Wei Yuchuan, Yu Hengyong, Wang Ge
Department of Radiation Oncology, Wake Forest University School of Medicine, Winston-Salem, NC 27157, USA.
Int J Biomed Imaging. 2011;2011:285130. doi: 10.1155/2011/285130. Epub 2010 Oct 26.
This paper provides auxiliary results for our general scheme of computed tomography. In 3D parallel-beam geometry, we first demonstrate that the inverse Fourier transform in different coordinate systems leads to different reconstruction formulas and explain why the Radon formula cannot directly work with truncated projection data. Also, we introduce a gamma coordinate system, analyze its properties, compute the Jacobian of the coordinate transform, and define weight functions for the inverse Fourier transform assuming a simple scanning model. Then, we generate Orlov's theorem and a weighted Radon formula from the inverse Fourier transform in the new system. Furthermore, we present the motion equation of the frequency plane and the conditions for sharp points of the instantaneous rotation axis. Our analysis on the motion of the frequency plane is related to the Frenet-Serret theorem in the differential geometry.
本文为我们的计算机断层扫描通用方案提供了辅助结果。在三维平行束几何中，我们首先证明在不同坐标系中的逆傅里叶变换会导致不同的重建公式，并解释为什么拉东公式不能直接处理截断投影数据。此外，我们引入一个伽马坐标系，分析其性质，计算坐标变换的雅可比行列式，并在假设一个简单扫描模型的情况下为逆傅里叶变换定义权重函数。然后，我们从新系统中的逆傅里叶变换生成奥尔洛夫定理和加权拉东公式。此外，我们给出了频率平面的运动方程以及瞬时旋转轴尖点的条件。我们对频率平面运动的分析与微分几何中的弗伦内 - 塞雷定理相关。
