Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds

TL;DR

This paper proves that on certain nontrapping manifolds, piecewise constant functions can be uniquely reconstructed from geodesic integrals, extending to higher dimensions under a foliation condition, with elementary proofs.

Contribution

It establishes new uniqueness results for geodesic X-ray transforms of piecewise constant functions on nontrapping manifolds, including higher-dimensional cases.

Findings

Unique determination of piecewise constant functions from geodesic integrals on 2D nontrapping manifolds.

Extension of uniqueness results to higher dimensions under foliation conditions.

Elementary proof techniques for the uniqueness theorems.

Abstract

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.