Iterated Tikhonov regularization with a general penalty term

Buccini, Alessandro; Donatelli, Marco; Reichel, Lothar

doi:10.1002/nla.2089

Cited by 50 publications

(33 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In many applications, the latter type has proven to obtain better results in terms of accuracy and stability than the classical Tikhonov method avoiding an accurate estimation of the parameter α. A generalization to the case where a matrix different from the identity is used in the IT iterations has recently been proposed in [8].…”

Section: Approximated Iterated Tikhonov Methodmentioning

confidence: 99%

A multigrid frame based method for image deblurring

Buccini¹,

Donatelli²

2020

etna

Self Cite

View full text Add to dashboard Cite

Iterative soft thresholding algorithms combine one step of a Landweber method (or accelerated variants) with one step of thresholding of the wavelet (framelet) coefficients. In this paper, we improve these methods by using the framelet multilevel decomposition for defining a multigrid deconvolution with grid transfer operators given by the low-pass filter of the frame. Assuming that an estimate of the noise level is available, we combine a recently proposed iterative method for 2-regularization with linear framelet denoising by soft-thresholding. This combination allows a fast frequency filtering in the Fourier domain and produces a sparse reconstruction in the wavelet domain. Moreover, its employment in a multigrid scheme ensures stable convergence and a reduced noise amplification. The proposed multigrid method is independent of the imposed boundary conditions, and the iterations can be easily projected onto a closed and convex set, e.g., the nonnegative cone. We study the convergence of the proposed algorithm and prove that it is a regularization method. Several numerical results prove that this approach is able to provide highly accurate reconstructions in several different scenarios without requiring the setting of any parameter.

show abstract

Section: Approximated Iterated Tikhonov Methodmentioning

confidence: 99%

A multigrid frame based method for image deblurring

Buccini¹,

Donatelli²

2020

etna

Self Cite

View full text Add to dashboard Cite

show abstract

“…where the noise level δ > 0 is known. Algorithm 1 is stopped with m = m(δ, y δ ) according to the discrepancy principle (3). For the stopping index m = m(δ, y δ ) ≥ 1,…”

Section: Sine Is An Order-optimal Regularisation Methodsmentioning

confidence: 99%

A conjugate-gradient-type rational Krylov subspace method for ill-posed problems

Grimm

2019

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…with a fixed 0 < ρ < c/2, where δ = η is the noise level and where |||·||| is the operator norm defined in (6).…”

Section: General Regularization Operator As H(cc * )mentioning

confidence: 99%

Generalized Structure Preserving Preconditioners for Frame-Based Image Deblurring

Bianchi

Buccini

2020

Mathematics

Self Cite

View full text Add to dashboard Cite

We are interested in fast and stable iterative regularization methods for image deblurring problems with space invariant blur. The associated coefficient matrix has a Block Toeplitz Toeplitz Blocks (BTTB) like structure plus a small rank correction depending on the boundary conditions imposed on the imaging model. In the literature, several strategies have been proposed in the attempt to define proper preconditioner for iterative regularization methods that involve such linear systems. Usually, the preconditioner is chosen to be a Block Circulant with Circulant Blocks (BCCB) matrix because it can be efficiently exploit Fast Fourier Transform (FFT) for any computation, including the (pseudo-)inversion. Nevertheless, for ill-conditioned problems, it is well known that BCCB preconditioners cannot provide a strong clustering of the eigenvalues. Moreover, in order to get an effective preconditioner, it is crucial to preserve the structure of the coefficient matrix.On the other hand, thresholding iterative methods have been recently successfully applied to image deblurring problems, exploiting the sparsity of the image in a proper wavelet domain. Motivated by the results of recent papers, we combine a nonstationary preconditioned iteration with the modified linearized Bregman algorithm (MLBA) and proper regularization operators.Several numerical experiments shows the performances of our methods in terms of quality of the restorations. * This is a preprint. † Member of INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico. ‡ Member of INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico. Partially founded by the Young Researcher Project "Reconstruction of sparse data" of the GNCS group of INdAM and by the AMIS (Algorithms e Models for Imaging Science) project of the Fondazione Sardegna.

show abstract

Iterated Tikhonov regularization with a general penalty term

Cited by 50 publications

References 16 publications

A multigrid frame based method for image deblurring

A multigrid frame based method for image deblurring

A conjugate-gradient-type rational Krylov subspace method for ill-posed problems

Generalized Structure Preserving Preconditioners for Frame-Based Image Deblurring

Contact Info

Product

Resources

About