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Spectrally Sparse Tensor Reconstruction in Optical Coherence Tomography Using Nuclear Norm Penalisation

Laboratoire de Mathématiques de Besançon, Université de Bourgogne-Franche Comté, 25030 Besançon, France
Laboratoire ERIC, UFR ASSP, Université Lyon 2, 69676 Bron, France
National Physical Laboratory, Teddington TW11 0LW, UK
The Alan Turing Institute, London NW1 2DB, UK
Institute for Intelligent Systems and Robotics, University of Sorbonne, CNRS, UMR 7222, 4 pl. Jussieu, 75005 Paris, France
FEMTO-ST Institute, University of Bourgogne-Franche Comté, CNRS, 25000 Besançon, France
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 628;
Received: 24 January 2020 / Revised: 3 April 2020 / Accepted: 7 April 2020 / Published: 18 April 2020
(This article belongs to the Special Issue New Trends in Machine Learning: Theory and Practice)
Reconstruction of 3D objects in various tomographic measurements is an important problem which can be naturally addressed within the mathematical framework of 3D tensors. In Optical Coherence Tomography, the reconstruction problem can be recast as a tensor completion problem. Following the seminal work of Candès et al., the approach followed in the present work is based on the assumption that the rank of the object to be reconstructed is naturally small, and we leverage this property by using a nuclear norm-type penalisation. In this paper, a detailed study of nuclear norm penalised reconstruction using the tubal Singular Value Decomposition of Kilmer et al. is proposed. In particular, we introduce a new, efficiently computable definition of the nuclear norm in the Kilmer et al. framework. We then present a theoretical analysis, which extends previous results by Koltchinskii Lounici and Tsybakov. Finally, this nuclear norm penalised reconstruction method is applied to real data reconstruction experiments in Optical Coherence Tomography (OCT). In particular, our numerical experiments illustrate the importance of penalisation for OCT reconstruction. View Full-Text
Keywords: tensor completion; tubal SVD; nuclear norm penalisation tensor completion; tubal SVD; nuclear norm penalisation
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Assoweh, M.I.; Chrétien, S.; Tamadazte, B. Spectrally Sparse Tensor Reconstruction in Optical Coherence Tomography Using Nuclear Norm Penalisation. Mathematics 2020, 8, 628.

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