Partial inversion of the 2D attenuated $X$-ray transform with data on an arc

TL;DR

This paper presents a method for inverting the 2D attenuated X-ray transform using data on an arc, reconstructing the function within the convex hull of the arc with known attenuation.

Contribution

It introduces a novel reconstruction technique based on the Hilbert transform of A-analytic functions for data restricted to an arc.

Findings

Reconstruction of functions on the convex hull from arc data

Use of A-analytic functions and Hilbert transform in inversion

Method applicable with known attenuation

Abstract

In two dimensions, we consider the problem of inversion of the attenuated $X$-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with $A$-analytic functions in the sense of Bukhgeim.