Preconditioned augmented Lagrangian method for mean curvature image deblurring

AIMS Mathematics

article

Shahbaz Ahmad¹  Faisal Fairag²  Adel M. Al-Mahdi³  Jamshaid ul Rahman¹ 

[1] ASSMS, Government College University;Department of Mathematics, King Fahd University of Petroleum & Minerals;PYP-Math, King Fahd University of Petroleum & Minerals

关键词: image deblurring;    augmented Lagrangian method;    mean curvature;    ill-posed problem;    Krylov subspace methods;    preconditioned matrix;   

DOI  :  10.3934/math.2022991

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

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.

【 授权许可】

CC BY