On tensor GMRES and Golub-Kahan methods via the T-product for color image processing

Guide, M. El; Ichi, Alaa El; Jbilou, Khalide; Sadaka, Rachid

doi:10.13001/ela.2021.5471

Cited by 24 publications

(13 citation statements)

References 31 publications

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“…The operator * described below denotes the t-product introduced in the seminal work by Kilmer and Martin [17]. This product has become ubiquitous in tensor literature applications; see, e.g., facial recognition [13], tomographic image reconstruction [24], video completion [32], image classification [21], and image deblurring [8,16,17,25,26,27,28]. Throughout this paper, A F denotes the Frobenius norm of a third order tensor A.…”

Section: Introductionmentioning

confidence: 99%

“…Computed examples presented in [25,26,27] illustrate the merits of tensorizing over matricizing or vectorizing ill-posed tensor equations. Solution methods described by El Guide et al [8] involve flattening, i.e., they reduce (1.1) to an equivalent equation involving a matrix and a vector. Structure preserving and other techniques for regularizing (1.1) by Tikhonov's approach with the t-product are described and compared in [25,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

“…These kinds of problems arise in color image and video restorations. Solution methods for (1.6) by Tikhonov regularization with the t-product Arnoldi (t-Arnoldi) and tGKB processes are discussed in [8,25,26,27].…”

Section: Introductionmentioning

confidence: 99%

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Tensor regularization by truncated iteration: a comparison of some solution methods for large-scale linear discrete ill-posed problem with a t-product

Ugwu¹,

Reichel²

2021

Preprint

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This paper describes and compares some structure preserving techniques for the solution of linear discrete ill-posed problems with the t-product. A new randomized tensor singular value decomposition (R-tSVD) with a t-product is presented for low tubal rank tensor approximations. Regularization of linear inverse problems by truncated tensor eigenvalue decomposition (T-tEVD), truncated tSVD (T-tSVD), randomized T-tSVD (RT-tSVD), t-product Golub-Kahan bidiagonalization (tGKB) process, and t-product Lanczos (t-Lanczos) process are considered. A solution method that is based on reusing tensor Krylov subspaces generated by the tGKB process is described. The regularization parameter is the number of iterations required by each method. The discrepancy principle is used to determine this parameter. Solution methods that are based on truncated iterations are compared with solution methods that combine Tikhonov regularization with the tGKB and t-Lanczos processes. Computed examples illustrate the performance of these methods when applied to image and gray-scale video restorations. Our new RT-tSVD method is seen to require less CPU time and yields restorations of higher quality than the T-tSVD method.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tensor regularization by truncated iteration: a comparison of some solution methods for large-scale linear discrete ill-posed problem with a t-product

Ugwu¹,

Reichel²

2021

Preprint

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“…Introduction. In the last decade, tensors become an important multilinear algebra tool involved in many modern problems such completion [9,18,24], principal component analysis [13], image processing [20,12,4] and others. The classical n-mode product leads to many concepts and developements when working with multidimensional data.…”

mentioning

confidence: 99%

“…The CP and the Tucker compressions were introduced as natural generalization of the classical singular value decomposition (SVD) for matrices; see [15,4,12,13,24]. In the last years, new tensor-tensor products such as cosine-product (c-product), using discrete cosine or T-product, using Fast Fourier Transform (FFT), were introduced for third-order tensors, studied and applied to image processing and other fields; see [19,1,25,13,20]. In the present paper, we generalize those tensor-tensor products for high-order tensors.…”

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confidence: 99%

Generalized L-product for hight order tensors with applications using GPU computation

Tbib¹,

Elalj²,

Hachimi³

et al. 2022

Preprint

Self Cite

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In this paper, we will present a generalization of the L-tensor product ( * L -product) including generalization of the well known tensor cosine and T-products that were defined for third-order tensors and based on fast Fourier transform and discrete cosine transform (DCT). We will give some applications on tensor completion. To solve some optimization problems linked with the problem of tensor completion, we will use the Proximal Gradient Algorithm (PGA) to solve some derived optimization problems. Numerical tests are given to show the effectiveness of the proposed methods and also present some tests using GPU computation.

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The tensor Golub–Kahan–Tikhonov method applied to the solution of ill‐posed problems with a t‐product structure

Reichel

Ugwu

2021

Numerical Linear Algebra App

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This paper discusses an application of partial tensor Golub-Kahan bidiagonalization to the solution of large-scale linear discrete ill-posed problems based on the t-product formalism for third-order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658). The solution methods presented first reduce a given (large-scale) problem to a problem of small size by application of a few steps of tensor Golub-Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third-order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third-order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill-posed problems defined by a matrix operator to linear discrete ill-posed problems defined by a third-order tensor operator. An interlacing property of singular tubes for third-order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t-product approach, in comparison with solution methods that are based on matricization of the tensor equation. K E Y W O R D Sdiscrepancy principle, discrete ill-posed problem, t-product, tensor Golub-Kahan bidiagonalization, Tikhonov regularization 1 where  = [a ijk ] 𝓁,m,n i,j,k=1 ∈ R 𝓁×m×n is a third-order tensor, ⃗  ∈ R m×1×n and ⃗  ∈ R 𝓁×1×n are lateral slices of third-order tensors and may be thought of as laterally oriented matrices. The operation * denotes the tensor t-product introduced in the seminal work by Kilmer and Martin 1 and applied to image deblurring problems by Kilmer et al. 1,2 The t-product between

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On tensor GMRES and Golub-Kahan methods via the T-product for color image processing

Cited by 24 publications

References 31 publications

Tensor regularization by truncated iteration: a comparison of some solution methods for large-scale linear discrete ill-posed problem with a t-product

Tensor regularization by truncated iteration: a comparison of some solution methods for large-scale linear discrete ill-posed problem with a t-product

Generalized L-product for hight order tensors with applications using GPU computation

The tensor Golub–Kahan–Tikhonov method applied to the solution of ill‐posed problems with a t‐product structure

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