On the Range Characterization of the two dimensional attenuated Doppler Transform

TL;DR

This paper characterizes the range of the 2D attenuated and non-attenuated X-ray transforms of vector fields, using a Hilbert transform linked to A-analytic functions, and identifies conditions for data indistinguishability.

Contribution

It provides a novel range characterization for the attenuated Doppler and X-ray transforms in the plane, connecting it with A-analytic functions and Hilbert transforms.

Findings

Range characterized via Hilbert transform and A-analytic functions

Necessary and sufficient conditions for data indistinguishability

Applicable to vector fields in the plane

Abstract

We characterize the range of the attenuated and non-attenuated $X$-ray transform of compactly supported vector fields in the plane. The characterization is in terms of a Hilbert transform associated with the $A$-analytic functions \`{a} la Bukhgeim. As an application we determine necessary and sufficient conditions for the attenuated Doppler and $X$-ray data to be mistaken for each other.