2D inverse problem with a Foliation Condition

TL;DR

This paper proves that functions can be reconstructed from X-ray transform data in domains with a foliation structure, facilitating inverse problems in such geometries.

Contribution

It establishes the possibility of function reconstruction in foliation-structured domains, expanding the understanding of inverse X-ray problems.

Findings

Reconstruction is possible in foliated domains.

Applicable to specific function classes.

Provides theoretical foundation for inverse problems in complex geometries.

Abstract

The inverse problem of X-tray transforms considers reconstructing functions from some data that are easier to measure, which is typically the integral of that function along geodesics. We prove that if the domain has a foliation structure, then this reconstruction process is possible on certain function classes.