Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images

Vladislav N. Kovalnogov; Ruslan V. Fedorov; Denis A. Demidov; Malyoshina A. Malyoshina; Theodore E. Simos; Vasilios N. Katsikis; Spyridon D. Mourtas; Romanos D. Sahas; Vladislav N. Kovalnogov; Ruslan V. Fedorov; Denis A. Demidov; Malyoshina A. Malyoshina; Theodore E. Simos; Vasilios N. Katsikis; Spyridon D. Mourtas; Romanos D. Sahas

doi:10.3934/math.2023733

AIMS Mathematics

2023, Volume 8, Issue 6: 14321-14339. doi: 10.3934/math.2023733

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Research article Special Issues

Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images

1.
Laboratory of Interdisciplinary Problems in Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung City 40402, Taiwan
3.
Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref, 32093 Kuwait
4.
Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
5.
Data Recovery Key Laboratory of Sichun Province, Neijing Normal Univ., Neijiang 641100, China
6.
Section of Mathematics, Dept. of Civil Engineering, Democritus Univ. of Thrace, Xanthi 67100, Greece
7.
Department of Economics, Division of Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian University of Athens, Sofokleous 1 Street, 10559 Athens, Greece
8.
Laboratory "Hybrid Methods of Modelling and Optimization in Complex Systems", Siberian Federal University, Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia

Received: 05 March 2023 Revised: 05 April 2023 Accepted: 06 April 2023 Published: 18 April 2023
MSC : 65F20, 68T05

The importance of quaternions in a variety of fields, such as physics, engineering and computer science, renders the effective solution of the time-varying quaternion matrix linear equation (TV-QLME) an equally important and interesting task. Zeroing neural networks (ZNN) have seen great success in solving TV problems in the real and complex domains, while quaternions and matrices of quaternions may be readily represented as either a complex or a real matrix, of magnified size. On that account, three new ZNN models are developed and the TV-QLME is solved directly in the quaternion domain as well as indirectly in the complex and real domains for matrices of arbitrary dimension. The models perform admirably in four simulation experiments and two practical applications concerning color restoration of images.
- quaternion,
- linear matrix equation,
- dynamical system,
- zeroing neural network,
- image restoration
Citation: Vladislav N. Kovalnogov, Ruslan V. Fedorov, Denis A. Demidov, Malyoshina A. Malyoshina, Theodore E. Simos, Vasilios N. Katsikis, Spyridon D. Mourtas, Romanos D. Sahas. Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images[J]. AIMS Mathematics, 2023, 8(6): 14321-14339. doi: 10.3934/math.2023733

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Abstract

The importance of quaternions in a variety of fields, such as physics, engineering and computer science, renders the effective solution of the time-varying quaternion matrix linear equation (TV-QLME) an equally important and interesting task. Zeroing neural networks (ZNN) have seen great success in solving TV problems in the real and complex domains, while quaternions and matrices of quaternions may be readily represented as either a complex or a real matrix, of magnified size. On that account, three new ZNN models are developed and the TV-QLME is solved directly in the quaternion domain as well as indirectly in the complex and real domains for matrices of arbitrary dimension. The models perform admirably in four simulation experiments and two practical applications concerning color restoration of images.

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