Polarimetric neutron tomography of magnetic fields: uniqueness of solution and reconstruction
Polarimetric neutron tomography of magnetic fields: uniqueness of solution and reconstruction
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DOI：
10.1088/1361-6420/ab44e0
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发表时间：
2020-04-01
期刊：
影响因子：
2.1
通讯作者：
Schmidt, Soren
中科院分区：
文献类型：
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作者：
Desai, Naeem M.;Lionheart, William R. B.;Schmidt, Soren
我们考虑的问题，确定磁场的三维极化中子层析成像数据。我们看到，这是一个非阿贝尔射线变换的例子，并且该问题对于具有紧支撑的光滑磁场具有全局唯一解，对于不太光滑的磁场具有局部唯一解。我们推导出线性化的问题，并指出，衍生物是内射。我们继续表明，线性化的问题约为零磁场减少到平面氡变换，并提出了一个修改后的牛顿-Kantarovich方法（MNKM）的数值解的非线性问题，其中的正向问题重新解决，但相同的衍生物是每次使用。数值实验表明，MNKM工程足够小的领域（或足够大的速度），我们展示了一个例子，它无法重建一个切片的模拟数据集。最后，我们表明，作为一个优化问题，逆问题是非凸的，所以我们预计基于梯度的方法可能会失败。
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.