A Tensor Regularized Nuclear Norm Method for Image and Video Completion

Bentbib, Abdeslem Hafid; Hachimi, Abderrahmane El; Jbilou, Khalide; Ratnani, Ahmed

doi:10.1007/s10957-021-01947-3

Cited by 13 publications

(9 citation statements)

References 23 publications

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“…However, those decompositions suffer from the high computational cost for large problems. In the recent years new tensor decompositions of the third-order case and based on tensor-tensor product using the Fourier domain such as the t-product [12] and cosine-product (c-product) [2], have been defined and used for many image processing applications; see [1,3,9,13]. In this section we will try to remind the most important results of those types of tensor-tensor product.…”

Section: Definitions and Notationsmentioning

confidence: 99%

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

Tbib¹,

Elalj²,

Hachimi³

et al. 2022

Preprint

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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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Section: Definitions and Notationsmentioning

confidence: 99%

“…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%

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

Tbib¹,

Elalj²,

Hachimi³

et al. 2022

Preprint

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“…This concept was generalized to functions of third-order tensors in [23], based on the tensor t-product formalism [5,19,20]; see also [25] for a further extension to so-called generalized tensor functions, which are functions of tensors with non-square faces. Functions (and generalized functions) of tensors have applications in deblurring of color images [28], tensor neural networks [24,27], multilinear dynamical systems [14], and the computation of the tensor nuclear norm [4].…”

Section: Introductionmentioning

confidence: 99%

The Fréchet derivative of the tensor t-function

Lund¹,

Schweitzer²

2023

Preprint

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The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in [K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numer. Linear Algebra Appl. 27 (3), e2288]. In this work, we investigate properties of the Fréchet derivative of the tensor t-function and derive algorithms for its efficient numerical computation. Applications in condition number estimation and nuclear norm minimization are explored. Numerical experiments implemented by the t-Frechet toolbox hosted at https://gitlab.com/katlund/t-frechet illustrate properties of the t-function Fréchet derivative as well as the efficiency and accuracy of the proposed algorithms.

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“…There are two essential monographs [40,56] and plenty of papers [1,16,21,26,27,34,42,54,66] to summarize the properties and varieties of TLS method. Since the TLS was proposed, it has gained wide attention and applications in signal processing [11], data mining [24,32] and image processing [5,14,40,57].…”

mentioning

confidence: 99%

“…Tensors have an innate advantage in the regression analysis [22,31,32,35,50,67]. There are plenty of essential applications in tensor images and video modeling [3,4,5]. On the one hand, using tensor structures to store data can preserve the spatial structure properties of higher-order data as much as possible [68].…”

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confidence: 99%

Tensor Regularized Total Least Squares Methods with Applications to Image and Video Deblurring

Han¹,

Wei²

2022

Preprint

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Total least squares (TLS) is an effective method for solving linear equations with the situations, when noise is not just in observation matrices but also in mapping matrices. Moreover, the Tikhonov regularization is widely used in plenty of ill-posed problems. In this paper, we extend the regularized total least squares (RTLS) method from the matrix form due to Golub, Hansen and O'Leary, to the tensor form proposing the tensor regularized total least squares (TR-TLS) method for solving ill-conditioned tensor systems of equations. Properties and algorithms about the solution of the TR-TLS problem, which might be similar to those of the RTLS, are also presented and proved. Based on this method, some applications in image and video deblurring are explored. Numerical examples illustrate the TR-TLS, compared with the existing methods.

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A Tensor Regularized Nuclear Norm Method for Image and Video Completion

Cited by 13 publications

References 23 publications

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

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

The Fréchet derivative of the tensor t-function

Tensor Regularized Total Least Squares Methods with Applications to Image and Video Deblurring

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