Convergence Analysis of Pixel-Driven Radon and Fanbeam Transforms
Abstract
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.
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© 2021, Society for Industrial and Applied Mathematics.
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Submitted: 6 April 2020
Accepted: 19 February 2021
Published online: 25 May 2021
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Austrian Science Fund https://doi.org/10.13039/501100002428 : W1244
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659
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