Convergent Data-Driven Regularizations for CT Reconstruction

doi:10.1007/s42967-023-00333-2

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (2): 1342-1368.doi: 10.1007/s42967-023-00333-2

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Convergent Data-Driven Regularizations for CT Reconstruction

Samira Kabri¹, Alexander Auras², Danilo Riccio³, Hartmut Bauermeister², Martin Benning^3,4, Michael Moeller², Martin Burger^1,5

1. Helmholtz Imaging, Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany;
2. Institute for Vision and Graphics, University of Siegen, Adolf-Reichwein-Stra?e 2a, 57076 Siegen, Germany;
3. School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK;
4. The Alan Turing Institute, British Library, 96 Euston Rd, London NW1 2DB, UK;
5. Fachbereich Mathematik, Universit?t Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany

Received:2022-12-13 Revised:2023-08-07 Accepted:2023-09-29 Online:2024-02-23 Published:2024-02-23
Contact: Samira Kabri,E-mail:samira.kabri@desy.de;Alexander Auras,E-mail:alexander.auras@uni-siegen.de;Danilo Riccio,E-mail:s.d.riccio@qmul.ac.uk;Hartmut Bauermeister,E-mail:hartmut.bauermeister@uni-siegen.de;Martin Benning,E-mail:m.benning@qmul.ac.uk;Michael Moeller,E-mail:michael.moeller@uni-siegen.de;Martin Burger,E-mail:martin.burger@desy.de E-mail:samira.kabri@desy.de;alexander.auras@uni-siegen.de;s.d.riccio@qmul.ac.uk;hartmut.bauermeister@uni-siegen.de;m.benning@qmul.ac.uk;michael.moeller@uni-siegen.de;martin.burger@desy.de

Abstract

Abstract: The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naïve) solution does not depend on the measured data continuously, regularization is needed to reestablish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: one generic linear regularization that learns how to manipulate the singular values of the linear operator in an extension of our previous work, and one tailored approach in the Fourier domain that is specific to CT-reconstruction. We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typically much smoother than the training data they were trained on. Finally, we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically, discuss their advantages and disadvantages and investigate the effect of discretization errors at different resolutions.

Key words: Inverse problems, Regularization, Computerized tomography (CT), Machine learning

Samira Kabri, Alexander Auras, Danilo Riccio, Hartmut Bauermeister, Martin Benning, Michael Moeller, Martin Burger. Convergent Data-Driven Regularizations for CT Reconstruction[J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 1342-1368.

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Convergent Data-Driven Regularizations for CT Reconstruction

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