On the preconditioning of the primal form of TFOV-based image deblurring model

To address the staircasing problem in deblurred images generated by a simple total variation (TV) based model, one approach is to use the total fractional-order variation (TFOV) image deblurring model. However, the discretization of the Euler-Lagrange equations for the TFOV-based model results in a nonlinear ill-conditioned system, which adversely influences the performance of computational methods like Krylov subspace algorithms (e.g., Generalized Minimal Residual, Conjugate Gradient). To address this challenge, three novel preconditioned matrices are proposed to improve the conditioning of the primal model when using the conjugate gradient method. These matrices are designed based on circulant approximations of the matrix associated with blurring kernel. Experimental evaluations demonstrate the effectiveness of the proposed preconditioned matrices in enhancing the convergence and accuracy of the conjugate gradient method for solving the primal form of the TFOV-based image deblurring model. The results highlight the importance of appropriate preconditioning strategies in achieving robust and high-quality image deblurring using the TFOV approach.