Tensor factorization and its application to multidimensional seismic data recovery

Sacchi, Mauricio D.; Gao, Jianjun; Stanton, Aaron; Cheng, Jinkun

doi:10.1190/segam2015-5884504.1

Cited by 5 publications

(5 citation statements)

References 9 publications

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“…Those two operations along mode ‐ n are designated as follows:

\begin{matrix} true \\ right {boldD}_{(n)} & = & left {unfold}_{n} (bold-scriptD), \\ right bold-scriptD & = & left {fold}_{n} ({boldD}_{(n)}), n = 1, \dots, N . \end{matrix}

Figure shows the schematic representation of unfolding and folding for a third‐order tensor (Kolda and Bader ; Sacchi et al . ). The Hadamard product is defined as the element‐wise product of two matrices of the same size.…”

Section: Methodsmentioning

confidence: 97%

“…(2) Figure 1 shows the schematic representation of unfolding and folding for a third-order tensor (Kolda and Bader 2009;Sacchi et al 2015). The Hadamard product is defined as the elementwise product of two matrices of the same size.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…In fact, Sacchi (2011, 2012) adopted the HO-SVD for data denoising and five-dimensional seismic data reconstruction. Alternatively, one can utilize the parallel matrix factorization method to represent a tensor with low-rank unfoldings (Xu et al 2015;Sacchi et al 2015). Sidiropoulos et al (2017) provided a thorough overview of the applications of tensor techniques to signal processing, statistical data analysis, data mining and machine learning.…”

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confidence: 99%

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Random noise attenuation via the randomized canonical polyadic decomposition

Gao

Sacchi

2019

Geophysical Prospecting

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A B S T R A C TTensor algebra provides a robust framework for multi-dimensional seismic data processing. A low-rank tensor can represent a noise-free seismic data volume. Additive random noise will increase the rank of the tensor. Hence, tensor rank-reduction techniques can be used to filter random noise. Our filtering method adopts the Candecomp/Parafac decomposition to approximates a N-dimensional seismic data volume via the superposition of rank-one tensors. Similar to the singular value decomposition for matrices, a low-rank Candecomp/Parafac decomposition can capture the signal and exclude random noise in situations where a low-rank tensor can represent the ideal noise-free seismic volume. The alternating least squares method is adopted to compute the Candecomp/Parafac decomposition with a provided target rank. This method involves solving a series of highly over-determined linear least-squares subproblems. To improve the efficiency of the alternating least squares algorithm, we uniformly randomly sample equations of the linear least-squares subproblems to reduce the size of the problem significantly. The computational overhead is further reduced by avoiding unfolding and folding large dense tensors. We investigate the applicability of the randomized Candecomp/Parafac decomposition for incoherent noise attenuation via experiments conducted on a synthetic dataset and field data seismic volumes. We also compare the proposed algorithm (randomized Candecomp/Parafac decomposition) against multi-dimensional singular spectrum analysis and classical f − xy prediction filtering. We conclude the proposed approach can achieve slightly better denoising performance in terms of signal-to-noise ratio enhancement than traditional methods, but with a less computational cost.

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“…Those two operations along mode ‐ n are designated as follows:

\begin{matrix} true \\ right {boldD}_{(n)} & = & left {unfold}_{n} (bold-scriptD), \\ right bold-scriptD & = & left {fold}_{n} ({boldD}_{(n)}), n = 1, \dots, N . \end{matrix}

Section: Methodsmentioning

confidence: 97%

Section: Preliminaries and Notationmentioning

confidence: 99%

mentioning

confidence: 99%

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Random noise attenuation via the randomized canonical polyadic decomposition

Gao

Sacchi

2019

Geophysical Prospecting

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“…The main problem with tensor recovery is that the spectrum is not a well defined quantity and several approaches have been proposed for defining it [4,5]. Different nuclear norms have also been proposed in the literature, based on the various definitions for the spectrum [6][7][8]. A very interesting approach was proposed in References [9,10].…”

Section: Motivations and Contributionsmentioning

confidence: 99%

“…Tensors play a fundamental role in statistics [24] and more recently in Machine Learning [13]. It can be used for multidimensional time series [25], analysis of seismic data [6], Hidden Markov Models [13], Gaussian Mixture based clustering [26], Phylogenetics [27], and much more. Tensor completion is however a more difficult problem from many points of view.…”

Section: Tensor Completionmentioning

confidence: 99%

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

2020

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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.

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