scholarly journals Performance of relaxed iterative methods for image deblurring problems

2019 ◽  
Vol 13 ◽  
pp. 174830261986173 ◽  
Author(s):  
Jae H Yun
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Total Variation ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Image Deblurring ◽  
Total Variation Regularization ◽  
L1 Norm ◽  
Split Bregman ◽  
Relaxation Parameters

In this paper, we consider performance of relaxation iterative methods for four types of image deblurring problems with different regularization terms. We first study how to apply relaxation iterative methods efficiently to the Tikhonov regularization problems, and then we propose how to find good preconditioners and near optimal relaxation parameters which are essential factors for fast convergence rate and computational efficiency of relaxation iterative methods. We next study efficient applications of relaxation iterative methods to Split Bregman method and the fixed point method for solving the L1-norm or total variation regularization problems. Lastly, we provide numerical experiments for four types of image deblurring problems to evaluate the efficiency of relaxation iterative methods by comparing their performances with those of Krylov subspace iterative methods. Numerical experiments show that the proposed techniques for finding preconditioners and near optimal relaxation parameters of relaxation iterative methods work well for image deblurring problems. For the L1-norm and total variation regularization problems, Split Bregman and fixed point methods using relaxation iterative methods perform quite well in terms of both peak signal to noise ratio values and execution time as compared with those using Krylov subspace methods.

Nonlocal Total Variation Based Image Deblurring Using Split Bregman Method and Fixed Point Iteration

2013 ◽  
Vol 333-335 ◽  
pp. 875-882 ◽  
Author(s):  
Dong Jie Tan ◽  
An Zhang
Keyword(s):  
Fixed Point ◽  
Total Variation ◽  
Image Deblurring ◽  
Benchmark Problems ◽  
Total Variation Regularization ◽  
Split Bregman Method ◽  
Split Bregman ◽  
Fixed Point Iteration ◽  
Nonlocal Total Variation ◽  
Nonlocal Regularization

Nonlocal regularization for image restoration is extensively studied in recent years. However, minimizing a nonlocal regularization problem is far more difficult than a local one and still challenging. In this paper, a novel nonlocal total variation based algorithm for image deblurring is presented. The core idea of this algorithm is to consider the latent image as the fixed point of the nonlocal total variation regularization functional. And a split Bregman method is proposed to solve the minimization problem in each fixed point iteration efficiently. By alternatively reconstructing a sharp image and updating the nonlocal gradient weights, the recovered image becomes more and more sharp. Experimental results on the benchmark problems are presented to show the efficiency and effectiveness of our algorithm.

Magnetic inversion to recover the subsurface block structures based on L1 norm and total variation regularization

2021 ◽  
Author(s):  
Mitsuru Utsugi
Keyword(s):  
Total Variation ◽  
Magnetic Data ◽  
Preconditioned Conjugate Gradient ◽  
Inversion Method ◽  
Total Variation Regularization ◽  
Subsurface Structure ◽  
L1 Norm ◽  
Magnetic Inversion ◽  
Original Definition ◽  
Primal Dual

Summary This paper presents a new sparse inversion method based on L1 norm regularization for 3D magnetic data. In isolation, L1 norm regularization yields model elements which are unconstrained by the input data to be exactly zero, leading to a sparse model with compact and focused structure. Here, we complement the L1 norm with a penalty minimizing total variation, the L1 norm of the model gradients; it is expected that the sharp boundaries of the subsurface structure are not compromised by incorporating this penalty. Although this penalty is widely used in the geophysical inversion studies, it is often replaced by an alternative quadratic penalty to ease solution of the penalized inversion problem; in this study, the original definition of the total variation, i.e., form of the L1 norm of the model gradients, is used. To solve the problem with this combined penalty of L1 norm and total variation, this study introduces alternative direction method of multipliers (ADMM), which is a primal-dual optimization algorithm that solves convex penalized problems based on the optimization of an augmented Lagrange function. To improve the computational efficiency of the algorithm to make this method applicable to large-scale magnetic inverse problems, this study applies matrix compression using the wavelet transform and the preconditioned conjugate gradient method. The inversion method is applied to both synthetic tests and real data, the synthetic tests demonstrate that, when subsurface structure is blocky, it can be reproduced almost perfectly.

