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

锥束变换在标量函数情况下的反演一直受到人们的关注，在过去的十年里，人们推导出了各种有效的反演公式。从实际应用的角度来看，锥形束的几何形状具有最大的相关性。尽管矢量场层析成像在医学成像、海洋学和等离子体物理等领域有许多有趣的应用，但矢量场的锥束变换的研究要少得多。减少研究的一个原因可能部分是因为这是一个困难得多的问题。在Schuster教授和Kazantsev博士最近的关键结果的基础上，得到了矢量场的锥束变换的反演公式。Katsevich和Schuster在2013年。该公式由两个相加项组成。第一项是滤波反投影形式，而第二项更复杂，涉及到数值上非常昂贵的四维积分。我们建议更详细地研究公式，并利用公式中内置的灵活性来简化第二项。然后，我们建议用计算机代码实现该公式，并在不同类型矢量场的数值算例中测试其性能。

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.