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) 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.
Highlights
Neutron tomography is widely used in science and industry; it provides important complementary information to that given by x-rays as neutrons have zero electrical charge and can penetrate deeply into massive samples, see [18]
We show that the problem can be formulated as a non-Abelian ray transform and that sufficiently smooth magnetic fields are uniquely determined by polarimetric neutron tomography data
We show that the linearized problem, for small magnetic fields, has a unique solution for less regular fields and propose an iterative reconstruction algorithm which we have implemented and tested on simulated data. (This formulation as a non-Abelian ray transform and the consequent uniqueness result for non-linear and linearized problem was first presented in the conference talk [9] and in more detail in the thesis [3].) This new theoretical framework lays the foundation for practical 3D polarimetric neutron tomography of magnetic fields (PNTMF) that will facilitate the imaging of magnetic domains in metal samples and the corresponding design of magnetic materials
Summary
Neutron tomography is widely used in science and industry; it provides important complementary information to that given by x-rays as neutrons have zero electrical charge and can penetrate deeply into massive samples, see [18]. In contrast to initial proposals, the setup used does not measure the full spin rotation matrix but only a single diagonal element While this is incomplete data the method has been applied for strongly oriented fields and high symmetry cases providing significant a priori knowledge for analysis, e.g. through field modelling and simulation matching to data. (This formulation as a non-Abelian ray transform and the consequent uniqueness result for non-linear and linearized problem was first presented in the conference talk [9] and in more detail in the thesis [3].) This new theoretical framework lays the foundation for practical 3D polarimetric neutron tomography of magnetic fields (PNTMF) that will facilitate the imaging of magnetic domains in metal samples and the corresponding design of magnetic materials We show that the linearized problem, for small magnetic fields, has a unique solution for less regular fields and propose an iterative reconstruction algorithm which we have implemented and tested on simulated data. (This formulation as a non-Abelian ray transform and the consequent uniqueness result for non-linear and linearized problem was first presented in the conference talk [9] and in more detail in the thesis [3].) This new theoretical framework lays the foundation for practical 3D polarimetric neutron tomography of magnetic fields (PNTMF) that will facilitate the imaging of magnetic domains in metal samples and the corresponding design of magnetic materials
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