# Reconstruction of piecewise constant functions from X-ray data

## TL;DR

This paper demonstrates that piecewise constant functions on certain Riemannian manifolds can be reconstructed from geodesic X-ray data, providing explicit formulas and analyzing stability.

## Contribution

It extends the injectivity proof for geodesic X-ray transforms to piecewise constant functions on nontrapping manifolds and improves results for simple manifolds with tiled functions.

## Key findings

- Reconstruction is possible on nontrapping Riemannian manifolds with convex boundary.
- Explicit boundary formulas are derived for the reconstruction.
- Stability of the method is analyzed.

## Abstract

We show that on a two-dimensional compact nontrapping Riemannian manifold with strictly convex boundary, a piecewise constant function can be recovered from its integrals over geodesics. We adapt the injectivity proof which uses variations through geodesics to recover the function and we improve this result when the manifold is simple and the function is constant on tiles with geodesic edges, showing that the Jacobi fields of these variations are sufficient. We give also explicit formulas for the values near the boundary. We finally study the stability of the reconstruction method.