An effective double direction method for solving convex-constrained nonlinear equations with image restoration

Muhammad Abdullahi; Abdullah Al-Yaari; Auwal Bala Abubakar; Abubakar Sani Halilu; Muhamad Afendee Muhamed; Muhammad Abdullahi; Abdullah Al-Yaari; Auwal Bala Abubakar; Abubakar Sani Halilu; Muhamad Afendee Muhamed

doi:10.3934/math.20251273

AIMS Mathematics

2025, Volume 10, Issue 12: 28926-28953. doi: 10.3934/math.20251273

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Research article

An effective double direction method for solving convex-constrained nonlinear equations with image restoration

1.
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China
2.
Mathematical Innovation and Applications Research Group, Department of Mathematics, Sule Lamido University Kafin Hausa, PMB 048, Nigeria
3.
Department of Art and Science, George Mason University, Songdomunhwa-ro 119-4 Yeonsu-gu, Incheon 21985, Korea
4.
Numerical Optimization Research Group, Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University Kano, Kano, Nigeria
5.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa-0204, South Africa
6.
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu 21300, Malaysia

Received: 04 August 2025 Revised: 15 September 2025 Accepted: 13 October 2025 Published: 10 December 2025
MSC : 65K05, 90C52, 90C56, 94A08

The restoration of medical imaging has recently gained significant attention due to its critical role in the medical sciences. This research introduces two efficient double-direction approaches for solving large-scale monotone nonlinear equations and addressing medical image restoration problems. The acceleration parameter in the first algorithm was derived by approximating the Jacobian matrix with a diagonal matrix, while the second algorithm integrates a correction parameter using the Picard-Mann hybrid iterative procedure. The study established the descent condition and demonstrated both the global convergence and R-linear convergence rate of the proposed method under favorable conditions. Numerical experiments showed that the procedure significantly enhances the numerical performance of the proposed methods. Additionally, one of the techniques was successfully applied to restore blurred medical images, highlighting its practical applicability in the field of medical imaging.
- medical image,
- convex-constrained,
- acceleration parameter,
- Picard-Mann,
- iterative procedure
Citation: Muhammad Abdullahi, Abdullah Al-Yaari, Auwal Bala Abubakar, Abubakar Sani Halilu, Muhamad Afendee Muhamed. An effective double direction method for solving convex-constrained nonlinear equations with image restoration[J]. AIMS Mathematics, 2025, 10(12): 28926-28953. doi: 10.3934/math.20251273

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Abstract

The restoration of medical imaging has recently gained significant attention due to its critical role in the medical sciences. This research introduces two efficient double-direction approaches for solving large-scale monotone nonlinear equations and addressing medical image restoration problems. The acceleration parameter in the first algorithm was derived by approximating the Jacobian matrix with a diagonal matrix, while the second algorithm integrates a correction parameter using the Picard-Mann hybrid iterative procedure. The study established the descent condition and demonstrated both the global convergence and R-linear convergence rate of the proposed method under favorable conditions. Numerical experiments showed that the procedure significantly enhances the numerical performance of the proposed methods. Additionally, one of the techniques was successfully applied to restore blurred medical images, highlighting its practical applicability in the field of medical imaging.

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