The Inversion of Ray Transforms on a Conformal Class of Curves

TL;DR

This paper presents a unified complex-analytic method for inverting ray transforms over a broad class of curves in Euclidean space, including attenuated transforms, with explicit formulas.

Contribution

It introduces a novel complexification approach to invert ray transforms on a conformal class of curves, extending previous methods.

Findings

Explicit inversion formulas derived for unattenuated ray transforms.

Extension of inversion techniques to attenuated ray transforms.

Unified framework applicable to a wide class of curves.

Abstract

We introduce a technique for recovering a sufficiently smooth function from its ray transform over a wide class of curves in a general region of Euclidean space. The method is based on a complexification of the underlying vector fields defining the initial transport and recasting the problem in terms of complex-analytic function theory. Explicit inversion formulae are then given in a unified form. The method is then used to give inversion formulae for the attenuated ray transform.