An inversion of the conical Radon transform arising in the Compton camera with helical movement
An inversion of the conical Radon transform arising in the Compton camera with helical movement
复制标题
康普顿相机中螺旋运动的圆锥拉东变换的反演
DOI：
10.1007/s13534-019-00106-y
复制
发表时间：
2019
影响因子：
4.6
通讯作者：
Kiwoon Kwon
中科院分区：
文献类型：
--
作者：
Kiwoon Kwon
自从康普顿照相机首次被引入以来，各种类型的锥形拉东变换已经被研究。在这里，我们推导出圆锥Radon变换的反演公式，其中积分圆锥沿着三维空间中的曲线（如螺旋线）移动。本文将沿着这条三维曲线，对康普顿成像的螺旋运动进行详细的反演。反演公式包括Hilbert变换和Radon变换。对于具有螺旋运动的康普顿成像的反演，需要对康普顿摄像机圆锥的顶点与中心轴的内积进行希尔伯特变换的反演。然而，内积函数不是单调的。因此，我们应该用与内积函数相关的某个单调函数上的Riemann-Stieltjes积分来代替Hilbert变换。我们将Riemann-Stieltjes积分表示为不相交区间的可数并集上的常规Riemann积分，其端点可以使用牛顿法计算。对于Radon变换的逆变换，采用了三维滤波反投影。在数值实现方面，我们解析计算了有限球特征函数的Hilbert变换和Radon变换。当密度函数由一个球或三个重叠球的特征函数给出时，给出了数值试验。
Since the Compton camera was first introduced, various types of conical Radon transforms have been examined. Here, we derive the inversion formula for the conical Radon transform, where the cone of integration moves along a curve in three-dimensional space such as a helix. Along this three-dimensional curve, a detailed inversion formula for helical movement will be treated for Compton imaging in this paper. The inversion formula includes Hilbert transform and Radon transform. For the inversion of Compton imaging with helical movement, it is necessary to invert Hilbert transform with respect to the inner product between the vertex and the central axis of the cone of the Compton camera. However, the inner product function is not monotone. Thus, we should replace the Hilbert transform by the Riemann–Stieltjes integral over a certain monotone function related with the inner product function. We represent the Riemann–Stieltjes integral as a conventional Riemann integral over a countable union of disjoint intervals, whose end points can be computed using the Newton method. For the inversion of Radon transform, three dimensional filtered backprojection is used. For the numerical implementation, we analytically compute the Hilbert transform and Radon transform of the characteristic function of finite balls. Numerical test is given, when the density function is given by a characteristic function of a ball or three overlapping balls.
一类 Radon 变换的 3D 康普顿散射成像和轮廓重建
影响因子：
2.1
作者：
G. Rigaud;B. N. Hahn
通讯作者：
B. N. Hahn
康普顿相机成像和锥体变换：简要概述
影响因子：
2.1
作者：
Terzioglu, Fatma;Kuchment, Peter;Kunyansky, Leonid
通讯作者：
Kunyansky, Leonid