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

Abstract

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.

Tensor Methods in Biomedical Image Analysis

“…Tensor decomposition methods studied in this paper are limited to CP, Tucker, higher order singular value decomposition (HOSVD), TT, and TR decompositions. Some other tensor decomposition methods, such as block term decomposition (BTD) and tensor singular value decomposition (t-SVD) which have been also exploited for biomedical image analysis,[ 40 41 42 43 44 45 ] have not been reviewed in this paper. In addition, since this paper focuses on biomedical image processing methods, the tensor-based signal processing approaches have not been reviewed here.…”