We study the inversion of the conical Radon transform which integrates a function on the surface of a cone. The conical Radon transform recently got significant attention due to its relevance in various imaging applications such as Compton camera imaging and single scattering optical tomography. The unrestricted conical Radon transform is over-determined because the manifold of all cones depends on six variables: the center position, the axis orientation and the opening angle of the cone. In this work, we consider a particular restricted conical Radon transform using triple linear sensor of finite length where integrals over a three-dimensional set of cones are collected, determined by a one-dimensional vertex set, a one-dimensional set of central axes, and a one-dimensional set of opening angle. As the main result in this paper we derive an analytic inversion formula for the restricted conical Radon transform. Along that way we define a certain ray transform adapted to the triple line sensor for which we establish an analytic inversion formula.

我们研究了锥形Radon变换的反演，该变换在圆锥表面上集成了一个函数。锥形Radon变换最近在康普顿相机成像和单散射光学层析成像等各种成像应用中引起了广泛关注。由于所有圆锥体的流形都取决于六个变量：圆锥体的中心位置，轴方向和张角，所以无限制的圆锥Radon变换是超定的。在这项工作中，我们考虑使用有限长度的三重线性传感器进行特殊的受限圆锥Radon变换，其中收集一维顶点集，一维中心轴集所确定的三维圆锥集上的积分，一维的张角 作为本文的主要结果，我们导出了约束圆锥Radon变换的解析反演公式。通过这种方式，我们定义了适用于三线传感器的特定射线变换，为此我们建立了一个解析反演公式。