Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging

Kilmer, Misha E.; Braman, Karen; Hao, Ning; Hoover, Randy C.

doi:10.1137/110837711

Cited by 875 publications

(820 citation statements)

References 16 publications

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“…Using the fact that a block circulant matrix can be block-diagonalized by the Discrete Fourier Transform (DFT) [20, §4.7.7], the t-product is computable in the Fourier domain [30]. Specifically, we can compute Y = D * H by applying the DFT along tube fibers of D and H:…”

Section: Tensor Factorization Via T-productmentioning

confidence: 99%

“…Recent work by Kilmer et al [30] sets up a new theoretical framework which facilitates a straightforward extension of matrix factorizations to third-order tensors based on a new tensor multiplication definition, called the t-product. The motivation for our work is to use the t-product as a natural extension for the dictionary learning problem and image reconstruction in a third-order tensor formulation with the factorization based on the framework in [29] and [30].…”

Section: Introductionmentioning

confidence: 99%

“…The motivation for our work is to use the t-product as a natural extension for the dictionary learning problem and image reconstruction in a third-order tensor formulation with the factorization based on the framework in [29] and [30].…”

Section: Introductionmentioning

confidence: 99%

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A tensor-based dictionary learning approach to tomographic image reconstruction

2016

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We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that a) we use a third-order tensor representation for our images and b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.

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Section: Tensor Factorization Via T-productmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A tensor-based dictionary learning approach to tomographic image reconstruction

2016

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“…In this section, we review the t-product proposed by Kilmer and Martin (2011b) and further analyzed by Braman (2010) and Kilmer et al (2013). We will focus on 3D tensors for ease of exposition and interpretation.…”

Section: The T-productmentioning

confidence: 99%

5D seismic data completion and denoising using a novel class of tensor decompositions

et al. 2015

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We have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data. Most seismic surveys fail to cover all possible source-receiver combinations, leading to missing data especially in the midpoint-offset domain. This undersampling can complicate certain data processing steps such as amplitude-variationwith-offset analysis and migration. Data interpolation can mitigate the impact of missing traces. We considered the prestack data as a 5D multidimensional array or otherwise referred to as a 5D tensor. Using synthetic data sets, we first found that prestack data can be well approximated by a lowrank tensor under a recently proposed framework for tensor singular value decomposition (tSVD). Under this low-rank assumption, we proposed a complexity-penalized algorithm for the recovery of missing traces and data denoising. In this algorithm, the complexity regularization was controlled by tuning a single regularization parameter using a statistical test. We tested the performance of the proposed algorithm on synthetic and real data to show that missing data can be reliably recovered under heavy downsampling. In addition, we demonstrated that compressibility, i.e., approximation of the data by a low-rank tensor, of seismic data under tSVD depended on the velocity model complexity and shot and receiver spacing. We further found that compressibility correlated with the recovery of missing data because high compressibility implied good recovery and vice versa.

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“…Inspired by the recently reported technique called t-product [7], [1], [6], we extend SRC to its tensor-based variant while still retaining our constrained subspace model. We demonstrate in experiments that tSRC outperforms SRC and its non-tensor variant (Yang's method) in classification accuracy.…”

Section: Introductionmentioning

confidence: 99%

Supervised classification via constrained subspace and tensor sparse representation

Liao

Maybank

Zhang

et al. 2017

2017 International Joint Conference on Neural Networks (IJCNN)

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Abstract-SRC, a supervised classifier via sparse representation, has rapidly gained popularity in recent years and can be adapted to a wide range of applications based on the sparse solution of a linear system. First, we offer an intuitive geometric model called constrained subspace to explain the mechanism of SRC. The constrained subspace model connects the dots of NN, NFL, NS, NM. Then, inspired from the constrained subspace model, we extend SRC to its tensor-based variant, which takes as input samples of high-order tensors which are elements of an algebraic ring. A tensor sparse representation is used for query tensors. We verify in our experiments on several publicly available databases that the tensor-based SRC called tSRC outperforms traditional SRC in classification accuracy. Although demonstrated for image recognition, tSRC is easily adapted to other applications involving underdetermined linear systems.

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Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging

Cited by 875 publications

References 16 publications

A tensor-based dictionary learning approach to tomographic image reconstruction

A tensor-based dictionary learning approach to tomographic image reconstruction

5D seismic data completion and denoising using a novel class of tensor decompositions

Supervised classification via constrained subspace and tensor sparse representation

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