Discrete non-abelian X-ray transforms

TL;DR

The paper defines a discrete non-abelian X-ray transform $\mathcal{S}$ for $GL(n,\mathbb{C})$-valued lattice functions and connects it to the continuous non-abelian and abelian discrete transforms. It develops non-overdetermined reconstruction strategies from restricted ray data, including layer-stripping along irrational and rational directions, and establishes two-dimensional slice formulas for higher dimensions. It then links the discrete transform to weighted discrete X-ray transforms and to continuous formulations through regularized delta functions and piecewise-constant functions, introducing a variant $\mathcal{S}^{\star}$. A key novel result is an exact non-overdetermined layer-stripping reconstruction for piecewise-constant functions from the continuous non-abelian X-ray transform, providing a new constructive pathway for numerical implementation and cross-domain insight between discrete and continuous non-abelian tomography.

Abstract

We define a discrete version of the non-abelian X-ray transform, going back in particular to Manakov, Zakharov (1981) and Strichartz (1982). We extend to this transform non-overdetermined reconstruction results obtained for the abelian case in the recent article by Novikov, Sharma (2025). In addition, we establish relations with the continuous non-abelian X-ray transform. In this respect, our results include an explicit and exact non-overdetermined layer-stripping reconstruction procedure for piecewise constant matrix-valued functions from their continuous non-abelian X-ray transform. To our knowledge, this result is new even for the classical X-ray transform.