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Error estimates and convergence rates for filtered back projection


Authors: Matthias Beckmann and Armin Iske
Journal: Math. Comp. 88 (2019), 801-835
MSC (2010): Primary 41A25; Secondary 94A20, 94A08
DOI: https://doi.org/10.1090/mcom/3343
Published electronically: April 30, 2018
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Abstract: We consider the approximation of target functions from fractional Sobolev spaces by the method of filtered back projection (FBP), which gives an inversion of the Radon transform. The objective of this paper is to analyse the intrinsic FBP approximation error which is incurred by the use of a low-pass filter with finite bandwidth. To this end, we prove $ \mathrm {L}^2$-error estimates on Sobolev spaces of fractional order. The obtained error bounds are affine-linear with respect to the distance between the filter's window function and the constant function $ 1$ in the $ \mathrm {L}^\infty $-norm. With assuming more regularity of the window function, we refine the error estimates to prove convergence for the FBP approximation in the $ \mathrm {L}^2$-norm as the filter's bandwidth goes to infinity. Further, we determine asymptotic convergence rates in terms of the bandwidth of the low-pass filter and the smoothness of the target function. Finally, we develop convergence rates for noisy data, where we first prove estimates for the data error, which we then combine with our estimates for the approximation error.


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Additional Information

Matthias Beckmann
Affiliation: Department of Mathematics, University of Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany
Email: matthias.beckmann@uni-hamburg.de

Armin Iske
Affiliation: Department of Mathematics, University of Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany
Email: armin.iske@uni-hamburg.de

DOI: https://doi.org/10.1090/mcom/3343
Keywords: Filtered back projection, error estimates, convergence rates, Sobolev functions
Received by editor(s): March 30, 2016
Received by editor(s) in revised form: February 10, 2017, and November 22, 2017
Published electronically: April 30, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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