Polarimetric neutron tomography of magnetic fields: Uniqueness of solution and reconstruction

We consider the problem of determination of a magnetic field from three dimensional polarimetric neutron tomography data. We see that this is an example of a non-Abelian ray transform and that the problem has a globally unique solution for smooth magnetic fields with compact support, and a locally unique solution for less smooth fields. We derive the linearization of the problem and note that the derivative is injective. We go on to show that the linearised problem about a zero magnetic field reduces to plane Radon transforms and suggest a modified Newton-Kantarovich method (MNKM) for the numerical solution of the non-linear problem, in which the forward problem is re-solved but the same derivative is used each time. Numerical experiments demonstrate that MNKM works for small enough fields (or large enough velocities) and we show an example where it fails to reconstruct a slice of the simulated data set. Lastly we show that, viewed as an optimization problem, the inverse problem is non-convex so we expect gradient based methods may fail.