Verification of Iterative Methods for the Linear Complementarity Problem

2016 ◽  
pp. 545-580 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Linear Complementarity Problem ◽  
Numerical Experiments ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Convergence Results ◽  
Fixed Point Principle ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

In the present chapter, we give an overview of iterative methods for linear complementarity problems (abbreviated as LCPs). We also introduce these iterative methods for the problems based on fixed-point principle. Next, we present some new properties of preconditioned iterative methods for solving the LCPs. Convergence results of the sequence generated by these methods and also the comparison analysis between classic Gauss-Seidel method and preconditioned Gauss-Seidel (PGS) method for LCPs are established under certain conditions. Finally, the efficiency of these methods is demonstrated by numerical experiments. These results show that the mentioned models are effective in actual implementation and competitive with each other.

scholarly journals An Adaptive Image Denoising Model Based on Tikhonov and TV Regularizations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Kui Liu ◽  
Jieqing Tan ◽  
Benyue Su
Keyword(s):  
Image Denoising ◽  
Total Variation ◽  
Tikhonov Regularization ◽  
Experimental Results ◽  
Total Variation Regularization ◽  
Split Bregman Method ◽  
Split Bregman ◽  
Gradient Information ◽  
Weighted Combination ◽  
Staircase Artifacts

To avoid the staircase artifacts, an adaptive image denoising model is proposed by the weighted combination of Tikhonov regularization and total variation regularization. In our model, Tikhonov regularization and total variation regularization can be adaptively selected based on the gradient information of the image. When the pixels belong to the smooth regions, Tikhonov regularization is adopted, which can eliminate the staircase artifacts. When the pixels locate at the edges, total variation regularization is selected, which can preserve the edges. We employ the split Bregman method to solve our model. Experimental results demonstrate that our model can obtain better performance than those of other models.

scholarly journals Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage

Algorithms ◽  
2019 ◽  
Vol 12 (10) ◽  
pp. 221
Author(s):  
Lin ◽  
Chen ◽  
Chen ◽  
Yu
Keyword(s):  
Total Variation ◽  
Impulse Noise ◽  
Signal To Noise Ratio ◽  
Image Deblurring ◽  
L1 Norm ◽  
Signal To Noise ◽  
Total Generalized Variation ◽  
First Order ◽  
Norm Constraint ◽  
Generalized Variation

Image deblurring under the background of impulse noise is a typically ill-posed inverse problem which attracted great attention in the fields of image processing and computer vision. The fast total variation deconvolution (FTVd) algorithm proved to be an effective way to solve this problem. However, it only considers sparsity of the first-order total variation, resulting in staircase artefacts. The L1 norm is adopted in the FTVd model to depict the sparsity of the impulse noise, while the L1 norm has limited capacity of depicting it. To overcome this limitation, we present a new algorithm based on the Lp-pseudo-norm and total generalized variation (TGV) regularization. The TGV regularization puts sparse constraints on both the first-order and second-order gradients of the image, effectively preserving the image edge while relieving undesirable artefacts. The Lp-pseudo-norm constraint is employed to replace the L1 norm constraint to depict the sparsity of the impulse noise more precisely. The alternating direction method of multipliers is adopted to solve the proposed model. In the numerical experiments, the proposed algorithm is compared with some state-of-the-art algorithms in terms of peak signal-to-noise ratio (PSNR), structural similarity (SSIM), signal-to-noise ratio (SNR), operation time, and visual effects to verify its superiority.

scholarly journals Infrared Image Deblurring via High-Order Total Variation and Lp-Pseudonorm Shrinkage

2020 ◽  
Vol 10 (7) ◽  
pp. 2533 ◽  
Author(s):  
Jingjing Yang ◽  
Yingpin Chen ◽  
Zhifeng Chen
Keyword(s):  
Total Variation ◽  
Infrared Image ◽  
Noise Pollution ◽  
Image Deblurring ◽  
High Order ◽  
L1 Norm ◽  
Alternating Direction ◽  
Image Blurring ◽  
Image Gradients ◽  
Staircase Artifacts

The quality of infrared images is affected by various degradation factors, such as image blurring and noise pollution. Anisotropic total variation (ATV) has been shown to be a good regularization approach for image deblurring. However, there are two main drawbacks in ATV. First, the conventional ATV regularization just considers the sparsity of the first-order image gradients, thus leading to staircase artifacts. Second, it employs the L1-norm to describe the sparsity of image gradients, while the L1-norm has a limited capacity of depicting the sparsity of sparse variables. To address these limitations of ATV, a high-order total variation is introduced in the ATV deblurring model and the Lp-pseudonorm is adopted to depict the sparsity of low- and high-order total variation. In this way, the recovered image can fit the image priors with clear edges and eliminate the staircase artifacts of the ATV model. The alternating direction method of multipliers is used to solve the proposed model. The experimental results demonstrate that the proposed method does not only remove blurs effectively but is also highly competitive against the state-of-the-art methods, both qualitatively and quantitatively.

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