三维康普顿散射层析成像
Three dimensional Compton scattering tomography
摘要：
本文提出了一种新的用于三维X射线康普顿成像中的电子密度重建的采集几何结构。这导致我们一个新的三维反问题，我们的目标是重建一个实值函数$f$（电子密度）从其积分主轴环面。本文证明了广义spindle torus变换在紧支撑于空心球上的光滑函数集上的内射性。这是通过一类Volterra积分算子的显式反演得到的，其解给出了$f$的调和系数的表达式。然后考虑多色源的情形，证明了一种新的spindle内变换、apple变换和apple内变换在紧支撑于空心球上的光滑函数集上的内射性.一个可能的物理模型，建议这两种源类型。我们还提供了模拟密度重建与不同水平的添加伪随机噪声和模型的系统误差，由于在我们的模拟中的入射和散射射线的衰减。
英文摘要：
We propose a new acquisition geometry for electron density reconstruction in three dimensional X-ray Compton imaging using a monochromatic source. This leads us to a new three dimensional inverse problem where we aim to reconstruct a real valued function $f$ (the electron density) from its integrals over spindle tori. We prove injectivity of a generalized spindle torus transform on the set of smooth functions compactly supported on a hollow ball. This is obtained through the explicit inversion of a class of Volterra integral operators, whose solutions give us an expression for the harmonic coefficients of $f$. The polychromatic source case is later considered, and we prove injectivity of a new spindle interior transform, apple transform and apple interior transform on the set of smooth functions compactly supported on a hollow ball. A possible physical model is suggested for both source types. We also provide simulated density reconstructions with varying levels of added pseudo random noise and model the systematic error due to the attenuation of the incoming and scattered rays in our simulation.
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