ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Abdeslem H. Bentbib,
Asmaa Khouia,
Hassane Sadok
S. 92 - 111 doi:10.1553/etna_vol55s92 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol55s92
Abstract: In this paper, we are interested in finding an approximate solution $ \hat{\mathcal{X}} $ of the tensor least-squares minimization problem$ \min_{\mathcal{X}}\left\|\mathcal{X}\times_1A^{(1)}\times_2A^{(2)}\times_3\cdots\times_NA^{(N)}-\mathcal{G}\right\|$, where $ \mathcal{G}\in \mathbb{R}^{J_1\times J_2\times \cdots \times J_N}$ and $ A^{(i)}\in \mathbb{R}^{J_i\times I_i} $ ($ i=1,\ldots,N $) are known and $ \mathcal{X}\in \mathbb{R}^{I_1\times I_2\times \cdots \times I_N} $ is the unknown tensor to be approximated. Our approach is based on two steps. Firstly, we apply the CP or HOSVD decomposition to the right-hand side tensor $ \mathcal{G} $. Secondly, we perform the well-known Golub-Kahan bidiagonalization for each coefficient matrix $ A^{(i)} $($ i=1,\ldots, N $) to obtain a reduced tensor least-squares minimization problem. This type of equations may appear in color image and video restorations as we described below. Some numerical tests are performed to show the effectiveness of our proposed method. Keywords: HOSVD, CP decomposition, color image restoration, video restoration, LSQR Published Online: 2021/11/04 11:14:02 Object Identifier: 0xc1aa5572 0x003cfb20 Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613. …
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |