Abstract

In this article we characterize the range of the attenuated and non-attenuated $X$-ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.