# Polarimetric Neutron Tomography of Magnetic Fields: Uniqueness of Solution and Reconstruction

## TL;DR

This paper investigates the uniqueness and reconstruction of magnetic fields using polarimetric neutron tomography, demonstrating theoretical guarantees, proposing a numerical method, and analyzing its limitations through experiments.

## Contribution

It establishes the uniqueness of solutions for magnetic field reconstruction, derives a linearized approach, and introduces a modified Newton method for numerical reconstruction.

## Key findings

- Unique solutions for smooth magnetic fields with compact support.
- MNKM algorithm effectively reconstructs small magnetic fields.
- Non-convexity of the inverse problem may cause gradient-based methods to fail.

## Abstract

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) type algorithm for the numerical solution of the non-linear problem, in which the forward problem is re-solved but the same derivative used each time. Numerical experiments demonstrate that MNKM works for small enough fields (or large enough velocities) and show an example where it fails to reconstruct a slice of the simulated data set. Lastly we show viewed as an optimization problem the inverse problem is non-convex so we expect gradient based methods may fail.