SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1399-1432, January 2021.
This paper presents a novel mathematical framework for understanding pixel-driven approaches for the parallel beam Radon transform as well as for the fanbeam transform, showing that with the correct discretization strategy, convergence---including rates---in the $L^2$ operator norm can be obtained. From these rates, suitable strategies are devised for discretization of the occurring domains/variables and are first established for the Radon transform. In particular, discretizing the detector in the same magnitude as the image pixels (which is standard practice) might not be ideal and, in fact, pixels that are asymptotically smaller than detectors lead to convergence. Possible adjustments to limited-angle and sparse-angle Radon transforms are discussed, and similar convergence results are shown. In the same vein, convergence results are readily extended to a novel pixel-driven approach to the fanbeam transform. Numerical aspects of the discretization scheme are discussed, and it is shown in particular that with the correct discretization strategy, the typical high-frequency artifacts can be avoided.

中文翻译：

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像素驱动Rad和扇束变换的收敛性分析

SIAM数值分析杂志，第59卷，第3期，第1399-1432页，2021年1月。
本文提出了一个新颖的数学框架，用于理解像素驱动的平行束Radon变换以及扇束变换的方法，表明采用正确的离散化策略，$ L ^ 2中的收敛（包括速率）可以获得$运算符范数。从这些速率中，设计出适当的策略来离散化所出现的域/变量，并首先为Radon变换建立该策略。特别是，以与图像像素相同的幅度离散检测器（这是标准做法）可能并不理想，实际上，渐近小于检测器的像素会导致会聚。讨论了对有限角度和稀疏角度Radon变换的可能调整，并显示了类似的收敛结果。同样，收敛结果很容易扩展到扇形变换的新型像素驱动方法。讨论了离散化方案的数值方面，并特别表明，通过正确的离散化策略，可以避免典型的高频伪像。