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

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.