Efficient multigrid method for dense regularization in fractional-order image deblurring

Shahbaz Ahmad; Shahid Saleem; Abid Iqbal; Saad Arif; Shahbaz Ahmad; Shahid Saleem; Abid Iqbal; Saad Arif

doi:10.3934/math.2026419

AIMS Mathematics

2026, Volume 11, Issue 4: 10159-10174. doi: 10.3934/math.2026419

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Efficient multigrid method for dense regularization in fractional-order image deblurring

1.
Abdus Salam School of Mathematical Sciences, Government College University, Lahore, 54600, Pakistan
2.
Department of Mathematics, The University of Chenab, Gujrat, 50700, Pakistan
3.
Department of Computer Engineering, College of Computer Sciences and Information Technology, King Faisal University, Al-Ahsa, 31982, Saudi Arabia
4.
Department of Mechanical Engineering, College of Engineering, King Faisal University, Al-Ahsa, 31982, Saudi Arabia

Received: 10 December 2025 Revised: 28 March 2026 Accepted: 31 March 2026 Published: 14 April 2026
MSC : 94A08, 65N55, 68U10, 65N12

Fractional-order models have emerged as powerful tools in real-world image processing, offering enhanced edge preservation and improved reconstruction quality compared to traditional integer-order methods. The total fractional-order variation (TFOV) model is employed in this work to enhance the quality of deblurred images, as it is well known for its ability to preserve edges and mitigate the staircase effect. However, the dense regularization matrix associated with the TFOV model poses challenges in the design of efficient numerical algorithms. To overcome this issue, a robust and efficient multigrid method specifically designed to address matrix density is proposed. This approach develops multigrid solvers tailored for image deblurring problems governed by TFOV regularization under Dirichlet boundary conditions. Using a Lagrange multiplier framework, the optimality system for the image deblurring problem is derived, linking the state, adjoint, and intensity variables. Multigrid techniques on staggered grids are explored, employing a coarsening factor of three to generate a nested hierarchy that facilitates simplified intergrid transfer operations. Within this framework, a preconditioned conjugate gradient method is used as a smoother. Numerical experiments confirm the accuracy and efficiency of the proposed multigrid approach. These results highlight the practical viability of multigrid solvers for dense fractional regularization and reinforce their usefulness in large-scale image restoration tasks.
- image deblurring,
- fractional-order variation,
- multigrid methods,
- inverse problems,
- image restoration,
- preconditioned conjugate gradient,
- staggered grids,
- edge preservation
Citation: Shahbaz Ahmad, Shahid Saleem, Abid Iqbal, Saad Arif. Efficient multigrid method for dense regularization in fractional-order image deblurring[J]. AIMS Mathematics, 2026, 11(4): 10159-10174. doi: 10.3934/math.2026419

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Abstract

Fractional-order models have emerged as powerful tools in real-world image processing, offering enhanced edge preservation and improved reconstruction quality compared to traditional integer-order methods. The total fractional-order variation (TFOV) model is employed in this work to enhance the quality of deblurred images, as it is well known for its ability to preserve edges and mitigate the staircase effect. However, the dense regularization matrix associated with the TFOV model poses challenges in the design of efficient numerical algorithms. To overcome this issue, a robust and efficient multigrid method specifically designed to address matrix density is proposed. This approach develops multigrid solvers tailored for image deblurring problems governed by TFOV regularization under Dirichlet boundary conditions. Using a Lagrange multiplier framework, the optimality system for the image deblurring problem is derived, linking the state, adjoint, and intensity variables. Multigrid techniques on staggered grids are explored, employing a coarsening factor of three to generate a nested hierarchy that facilitates simplified intergrid transfer operations. Within this framework, a preconditioned conjugate gradient method is used as a smoother. Numerical experiments confirm the accuracy and efficiency of the proposed multigrid approach. These results highlight the practical viability of multigrid solvers for dense fractional regularization and reinforce their usefulness in large-scale image restoration tasks.

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