University of Oulu

Simon Arridge, Pascal Fernsel, Andreas Hauptmann. Joint reconstruction and low-rank decomposition for dynamic inverse problems. Inverse Problems and Imaging, 2022, 16 (3) : 483-523. doi: 10.3934/ipi.2021059

Joint reconstruction and low-rank decomposition for dynamic inverse problems

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Author: Arridge, Simon1; Fernsel, Pascal2; Hauptmann, Andreas3,1
Organizations: 1Department of Computer Science, University College London, London WC1E 6BT, United Kingdom
2Center for Industrial Mathematics, University of Bremen 28359 Bremen, Germany
3Department of Mathematical Sciences, University of Oulu, 90014 Oulu, Finland
Format: article
Version: accepted version
Access: embargoed
Persistent link:
Language: English
Published: American Institute of Mathematical Sciences, 2022
Publish Date: 2022-10-31


A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work, we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features. For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up. The proposed methods are evaluated for three simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis.

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Series: Inverse problems and imaging
ISSN: 1930-8337
ISSN-E: 1930-8345
ISSN-L: 1930-8337
Volume: 16
Issue: 3
Pages: 483 - 523
DOI: 10.3934/ipi.2021059
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: Academy of Finland project 338408, 336796 DFG, German Research Foundation 281474342/GRK2224/1. EPSRC - EP/N022750/1 and EP/M020533/1.
Academy of Finland Grant Number: 338408
Detailed Information: 338408 (Academy of Finland Funding decision)
336796 (Academy of Finland Funding decision)
Copyright information: © 2021 IOP Publishing Ltd. This is a peer-reviewed, un-copyedited version of an article accepted for publication/published in Inverse Problems. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at