Preconditioned augmented Lagrangian method for mean curvature image deblurring

Shahbaz Ahmad; Faisal Fairag; Adel M. Al-Mahdi; Jamshaid ul Rahman; Shahbaz Ahmad; Faisal Fairag; Adel M. Al-Mahdi; Jamshaid ul Rahman

doi:10.3934/math.2022991

AIMS Mathematics

2022, Volume 7, Issue 10: 17989-18009. doi: 10.3934/math.2022991

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Preconditioned augmented Lagrangian method for mean curvature image deblurring

1.
ASSMS, Government College University, Lahore 54000, Pakistan
2.
Department of Mathematics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
3.
PYP-Math, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

Received: 20 April 2022 Revised: 07 July 2022 Accepted: 21 July 2022 Published: 05 August 2022
MSC : 68U10, 94A08, 65K10, 65N12

Image deblurring models with a mean curvature functional has been widely used to preserve edges and remove the staircase effect in the resulting images. However, the Euler-Lagrange equations of a mean curvature model can be used to solve fourth-order non-linear integro-differential equations. Furthermore, the discretization of fourth-order non-linear integro-differential equations produces an ill-conditioned system so that the numerical schemes like Krylov subspace methods (conjugate gradient etc.) have slow convergence. In this paper, we propose an augmented Lagrangian method for a mean curvature-based primal form of the image deblurring problem. A new circulant preconditioned matrix is introduced to overcome the problem of slow convergence when employing a conjugate gradient method inside of the augmented Lagrangian method. By using the proposed new preconditioner fast convergence has been observed in the numerical results. Moreover, a comparison with the existing numerical methods further reveal the effectiveness of the preconditioned augmented Lagrangian method.
- image deblurring,
- augmented Lagrangian method,
- mean curvature,
- ill-posed problem,
- Krylov subspace methods,
- preconditioned matrix
Citation: Shahbaz Ahmad, Faisal Fairag, Adel M. Al-Mahdi, Jamshaid ul Rahman. Preconditioned augmented Lagrangian method for mean curvature image deblurring[J]. AIMS Mathematics, 2022, 7(10): 17989-18009. doi: 10.3934/math.2022991

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Abstract

Image deblurring models with a mean curvature functional has been widely used to preserve edges and remove the staircase effect in the resulting images. However, the Euler-Lagrange equations of a mean curvature model can be used to solve fourth-order non-linear integro-differential equations. Furthermore, the discretization of fourth-order non-linear integro-differential equations produces an ill-conditioned system so that the numerical schemes like Krylov subspace methods (conjugate gradient etc.) have slow convergence. In this paper, we propose an augmented Lagrangian method for a mean curvature-based primal form of the image deblurring problem. A new circulant preconditioned matrix is introduced to overcome the problem of slow convergence when employing a conjugate gradient method inside of the augmented Lagrangian method. By using the proposed new preconditioner fast convergence has been observed in the numerical results. Moreover, a comparison with the existing numerical methods further reveal the effectiveness of the preconditioned augmented Lagrangian method.

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