Two new preconditioners for mean curvature-based image deblurring problem

Shahbaz Ahmad; Adel M. Al-Mahdi; Rashad Ahmed; Shahbaz Ahmad; Adel M. Al-Mahdi; Rashad Ahmed

doi:10.3934/math.2021802

AIMS Mathematics

2021, Volume 6, Issue 12: 13824-13844. doi: 10.3934/math.2021802

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Research article

Two new preconditioners for mean curvature-based image deblurring problem

1.
Department of Mathematics & Statistic, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia
2.
The Preparatory Year Program and The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
3.
Research Institute, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia

Received: 30 July 2021 Accepted: 21 September 2021 Published: 27 September 2021
MSC : 65N06, 65N12, 68U10, 94A08

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.
- image deblurring,
- mean curvature,
- ill-posed problem,
- numerical analysis,
- preconditioning
Citation: Shahbaz Ahmad, Adel M. Al-Mahdi, Rashad Ahmed. Two new preconditioners for mean curvature-based image deblurring problem[J]. AIMS Mathematics, 2021, 6(12): 13824-13844. doi: 10.3934/math.2021802

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Abstract

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.

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