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22 pages, 15185 KB

Open AccessArticle

Low Tensor Rank Constrained Image Inpainting Using a Novel Arrangement Scheme

by Shuli Ma, Youchen Fan, Shengliang Fang, Weichao Yang and Li Li

Appl. Sci. 2025, 15(1), 322; https://doi.org/10.3390/app15010322 - 31 Dec 2024

Viewed by 1163

Employing low tensor rank decomposition in image inpainting has attracted increasing attention. This study exploited novel tensor arrangement schemes to transform an image (a low-order tensor) to a higher-order tensor without changing the total number of pixels. The developed arrangement schemes enhanced the low rankness of images under three tensor decomposition methods: matrix SVD, tensor train (TT) decomposition, and tensor singular value decomposition (t-SVD). By exploiting the schemes, we solved the image inpainting problem with three low-rank constrained models that use the matrix rank, TT rank, and tubal rank as constrained priors. The tensor tubal rank and tensor train multi-rank were developed from t-SVD and TT decomposition, respectively. Then, ADMM algorithms were efficiently exploited for solving the three models. Experimental results demonstrate that our methods are effective for image inpainting and superior to numerous close methods. Full article

(This article belongs to the Special Issue AI-Based Image Processing: 2nd Edition)

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46 pages, 25331 KB

Open AccessArticle

Guaranteed Robust Tensor Completion via ∗_L-SVD with Applications to Remote Sensing Data

by Andong Wang, Guoxu Zhou and Qibin Zhao

Remote Sens. 2021, 13(18), 3671; https://doi.org/10.3390/rs13183671 - 14 Sep 2021

Cited by 7 | Viewed by 4516

Abstract

This paper conducts a rigorous analysis for the problem of robust tensor completion, which aims at recovering an unknown three-way tensor from incomplete observations corrupted by gross sparse outliers and small dense noises simultaneously due to various reasons such as sensor dead pixels, communication loss, electromagnetic interferences, cloud shadows, etc. To estimate the underlying tensor, a new penalized least squares estimator is first formulated by exploiting the low rankness of the signal tensor within the framework of tensor

*_{L}

-Singular Value Decomposition (

*_{L}

-SVD) and leveraging the sparse structure of the outlier tensor. Then, an algorithm based on the Alternating Direction Method of Multipliers (ADMM) is designed to compute the estimator in an efficient way. Statistically, the non-asymptotic upper bound on the estimation error is established and further proved to be optimal (up to a log factor) in a minimax sense. Simulation studies on synthetic data demonstrate that the proposed error bound can predict the scaling behavior of the estimation error with problem parameters (i.e., tubal rank of the underlying tensor, sparsity of the outliers, and the number of uncorrupted observations). Both the effectiveness and efficiency of the proposed algorithm are evaluated through experiments for robust completion on seven different types of remote sensing data. Full article

(This article belongs to the Special Issue Remote Sensing Image Denoising, Restoration and Reconstruction)

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17 pages, 1694 KB

Open AccessArticle

A Multidimensional Principal Component Analysis via the C-Product Golub–Kahan–SVD for Classification and Face Recognition

by Mustapha Hached, Khalide Jbilou, Christos Koukouvinos and Marilena Mitrouli

Mathematics 2021, 9(11), 1249; https://doi.org/10.3390/math9111249 - 29 May 2021

Cited by 13 | Viewed by 2982

Abstract

Face recognition and identification are very important applications in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to the mere structure of the databases, for example in the case of color images. Nevertheless, even though various authors proposed factorization strategies for tensors, the size of the considered tensors can pose some serious issues. Indeed, the most demanding part of the computational effort in recognition or identification problems resides in the training process. When only a few features are needed to construct the projection space, there is no need to compute a SVD on the whole data. Two versions of the tensor Golub–Kahan algorithm are considered in this manuscript, as an alternative to the classical use of the tensor SVD which is based on truncated strategies. In this paper, we consider the Tensor Tubal Golub–Kahan Principal Component Analysis method which purpose it to extract the main features of images using the tensor singular value decomposition (SVD) based on the tensor cosine product that uses the discrete cosine transform. This approach is applied for classification and face recognition and numerical tests show its effectiveness. Full article

(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)

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31 pages, 4841 KB

Open AccessArticle

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

by Mohamed Ibrahim Assoweh, Stéphane Chrétien and Brahim Tamadazte

Mathematics 2020, 8(4), 628; https://doi.org/10.3390/math8040628 - 18 Apr 2020

Cited by 1 | Viewed by 3391

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. Full article

(This article belongs to the Special Issue New Trends in Machine Learning: Theory and Practice)

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