Several types of conical Radon transforms have been studied since the introduction of the Compton camera. Several factors of a cone of integration can be considered as variables, for example, a vertex, a central axis, and an opening angle. In this paper, we study the conical Radon transform with a fixed central axis and opening angle. Furthermore, we consider the attenuation effect in the conical Radon transform because it allows us to obtain a high-quality reconstruction image. We construct a nonseparable Hilbert space and its maximal orthonormal set. This maximal orthonormal set comprises the eigenfunctions of the attenuated conical Radon transform, i.e. singular value decomposition (SVD). Finally, the inversion formula of the attenuated conical Radon transform is deduced from the SVD.

自从康普顿相机推出以来，已经研究了几种类型的锥形 Radon 变换。积分锥的几个因素可以被视为变量，例如，顶点、中心轴和张角。在本文中，我们研究了具有固定中心轴和张角的锥形Radon变换。此外，我们考虑了锥形 Radon 变换中的衰减效应，因为它允许我们获得高质量的重建图像。我们构造了一个不可分离的希尔伯特空间及其最大正交集。该最大正交集包括衰减圆锥形Radon变换的特征函数，即奇异值分解(SVD)。最后，从SVD推导出衰减圆锥Radon变换的反演公式。