Ray Transforms on a Conformal Class of Curves

TL;DR

This paper presents a novel technique for inverting ray transforms over a broad class of curves in Euclidean space, using complex analysis to derive explicit inversion formulas, including for attenuated transforms.

Contribution

It introduces a unified complex-analytic approach for inverting ray transforms on a conformal class of curves, extending previous methods to more general curve families.

Findings

Derived explicit inversion formulas for the ray transform.

Extended inversion techniques to attenuated ray transforms.

Unified approach 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.