Radon transforms on generalized Cormack’s curves and a new Compton scatter tomography modality
Radon transforms on generalized Cormack’s curves and a new Compton scatter tomography modality
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氡气在广义科马克曲线和新的康普顿散射断层扫描模式上进行变换
DOI：
10.1088/0266-5611/27/12/125001
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发表时间：
2011
期刊：
影响因子：
2.1
通讯作者：
M. Nguyen
中科院分区：
文献类型：
--
作者：
T. Truong;M. Nguyen
在他1981年的开创性工作中，Cormack建立了在平面上的两个显著曲线族上定义的Radon变换是可逆的，并且通过圆调和分解承认显式的反演公式。本文给出了寻找具有相同性质的更大类曲线的一个充分条件。我们证明了这些广义Cormack曲线是由一个非线性一阶微分方程的解给出的，它在几何反演下是不变的。推导了相应Radon变换的解析逆公式，并给出了Radon变换的一些主要性质。有趣的是，在这些广义的Cormack曲线中，有一些圆与以坐标原点为中心的半径固定的圆正交。提出了一种新的康普顿散射层析成像模态可以用在这些圆上定义的Radon变换来建模。
In his seminal work of 1981, Cormack established that Radon transforms defined on two remarkable families of curves in the plane are invertible and admit explicit inversion formulas via circular harmonic decomposition. A sufficient condition for finding larger classes of curves enjoying the same property is given in this paper. We show that these generalized Cormack’s curves are given by the solutions of a nonlinear first-order differential equation, which is invariant under geometric inversion. A derivation of the analytic inverse formula of the corresponding Radon transforms, as well as some of their main properties, are worked out. Interestingly, among these generalized Cormack’s curves are circles orthogonal to a circle of fixed radius centered at the origin of coordinates. It is suggested that a novel Compton scatter tomography modality may be modeled by a Radon transform defined on these circles.