Convergence of Ray- and Pixel-Driven Discretization Frameworks in the Strong Operator Topology

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 what we refer to as the ray-driven Radon transform with what we refer to as 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 an interpretation of ray- and pixel-driven discretizations as `convolutional methods', a special class of finite-rank operators. 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. Numerical experiments corroborate these theoretical findings.