On the $X$-ray transform of planar symmetric 2-tensors

TL;DR

This paper investigates the attenuated X-ray transform of planar symmetric 2-tensors, characterizing its range and providing a method to reconstruct tensors with identical X-ray data using Hilbert transforms and A-analytic maps.

Contribution

It offers a novel characterization of the transform's range and a reconstruction method for 2-tensors in the Euclidean plane.

Findings

Range characterized via Hilbert-transform and A-analytic maps

Reconstruction of tensors with identical X-ray data

Theoretical framework for tensor tomography in planar domains

Abstract

In this paper we study the attenuated $X$-ray transform of 2-tensors supported in strictly convex bounded subsets in the Euclidean plane. We characterize its range and reconstruct all possible 2-tensors yielding identical $X$-ray data. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.