Reconstruction formulas for X-ray transforms in negative curvature

TL;DR

This paper derives new reconstruction formulas for the geodesic X-ray transform on negatively curved surfaces with trapping, extending previous formulas to more complex geometries, and provides numerical validation.

Contribution

It generalizes Pestov-Uhlmann formulas to surfaces with trapping, involving Fredholm equations and new estimates for the normal operator.

Findings

Formulas invert the X-ray transform on negatively curved surfaces

Extension to surfaces with trapping and infinite geodesics

Numerical examples demonstrating the formulas' effectiveness

Abstract

We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it $I_0$) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [Pestov-Uhlmann, IMRN '04] (established for simple surfaces) to cases allowing geodesics with infinite length on surfaces with trapping. Such formulas take the form of Fredholm equations, where the analysis of error operators requires deriving new estimates for the normal operator $\Pi_0 = I_0^* I_0$. Numerical examples are provided at the end.