Three Dimensional Polarimetric Neutron Tomography of Magnetic Fields

Through the use of Time-of-Flight Three Dimensional Polarimetric Neutron Tomography (ToF 3DPNT) we have for the first time successfully demonstrated a technique capable of measuring and reconstructing three dimensional magnetic field strengths and directions unobtrusively and non-destructively with the potential to probe the interior of bulk samples which is not amenable otherwise. Using a pioneering polarimetric set-up for ToF neutron instrumentation in combination with a newly developed tailored reconstruction algorithm, the magnetic field generated by a current carrying solenoid has been measured and reconstructed, thereby providing the proof-of-principle of a technique able to reveal hitherto unobtainable information on the magnetic fields in the bulk of materials and devices, due to a high degree of penetration into many materials, including metals, and the sensitivity of neutron polarisation to magnetic fields. The technique puts the potential of the ToF time structure of pulsed neutron sources to full use in order to optimise the recorded information quality and reduce measurement time.Comment: 12 pages, 4 figure

Inversion of the Momenta Doppler Transform in two dimensions

We introduce an analytic method which stably reconstructs both components of a (sufficiently) smooth, real valued, vector field compactly supported in the plane from knowledge of its Doppler transform and its first moment Doppler transform. The method of proof is constructive. Numerical inversion results indicate robustness of the method.Comment: 18 pages, 12 figure

On the XX-ray transform of symmetric higher order tensors

In this article we characterize the range of the attenuated and non-attenuated $X$-ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1503.0432

On the inversion of the momenta ray transform of symmetric tensors in the plane

We present a reconstruction method which stably recovers some sufficiently smooth, real valued, symmetric tensor fields compactly supported in the Euclidean plane, from knowledge of their non/attenuated momenta ray transform. The reconstruction method extends Bukhgeim's $A$-analytic theory from an equation to a system.Comment: 20 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2307.1075

On the XX-ray transform of planar symmetric 2-tensors

In this paper we study the attenuated $X$-ray transform of 2-tensors supported in strictly convex bounded subsets in the Euclidean plane. We characterize its range and reconstruct all possible 2-tensors yielding identical $X$-ray data. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.Comment: 1 figure. arXiv admin note: text overlap with arXiv:1411.492

Improved 2-D vector field reconstruction using virtual sensors and the Radon transform

This paper describes a method that allows one to recover both components of a 2-D vector field based on boundary information only, by solving a system of linear equations. The analysis is carried out in the digital domain and takes advantage of the redundancy in the boundary data, since these may be viewed as weighted sums of the local vector field’s Cartesian components. Furthermore, a sampling of lines is used in order to combine the available measurements along continuous tracing lines with the digitised 2-D space where the solution is sought. A significant enhancement in the performance of the proposed algorithm is achieved by using, apart from real data, also boundary data obtained at virtual sensors. The potential of the proposed method is demonstrated by presenting an example of vector field reconstruction

Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

In this article we study the fan-beam Radon transform ${\cal D}_m$ of symmetrical solenoidal 2D tensor fields of arbitrary rank $m$ in a unit disc $\mathbb D$ as the operator, acting from the object space ${\mathbf L}_{2}(\mathbb D; {\bf S}_m)$ to the data space $L_2([0,2\pi)\times[0,2\pi)).$ The orthogonal polynomial basis ${\bf s}^{(\pm m)}_{n,k}$ of solenoidal tensor fields on the disc $\mathbb D$ was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator ${\cal D}_m$ was obtained. The inversion formula for the fan-beam tensor transform ${\cal D}_m$ follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.Comment: LaTeX, 37 pages with 5 figure