A preconditioned conjugate gradient method for multiplicative half-quadratic image restoration (original) (raw)
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Periodically preconditioned conjugate gradient-restoration algorithm
Journal of Optimization Theory and Applications, 1991
In this paper, the problem of minimizing a nonlinear function f ( x ) subject to a nonlinear constraint q~(x)= 0 is considered, where f is a scalar, x is an n-vector, and ~ is a q-vector, with q < n. A conjugate gradient-restoration algorithm similar to those developed by Miele et aL (Refs. 1 and 2) is employed. This particular algorithm consists of a sequence of conjugate gradient-restoration cycles. The conjugate gradient portion of each cycle is based upon a conjugate gradient algorithm that is derived for the special case of a quadratic function subject to linear constraints. This portion of the cycle involves a single step and is designed to decrease the value of the function while satisfying the constraints to first order. The restoration portion of each cycle involves one or more iterations and is designed to restore the norm of the constraint function to within a predetermined tolerance about zero.
On image restoration problems using new conjugate gradient methods
The Indonesian Journal of Electrical Engineering and Computer Science (IJEECS), 2023
The nonlinear conjugate gradient algorithm is one of the effective algorithms for optimization since it has low storage and simple structure properties. The coefficient conjugate is the basis of conjugate gradient algorithms with the desirable conjugate property. In this manuscript, we have derived a new second order information for the Hessian from objective function, which can give a new search direction. Based on new search direction, we have proposed the update formula interesting and nonlinear conjugate gradient method. Under wolfe line search and mild assumptions on objective function, the method possess sufficient descent property and are always globally convergent. Numerical results show that the method is effective and competitive to recover the original image from an image corrupted by impulse noise.
Modified three-term conjugate gradient algorithm and its applications in image restoration
Indonesian Journal of Electrical Engineering and Computer Science, 2022
In image restoration, the goal is often to bring back a high-quality version of an image from a lower-quality copy of it. In this article, we will investigate one kind of recovery issue, namely recovering photos that have been blurred by noise in digital photographs (sometimes known as "salt and pepper" noise). When subjected to noise at varying frequencies and intensities (30,50,70,90). In this paper, we used the conjugate gradient algorithm to Restorative images and remove noise from them, we developed the conjugate gradient algorithm with three limits using the conjugate condition of Dai and Liao, where the new algorithm achieved the conditions for descent and global convergence under some assumptions. According to the results of the numerical analysis, the recently created approach is unequivocally superior to both the fletcher and reeves (FR) method and the fletcher and reeves three-term (TTFR) metod. Use the structural similarity index measure (SSIM), which is used to measure image quality and the higher its value, the better the result. The original image was compared with all the noisy images and each according to the percentage of noise as well as the images processed with the four methods.
Preconditioners for image restoration by reblurring techniques
Journal of Computational and Applied Mathematics, 2013
It is well known that iterative algorithms for image deblurring that involve the normal equations show usually a slow convergence. A variant of the normal equations which replaces the conjugate transpose A H of the system matrix A with a new matrix is proposed. This approach, which is linked with regularization preconditioning theory and reblurring processes, can be applied to a wide set of iterative methods; here we examine Landweber, Steepest descent, Richardson-Lucy and Image Space Reconstruction Algorithm. Several computational tests show that this strategy leads to a significant improvement of the convergence speed of the methods. Moreover it can be naturally combined with other widely used acceleration techniques.
An Iterative Conjugate Gradient Regularization Method for Image Restoration
2009
Image restoration is an ill-posed inverse problem, which has been introduced the regularization method to suppress over-amplification. In this paper, we propose to apply the iterative regularization method to the image restoration problem and present a nested iterative method, called iterative conjugate gradient regularization method. Convergence properties are established in detail. Based on (6), we also simultaneously determine the regularization parameter based on the restored image at each step. Simulation results show that the proposed iterative regularization method is feasible and effective for image restoration.
Non-stationary Structure-Preserving Preconditioning for Image Restoration
Computational Methods for Inverse Problems in Imaging, 2019
Non-stationary regularizing preconditioners have recently been proposed for the acceleration of classical iterative methods for the solution of linear discrete ill-posed problems. This paper explores how these preconditioners can be combined with the flexible GMRES iterative method. A new structure-respecting strategy to construct a sequence of regularizing preconditioners is proposed. We show that flexible GMRES applied with these preconditioners is able to restore images that have been contaminated by strongly non-symmetric blur, while several other iterative methods fail to do this.
Mathematical Problems in Engineering, 2021
The conjugate gradient is a useful tool in solving large- and small-scale unconstrained optimization problems. In addition, the conjugate gradient method can be applied in many fields, such as engineering, medical research, and computer science. In this paper, a convex combination of two different search directions is proposed. The new combination satisfies the sufficient descent condition and the convergence analysis. Moreover, a new conjugate gradient formula is proposed. The new formula satisfies the convergence properties with the descent property related to Hestenes–Stiefel conjugate gradient formula. The numerical results show that the new search direction outperforms both two search directions, making it convex between them. The numerical result includes the number of iterations, function evaluations, and central processing unit time. Finally, we present some examples about image restoration as an application of the proposed conjugate gradient method.
Efficient Minimization Methods of Mixed l 2- l 1 and l 1- l 1 Norms for Image Restoration
Siam Journal on Scientific Computing, 2005
Image restoration problems are often solved by finding the minimizer of a suitable objective function. Usually this function consists of a data-fitting term and a regularization term. For the least squares solution, both the data-fitting and the regularization terms are in the 2 norm. In this paper, we consider the least absolute deviation (LAD) solution and the least mixed norm (LMN) solution. For the LAD solution, both the data-fitting and the regularization terms are in the 1 norm. For the LMN solution, the regularization term is in the 1 norm but the data-fitting term is in the 2 norm. Since images often have nonnegative intensity values, the proposed algorithms provide the option of taking into account the nonnegativity constraint. The LMN and LAD solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods. At each iteration of the interior point method, a structured linear system must be solved. The preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems. Experimental results are used to demonstrate the effectiveness of our approach. We also show the quality of the restored images, using the minimization of mixed 2-1 and 1-1 norms, is better than that using only the 2 norm.
Iterative Methods for Image Restoration
Although image restoration methods based on spectral filtering techniques are very efficient, they can be applied only to problems with fairly simple spatially invariant blurring operators. Iterative methods, however, are much more flexible; they can be very efficient for spatially invari- ant as well as spatially variant blurs, they can incorporate a variety of regularization techniques and boundary conditions, and they can more easily incorporate additional constraints, such as nonnegativity. This chapter describes a variety of iterative methods used in image restoration, with a particular emphasis on efficiency, convergence behavior, and implementation. Discussion of MATLAB software implementing the methods is also provided.