Discrete Non-Abelian X-Ray Transforms

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 particular, we contribute to reconstructing piecewise constant matrix-valued functions from their continuous non-abelian X-ray transform. Our main result in this direction consists of an explicit and exact non-overdetermined layer-stripping reconstruction procedure. To our knowledge, this result is new even for the classical X-ray transform.