Two new preconditioners for mean curvature-based image deblurring problem

Ahmad, Shahbaz; Al‐Mahdi, Adel M.; Ahmed, Rashad

doi:10.3934/math.2021802

Cited by 11 publications

(2 citation statements)

References 18 publications

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“…Hence, evaluating this product must be cheap. In the literature, several preconditioners 26 – 34 are developed for the nonlinear systems. In this study, we consider the following non-linear system of equations This system is derived by discretizing the EL equations associated with the TFOV based image deblurring problem.…”

Section: Introductionmentioning

confidence: 99%

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

Kim,

Ahmad

2023

Sci Rep

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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.

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Section: Introductionmentioning

confidence: 99%

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

Kim,

Ahmad

2023

Sci Rep

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“…To remove the stair case effects, two modifications to the TV regularization model have been used in the literature. The first approach is to higher the order of the derivatives in the TV regularization term, such as the mean curvature or a nonlinear combination of the first and second derivatives [40][41][42][43][44][45]. These modifications remove/reduce the staircase effects and they are effective but they are computationally expensive due to the increasing the order of the derivatives or due to the nonlinearity terms.…”

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confidence: 99%

Preconditioning Technique for an Image Deblurring Problem with the Total Fractional-Order Variation Model

Al-Mahdi

2023

MCA

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Total fractional-order variation (TFOV) in image deblurring problems can reduce/remove the staircase problems observed with the image deblurring technique by using the standard total variation (TV) model. However, the discretization of the Euler–Lagrange equations associated with the TFOV model generates a saddle point system of equations where the coefficient matrix of this system is dense and ill conditioned (it has a huge condition number). The ill-conditioned property leads to slowing of the convergence of any iterative method, such as Krylov subspace methods. One treatment for the slowness property is to apply the preconditioning technique. In this paper, we propose a block triangular preconditioner because we know that using the exact triangular preconditioner leads to a preconditioned matrix with exactly two distinct eigenvalues. This means that we need at most two iterations to converge to the exact solution. However, we cannot use the exact preconditioner because the Shur complement of our system is of the form S=K*K+λLα which is a huge and dense matrix. The first matrix, K*K, comes from the blurred operator, while the second one is from the TFOV regularization model. To overcome this difficulty, we propose two preconditioners based on the circulant and standard TV matrices. In our algorithm, we use the flexible preconditioned GMRES method for the outer iterations, the preconditioned conjugate gradient (PCG) method for the inner iterations, and the fixed point iteration (FPI) method to handle the nonlinearity. Fast convergence was found in the numerical results by using the proposed preconditioners.

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Optimizing convergence: GMRES preconditioner for Darcy flow problem in a fracture network

Ahmad,

Khan

2024

Comput Geosci

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Two new preconditioners for mean curvature-based image deblurring problem

Cited by 11 publications

References 18 publications

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

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

Preconditioning Technique for an Image Deblurring Problem with the Total Fractional-Order Variation Model

Optimizing convergence: GMRES preconditioner for Darcy flow problem in a fracture network

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