Development, analysis and implementation of exact inversion formulas for cone beam vector tomography

在过去的十年中，锥束变换在标量函数情况下的反演得到了广泛的关注，并推导出了各种有效的反演公式。考虑到实际应用，锥束几何形状具有最大的相关性。矢量场的锥束变换的研究要少得多，即使矢量场层析成像有各种有趣的应用，从医学成像，海洋学等离子体物理。研究较少的一个原因可能是，部分原因是这是一个更困难的问题。在Schuster教授和Kazantsev博士最近的一个关键结果的基础上，Schuster教授和Kazantsev博士得到了矢量场锥束变换的反演公式。Katsevich和Schuster在2013年。该公式由两个加法项组成。第一项是滤波反投影形式，而第二项更复杂，涉及四维积分，这在数值上非常昂贵。我们建议更详细地研究公式，并使用公式中内置的灵活性来简化第二项。然后，我们建议在计算机代码中实现该公式，并在不同类型的向量场的数值例子上测试其性能。

Inversion of the cone beam transform in the case of scalar functions has received much attention, and various efficient inversion formulas were derived during the past decade. The cone-beam geometry is of greatest relevance in view of practical applications. The cone-beam transform of vector fields has been studied much less, even though vector field tomography has a variety of interesting applications ranging from medical imaging to oceanography to plasma physics. One reason for less studies could be, in part, due to the fact that it is a much harder problem. Building on a recent key result of Prof. Schuster and Dr. Kazantsev, an inversion formula for the cone-beam transform of vector fields was obtained by Profs. Katsevich and Schuster in 2013. The formula consists of two additive terms. The first term is of the filtered-backprojection form, while the second term is more complicated and involves a four-dimensional integral which is numerically very costly. We propose to investigate the formula in more detail, and use the flexibilities built into the formula to simplify the second term. Then we propose to implement the formula in computer code and test its performance on numerical examples for different types of vector fields.