University of Oulu

Lunz, S., Hauptmann, A., Tarvainen, T., Schönlieb, C.-B., & Arridge, S. (2021). On Learned Operator Correction in Inverse Problems. SIAM Journal on Imaging Sciences, 14(1), 92–127.

On learned operator correction in inverse problems

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Author: Lunz, Sebastian1; Hauptmann, Andreas2,3; Tarvainen, Tanja4,3;
Organizations: 1University of Cambridge, Department of Applied Mathematics and Theoretical Physics
2University of Oulu, Research Unit of Mathematical Sciences
3University College London, Department of Computer Science
4University of Eastern Finland, Department of Applied Physics
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 2.7 MB)
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Language: English
Published: Society for Industrial and Applied Mathematics, 2021
Publish Date: 2021-02-26


We discuss the possibility of learning a data-driven explicit model correction for inverse problems and whether such a model correction can be used within a variational framework to obtain regularized reconstructions. This paper discusses the conceptual difficulty of learning such a forward model correction and proceeds to present a possible solution as a forward-adjoint correction that explicitly corrects in both data and solution spaces. We then derive conditions under which solutions to the variational problem with a learned correction converge to solutions obtained with the correct operator. The proposed approach is evaluated on an application to limited view photoacoustic tomography and compared to the established framework of the Bayesian approximation error method.

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Series: SIAM journal on imaging sciences
ISSN: 1936-4954
ISSN-E: 1936-4954
ISSN-L: 1936-4954
Volume: 14
Issue: 1
Pages: 92 - 127
DOI: 10.1137/20M1338460
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: Academy of Finland Projects 312123 and 312342 (Finnish Centre of Excellence in Inverse Modelling and Imaging, 2018–2025) and Projects 334817 and 314411, Jane and Aatos Erkko Foundation, British Heart Foundation grant NH/18/1/33511, CMIC-EPSRC platform grant (EP/M020533/1), EPSRC-Wellcome grant WT101957, Leverhulme Trust project on Breaking the non-convexity barrier, Philip Leverhulme Prize, EPSRC grants EP/S026045/1, EP/T003553/1, EP/T000864/1, EPSRC Centre Nr. EP/N014588/1, Wellcome Innovator Award RG98755, the RISE projectsCHiPS and NoMADS, Cantab Capital Institute for the Mathematics of Information, Alan Turing Institute, EPSRC grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training (Cambridge Centre for Analysis)
Academy of Finland Grant Number: 312123
Detailed Information: 312123 (Academy of Finland Funding decision)
312342 (Academy of Finland Funding decision)
334817 (Academy of Finland Funding decision)
314411 (Academy of Finland Funding decision)
Copyright information: © 2021 Society for Industrial and Applied Mathematics. Unauthorized reproduction of this article is prohibited.