Adaptive Arnoldi-Tikhonov regularization for image restoration

Novati, Paolo; Russo, Maria Rosaria

doi:10.1007/s11075-013-9712-0

Cited by 13 publications

(8 citation statements)

References 10 publications

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“…A permutation matrix P * ∈ R n×n that satisfies ( 22) is said to be optimal with respect to the matrix L, the vector x, and the norm • . Novati and Russo [13] considered the minimization problem (22) for the Euclidean vector norm, but other vector norms also can be used. We are particularly interested in the situation when the matrix L in ( 22) is the bidiagonal matrix…”

Section: A Majorization-minimization Methodsmentioning

confidence: 99%

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On the choice of regularization matrix for an ℓ2-ℓ minimization method for image restoration

Buccini

Huang

Reichel

et al. 2021

Applied Numerical Mathematics

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Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. To reduce this sensitivity, the original problem may be replaced by a minimization problem with a fidelity term and a regularization term. We consider minimization problems of this kind, in which the fidelity term is the square of the ℓ 2 -norm of a discrepancy and the regularization term is the q th power of the ℓq-norm of the size of the computed solution measured in some manner. We are interested in the situation when 0 < q ≤ 1, because such a choice of q promotes sparsity of the computed solution. The regularization term is determined by a regularization matrix. Novati and Russo let q = 2 and proposed in [P. Novati and M. R. Russo, Adaptive Arnoldi-Tikhonov regularization for image restoration, Numer. Algorithms, 65 (2014), pp. 745-757] a regularization matrix that is a finite difference approximation of a differential operator applied to the computed approximate solution after reordering. This gives a Tikhonov regularization problem in general form. We show that this choice of regularization matrix also is well suited for minimization problems with 0 < q ≤ 1. Applications to image restoration are presented.

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Section: A Majorization-minimization Methodsmentioning

confidence: 99%

“…An adaptive reordering method is described. The construction of this kind of regularization matrices for the minimization problem (4) is an adaption of a method proposed by Novati and Russo [13] for choosing a regularization matrix for Tikhonov regularization in general form.…”

Section: Introductionmentioning

confidence: 99%

On the choice of regularization matrix for an ℓ2-ℓ minimization method for image restoration

Buccini

Huang

Reichel

et al. 2021

Applied Numerical Mathematics

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“…[20]). Then, in [5,10,17] the method has been extended to work with a general L ∈ R P ×N and x 0 . Assuming x 0 = 0 (this assumption will hold throughout the paper), we consider the Krylov subspaces…”

Section: The Arnoldi-tikhonov Methodsmentioning

confidence: 99%

“…The AT method was first proposed in [4] with the basic aims of reducing the problem (4) (in the particular case L D I N and x 0 D 0) to a problem of much smaller dimension and to avoid the use of A T as in the Lanczos-type methods (see e.g., [8]). Then, in [7,9,10], the method has been extended to work with a general L 2 R P N and x 0 . Assuming x 0 D 0 (this assumption will hold throughout the paper), we consider the Krylov subspaces K m .A; b/ D span¹b; Ab; : : : ; A m 1 bº; m > 1:…”

Section: The Arnoldi-tikhonov Methodsmentioning

confidence: 99%

Embedded techniques for choosing the parameter in Tikhonov regularization

Gazzola

Novati

Russo

2014

Numer. Linear Algebra Appl.

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This paper introduces a new strategy for setting the regularization parameter when solving large-scale discrete ill-posed linear problems by means of the Arnoldi-Tikhonov method. This new rule is essentially based on the discrepancy principle, although no initial knowledge of the norm of the error that affects the right-hand side is assumed; an increasingly more accurate approximation of this quantity is recovered during the Arnoldi algorithm. Some theoretical estimates are derived in order to motivate our approach. Many numerical experiments, performed on classical test problems as well as image deblurring are presented.

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“…is to control the shape of the gauss pulse function, along with the increase of the ! 's values the ill-posed problems [6,7] become more and more difficult to deal with.…”

Section: Image Restorationmentioning

confidence: 99%

Several Mathematical Methods Applied in Image Compression and Restoration

Hongyu,

2014

TOCSJ

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Several mathematical methods are discussed in this paper, which are applied in image compression and restoration. Singular value decomposition (SVD) is used in compressing image. Conjugate gradients (CG) method and truncated Singular value decomposition (TSVD) regularization method are applied in image restoration. From the experience results we can see that those methods are effective in image compression and image restoration.

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Adaptive Arnoldi-Tikhonov regularization for image restoration

Cited by 13 publications

References 10 publications

On the choice of regularization matrix for an ℓ2-ℓ minimization method for image restoration

On the choice of regularization matrix for an ℓ2-ℓ minimization method for image restoration

Embedded techniques for choosing the parameter in Tikhonov regularization

Several Mathematical Methods Applied in Image Compression and Restoration

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