A generalized matrix Krylov subspace method for TV regularization

Bentbib, Abdeslem Hafid; Guide, M. El; Jbilou, Khalide

doi:10.1016/j.cam.2019.112405

Cited by 6 publications

(9 citation statements)

References 29 publications

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“…This setting ensures the improvement of the convergence speed of the algorithm. We compute the other regularization parameter λ using the estimated value of λ f and the GCV function (28).…”

Section: Resultsmentioning

confidence: 99%

“…An appropriate selection of the regularization parameters λ f and λ is important in the regularization. The well-known methods for this purpose are the L-curve [26] and the GCV method [27,28]; here, we consider the GCV one. It is a widely used and very successful predictive method for choosing the smoothing parameter.…”

Section: A Parameter Selection Methods For H 1 Regularizationmentioning

confidence: 99%

See 1 more Smart Citation

Regularized supervised Bayesian approach for image deconvolution with regularization parameter estimation

Laaziri

Raghay

Hakim

2020

EURASIP J. Adv. Signal Process.

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Image deconvolution consists in restoring a blurred and noisy image knowing its point spread function (PSF). This inverse problem is ill-posed and needs prior information to obtain a satisfactory solution. Bayesian inference approach with appropriate prior on the image, in particular with a Gaussian prior, has been used successfully. Supervised Bayesian approach with maximum a posteriori (MAP) estimation, a method that has been considered recently, is unstable and suffers from serious ringing artifacts in many applications. To overcome these drawbacks, we propose a regularized version where we minimize an energy functional combined by the mean square error with H 1 regularization term, and we consider the generalized cross validation (GCV) method, a widely used and very successful predictive approach, for choosing the smoothing parameter. Theoretically, we study the convergence behavior of the method and we give numerical tests to show its effectiveness.

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Section: Resultsmentioning

confidence: 99%

Section: A Parameter Selection Methods For H 1 Regularizationmentioning

confidence: 99%

Regularized supervised Bayesian approach for image deconvolution with regularization parameter estimation

Laaziri

Raghay

Hakim

2020

EURASIP J. Adv. Signal Process.

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“…To the best of our knowledge, Krylov methods have been coupled with ADMM only in [4,42]. In [4] the authors only consider the case in which the operator A can be expressed as the Kronecker product of two matrices and the regularization term is the TV norm. In this work we do not make any assumption on the structure of A and we only require R to be closed, proper, and convex.…”

Section: Introductionmentioning

confidence: 99%

Fast Alternating Direction Multipliers Method by Generalized Krylov Subspaces

Buccini

2021

J Sci Comput

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The Alternating Direction Multipliers Method (ADMM) is a very popular and powerful algorithm for the solution of many optimization problems. In the recent years it has been widely used for the solution of ill-posed inverse problems. However, one of its drawback is the possibly high computational cost, since at each iteration, it requires the solution of a large-scale least squares problem.In this work we propose a computationally attractive implementation of ADMM, with particular attention to ill-posed inverse problems. We signicantly decrease the computational cost by projecting the original large scale problem into a low-dimensional subspace by means of Generalized Krylov Subspaces (GKS). The dimension of the projection space is not an additional parameter of the method as it increases with the iterations. The construction of GKS allows for very fast computations, regardless of the increasing size of the problem. Several computed examples show the good performances of the proposed method.

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“…Second, the summation of s terms in

false(\overset{scriptP}{true} false)

can be performed independently and enables

false(\overset{scriptP}{true} false)

to reach superior parallel efficiency when implemented on modern high performance computing architectures. Some work has been done to exploit matrix equation structures for iterative methods to solve inverse problems of the form ; see, for example, other works . In this paper, we propose the structured FISTA (sFISTA) method.…”

Section: Introductionmentioning

confidence: 99%

“…Some work has been done to exploit matrix equation structures for iterative methods to solve inverse problems of the form (5); see, for example, other works. [18][19][20][21] In this paper, we propose the structured FISTA (sFISTA) method. It gains its efficiency by exploiting both the Kronecker product structure of A and the structures from K i and H i .…”

Section: Introductionmentioning

confidence: 99%

Structured FISTA for image restoration

Chen

Nagy

et al. 2019

Numerical Linear Algebra App

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Summary In this paper, we propose an efficient numerical scheme for solving some large‐scale ill‐posed linear inverse problems arising from image restoration. In order to accelerate the computation, two different hidden structures are exploited. First, the coefficient matrix is approximated as the sum of a small number of Kronecker products. This procedure not only introduces one more level of parallelism into the computation but also enables the usage of computationally intensive matrix–matrix multiplications in the subsequent optimization procedure. We then derive the corresponding Tikhonov regularized minimization model and extend the fast iterative shrinkage‐thresholding algorithm (FISTA) to solve the resulting optimization problem. Because the matrices appearing in the Kronecker product approximation are all structured matrices (Toeplitz, Hankel, etc.), we can further exploit their fast matrix–vector multiplication algorithms at each iteration. The proposed algorithm is thus called structured FISTA (sFISTA). In particular, we show that the approximation error introduced by sFISTA is well under control and sFISTA can reach the same image restoration accuracy level as FISTA. Finally, both the theoretical complexity analysis and some numerical results are provided to demonstrate the efficiency of sFISTA.

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A generalized matrix Krylov subspace method for TV regularization

Cited by 6 publications

References 29 publications

Regularized supervised Bayesian approach for image deconvolution with regularization parameter estimation

Regularized supervised Bayesian approach for image deconvolution with regularization parameter estimation

Fast Alternating Direction Multipliers Method by Generalized Krylov Subspaces

Structured FISTA for image restoration

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