Tomography is a central tool in medical applications, allowing doctors to investigate patients' interior features. The Radon transform (in two dimensions) is commonly used to model the measurement process in parallel-beam CT. Suitable discretization of the Radon transform and its adjoint (called the backprojection) is crucial. The most commonly used discretization approach combines the ray-driven Radon transform with the pixel-driven backprojection, as anecdotal reports describe these as showing the best approximation performance. However, there is little rigorous understanding of induced approximation errors. These methods involve three discretization parameters: the spatial-, detector-, and angular resolutions. Most commonly, balanced resolutions are used, i.e., the same (or similar) spatial- and detector resolutions are employed. We present a novel interpretation of ray- and pixel-driven discretizations as `convolutional methods'. This allows for a structured analysis that can explain observed behavior. In particular, we prove convergence in the strong operator topology of the ray-driven Radon transform and the pixel-driven backprojection under balanced resolutions, thus theoretically justifying this approach. In particular, with high enough resolutions one can approximate the Radon transform arbitrarily well.

翻译：层析成像在医学应用中是一项核心工具，使得医生能够探查患者内部特征。拉东变换（二维情形）通常用于模拟平行束CT中的测量过程。对拉东变换及其伴随算子（称为反投影）的恰当离散化至关重要。最常用的离散化方法将射线驱动的拉东变换与像素驱动的反投影相结合，因为经验性报告指出这种方法表现出最佳的近似性能。然而，对于由此产生的近似误差，目前缺乏严格的理论理解。这些方法涉及三个离散化参数：空间分辨率、探测器分辨率和角度分辨率。最常采用的是平衡分辨率，即使用相同（或相近）的空间分辨率与探测器分辨率。我们提出了一种将射线驱动与像素驱动离散化解释为“卷积方法”的新颖观点，这为解释观测到的行为提供了结构化分析框架。特别地，我们证明了在平衡分辨率条件下，射线驱动的拉东变换与像素驱动的反投影在强算子拓扑意义下收敛，从而从理论上验证了该方法的合理性。这意味着，当分辨率足够高时，可以任意精确地逼近拉东变换。