The inverse problem for the local geodesic ray transform

TL;DR

This paper demonstrates local and global invertibility of the geodesic X-ray transform on manifolds with convex boundary conditions, providing stability estimates and a reconstruction algorithm.

Contribution

It introduces new conditions for local and global invertibility of the geodesic X-ray transform, including a stable inversion formula and a layer stripping reconstruction method.

Findings

Local invertibility with a reconstruction formula

Global injectivity under convex foliation

Stability estimates and a reconstruction algorithm

Abstract

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.