Two-Level method for the total fractional-order variation model in image deblurring problem

Fairag, Faisal; Al‐Mahdi, Adel M.; Ahmad, Shahbaz

doi:10.1007/s11075-019-00845-0

Cited by 10 publications

(8 citation statements)

References 34 publications

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“…These studies have shown that, compared to first-order and second-order total variation methods, TFOV can more accurately and delicately represent image textures. More recently, Fairag et al [39] and Guo et al [30] have incorporated the TFOV model within the framework of image deblurring problems, further highlighting the applicability and potential of the TFOV-based approaches.…”

Section: Related Workmentioning

confidence: 99%

“…The values of α = 1.8, α = 1 × 10 −8 and β = 0.1 are chosen according to [39]. In all experiments, the stopping criterion for the numerical iterations is defined as b − Ax k < tol b , where x k = (v k , u k ) is the solution vector in the k-th iteration.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Here, we have compared our algorithms (OSCM and TSCM) with the TFOV-based methods by Fairag et al [39], in which they created the one-level method (OLM) and the two-level method (TLM) for the TFOV-based image deblurring problem. Two different kinds of images were used in this investigation: Goldhills (actual) and Cameraman (complicated).…”

Section: Examplementioning

confidence: 99%

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Total Fractional-Order Variation-Based Constraint Image Deblurring Problem

2023

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When deblurring an image, ensuring that the restored intensities are strictly non-negative is crucial. However, current numerical techniques often fail to consistently produce favorable results, leading to negative intensities that contribute to significant dark regions in the restored images. To address this, our study proposes a mathematical model for non-blind image deblurring based on total fractional-order variational principles. Our proposed model not only guarantees strictly positive intensity values but also imposes limits on the intensities within a specified range. By removing negative intensities or constraining them within the prescribed range, we can significantly enhance the quality of deblurred images. The key concept in this paper involves converting the constrained total fractional-order variational-based image deblurring problem into an unconstrained one through the introduction of the augmented Lagrangian method. To facilitate this conversion and improve convergence, we describe new numerical algorithms and introduce a novel circulant preconditioned matrix. This matrix effectively overcomes the slow convergence typically encountered when using the conjugate gradient method within the augmented Lagrangian framework. Our proposed approach is validated through computational tests, demonstrating its effectiveness and viability in practical applications.

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Section: Related Workmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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Total Fractional-Order Variation-Based Constraint Image Deblurring Problem

2023

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“…This concept is generalized to the ordinary derivative to fractional numbers [15]. This tool has been used in recent years in various articles on image processing as [16,17]. More details of this concept are discussed in the following sections.…”

Section: Introductionmentioning

confidence: 99%

Point spread function estimation for blind image deblurring problems based on framelet transform

Parvaz¹

2021

Preprint

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One of the most important issues in the image processing is the approximation of the image that has been lost due to the blurring process. These types of matters are divided into non-blind and blind problems. The second type of problem is more complex in terms of calculations than the first problems due to the unknown of original image and point spread function estimation. In the present paper, an algorithm based on coarse-to-fine iterative by l 0 − αl 1 regularization and framelet transform is introduced to approximate the spread function estimation. Framelet transfer improves the restored kernel due to the decomposition of the kernel to different frequencies. Also in the proposed model fraction gradient operator is used instead of ordinary gradient operator. The proposed method is investigated on different kinds of images such as text, face, natural. The output of the proposed method reflects the effectiveness of the proposed algorithm in restoring the images from blind problems.

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“…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. The second approach is to use the fractional-order derivatives in the TV regularization terms as shown in [46,47].…”

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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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Two-Level method for the total fractional-order variation model in image deblurring problem

Cited by 10 publications

References 34 publications

Total Fractional-Order Variation-Based Constraint Image Deblurring Problem

Total Fractional-Order Variation-Based Constraint Image Deblurring Problem

Point spread function estimation for blind image deblurring problems based on framelet transform

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

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