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Open AccessArticle

Least-Square-Based Three-Term Conjugate Gradient Projection Method for 1-Norm Problems with Application to Compressed Sensing

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KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
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Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
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Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
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Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria
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Department of Mathematics, Usmanu Danfodiyo University, Sokoto 840004, Nigeria
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Faculty of Natural Sciences II, Institute of Mathematics, Martin Luther University Halle-Wittenberg, 06099 Halle, Germany
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 602; https://doi.org/10.3390/math8040602
Received: 27 February 2020 / Revised: 7 April 2020 / Accepted: 9 April 2020 / Published: 15 April 2020
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems 2020)
In this paper, we propose, analyze, and test an alternative method for solving the 1 -norm regularization problem for recovering sparse signals and blurred images in compressive sensing. The method is motivated by the recent proposed nonlinear conjugate gradient method of Tang, Li and Cui [Journal of Inequalities and Applications, 2020(1), 27] designed based on the least-squares technique. The proposed method aims to minimize a non-smooth minimization problem consisting of a least-squares data fitting term and an 1 -norm regularization term. The search directions generated by the proposed method are descent directions. In addition, under the monotonicity and Lipschitz continuity assumption, we establish the global convergence of the method. Preliminary numerical results are reported to show the efficiency of the proposed method in practical computation. View Full-Text
Keywords: image processing; compressed sensing; 1-norm regularization; nonlinear equations; conjugate gradient method; projection method; global convergence image processing; compressed sensing; 1-norm regularization; nonlinear equations; conjugate gradient method; projection method; global convergence
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Hassan Ibrahim, A.; Kumam, P.; Abubakar, A.B.; Abubakar, J.; Muhammad, A.B. Least-Square-Based Three-Term Conjugate Gradient Projection Method for 1-Norm Problems with Application to Compressed Sensing. Mathematics 2020, 8, 602.

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