An $$\ell ^2-\ell ^q$$ Regularization Method for Large Discrete Ill-Posed Problems

Buccini, Alessandro; Reichel, Lothar

doi:10.1007/s10915-018-0816-5

Cited by 37 publications

(54 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We conclude this section with a discussion on the application of framelets to represent the solution. Many solutions of interest have a sparse representation in terms of framelets; see, e.g., [8,12,13,26]. Framelets are frames with local support.…”

Section: The Linearized Bregman Algorithmmentioning

confidence: 99%

Linearized Krylov subspace Bregman iteration with nonnegativity constraint

2020

Self Cite

View full text Add to dashboard Cite

Bregman-type iterative methods have received considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they deliver. However, these iterative methods may require a large number of iterations and this reduces their usefulness. This paper develops a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. The projection reduces the computational effort required for each iteration. A variant of this solution method, in which nonnegativity of each computed iterate is imposed, also is described. Extensive numerical examples illustrate the performance of the proposed methods.

show abstract

Section: The Linearized Bregman Algorithmmentioning

confidence: 99%

Linearized Krylov subspace Bregman iteration with nonnegativity constraint

2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the present paper, we are primarily concerned with the case when both p, q < 1 in (5), though the method described also can be applied when 1 ≤ p < 2 or 1 ≤ q < 2. The following description is very similar to the one provided in [2,19]. We present it here for the convenience of the reader.…”

Section: A Majorization-minimization Methodsmentioning

confidence: 99%

“…Impulse noise is modeled by letting a certain percentage of randomly chosen entries of b be randomly chosen uniformly distributed integers in the interval [0, 255]; cf. (2). We refer to this percentage as the noise level and denote it by σ.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…We turn to the choice of p. The value of p should depend on the type of noise in the data b δ . For white Gaussian noise, p = 2 is appropriate and a method for determining µ for this kind of noise, based on the discrepancy principle is described in [2]. This method requires that a fairly accurate estimate of the norm of the noise be available and allows 0 < q < 1.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Anℓp-ℓqminimization method with cross-validation for the restoration of impulse noise co

Buccini

Reichel

2020

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Discrete ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. Regularization aims to reduce this sensitivity. Many regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term and a q-norm to measure the regularization term has received considerable attention. The relative importance of these terms is determined by a regularization parameter. When the perturbation in the available data is made up of impulse noise and a sparse solution is desired, it often is beneficial to let 0 < p, q < 1. Then the p-and q-norms are not norms. The choice of a suitable regularization parameter is crucial for the quality of the computed solution. It therefore is important to develop methods for determining this parameter automatically, without user-interaction. However, the latter has so far not received much attention when the data is contaminated by impulse noise. This paper discusses two approaches based on cross validation for determining the regularization parameter in this situation. Computed examples that illustrate the performance of these approaches when applied to the restoration of impulse noise contaminated images are presented.

show abstract

“…• ADMM with unknown boundary conditions (ADMM-UBC) which uses a Total Variation penalty term; see [1]; • The 2 − q coupled with the discrepancy principle described in [9];…”

Section: Numerical Examplesmentioning

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

An $$\ell ^2-\ell ^q$$ Regularization Method for Large Discrete Ill-Posed Problems

Cited by 37 publications

References 24 publications

Linearized Krylov subspace Bregman iteration with nonnegativity constraint

Linearized Krylov subspace Bregman iteration with nonnegativity constraint

Anℓp-ℓqminimization method with cross-validation for the restoration of impulse noise co

A multigrid frame based method for image deblurring

Contact Info

Product

Resources

About