星变换的反演
Inversion of the star transform
摘要：
我们将星变换定义为我们在以前的工作中引入的破碎射线变换的推广。使用星形变换的优点包括（从相同的数据）单独地和同时地重建介质的吸收和散射系数的可能性，以及利用散射辐射的可能性，在常规X射线断层摄影的情况下，散射辐射被丢弃。本文从物理原理出发，推导了星变换，讨论了星变换的数学性质，分析了星变换反演的数值稳定性。特别是，研究表明，仅对于涉及奇数条射线的配置，才能获得星变换的稳定反演。几个计算效率的反演算法推导和测试数值。
英文摘要：
We define the star transform as a generalization of the broken ray transform introduced by us in previous work. The advantages of using the star transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium separately and simultaneously (from the same data) and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is discarded. In this paper, we derive the star transform from physical principles, discuss its mathematical properties and analyze numerical stability of inversion. In particular, it is shown that stable inversion of the star transform can be obtained only for configurations involving odd number of rays. Several computationally-efficient inversion algorithms are derived and tested numerically.
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