The Anti-Reflective Transform and Regularization by Filtering

Aricò, Antonio; Donatelli, Marco; Nagy, James G.; Serra‐Capizzano, Stefano

doi:10.1007/978-94-007-0602-6_1

Cited by 16 publications

(29 citation statements)

References 13 publications

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“…The matrix L AR is not normal and therefore cannot be diagonalized by a unitary similarity transformation. Nevertheless, it is shown in [1] that the eigenvector matrix of L AR is a modification of a unitary matrix with a structure that makes fast diagonalization possible. Specifically, the eigenvector matrix of L AR can be chosen as…”

Section: Antireflective Boundary Conditionsmentioning

confidence: 99%

“…It is shown in [1] that the inverse of (21) has a structure similar to that of T AR . For the computations of the present paper, we only need to be able to evaluate matrix-vector products with the inverse rapidly.…”

Section: Antireflective Boundary Conditionsmentioning

confidence: 99%

“…Moreover, the matrix T −1 AR is explicitly known (see [1]) and, therefore, the determination of T −T AR e j does not require any computations. Let the columns of the matrix Q N (DX T T X ) form an orthonormal basis for…”

Section: −Tmentioning

confidence: 99%

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Square smoothing regularization matrices with accurate boundary conditions

Donatelli

Reichel

2014

Journal of Computational and Applied Mathematics

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This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. The present paper describes the construction of square regularization matrices from finite difference equations with a focus on the boundary conditions. The regularization matrices considered have a structure that makes them easy to apply in iterative methods, including methods based on the Arnoldi process. Numerical examples illustrate the properties and effectiveness of the regularization matrices described.

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Section: Antireflective Boundary Conditionsmentioning

confidence: 99%

Section: Antireflective Boundary Conditionsmentioning

confidence: 99%

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Square smoothing regularization matrices with accurate boundary conditions

Donatelli

Reichel

2014

Journal of Computational and Applied Mathematics

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“…This matrix has a block structure that is diagonalized by the discrete sine (I) transform (DST), which may be implemented essentially through use of the fast Fourier transform; see SerraCapizzano [20]. A linear transformation may be defined to compute the eigenvalues of A using the components of G and K without explicit construction of A, defined mostly through use of the DST; see [20,1] for details. However, it does not follow from the definition of the boundary conditions that the bound (14) holds for symmetric, non-negative, and suitably bounded kernels.…”

Section: Antireflective Boundary Conditionsmentioning

confidence: 99%

“…The blurring of a digital image is often modeled by a linear convolution operation, g (x, y) = (k * f ) (x, y) = Ω k (x − ψ, y − ξ) f (ψ, ξ) dψdξ, (x, y) ∈ Ω, (1) where f represents the exact image, g is the resulting blurred image, k is the kernel function of the blurring operator, and Ω is a rectangular region in R 2 . In the context of digital image restoration, the function k is often referred to as a point spread function (PSF).…”

Section: Introductionmentioning

confidence: 99%

Extensions of the Justen–Ramlau blind deconvolution method

Hearn

Reichel

2013

Adv Comput Math

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Blind deconvolution problems arise in many image restoration applications. Most available blind deconvolution methods are iterative. Recently, Justen and Ramlau proposed a novel non-iterative blind deconvolution method. The method was derived under the assumption of periodic boundary conditions. These boundary conditions may introduce oscillatory artifacts into the computed restoration. We describe extensions of the Justen-Ramlau method that allow the use of Neumann and antireflective boundary conditions.

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Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

Gazzola

Landman

2020

GAMM-Mitteilungen

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Large-scale linear systems coming from suitable discretizations of linear inverse problems are challenging to solve. Indeed, since they are inherently ill-posed, appropriate regularization should be applied; since they are large-scale, well-established direct regularization methods (such as Tikhonov regularization) cannot often be straightforwardly employed, and iterative linear solvers should be exploited. Moreover, every regularization method crucially depends on the choice of one or more regularization parameters, which should be suitably tuned. The aim of this paper is twofold: (a) survey some well-established regularizing projection methods based on Krylov subspace methods (with a particular emphasis on methods based on the Golub-Kahan bidiagonalization algorithm), and the so-called hybrid approaches (which combine Tikhonov regularization and projection onto Krylov subspaces of increasing dimension); (b) introduce a new principled and adaptive algorithmic approach for regularization similar to specific instances of hybrid methods. In particular, the new strategy provides reliable parameter choice rules by leveraging the framework of bilevel optimization, and the links between Gauss quadrature and Golub-Kahan bidiagonalization. Numerical tests modeling inverse problems in imaging illustrate the performance of existing regularizing Krylov methods, and validate the new algorithms. K E Y W O R D S hybrid methods, imaging problems, Krylov subspace methods, large-scale linear inverse problems, regularization parameter choice rules, Tikhonov regularization 1 INTRODUCTION This paper considers linear, large-scale, discrete ill-posed problems of the form Ax true + e = b true + e = b , (1) where the matrix A ∈ R m×n represents a forward mapping that is ill-conditioned with ill-determined rank (ie, the singular values of A decay and cluster at zero without an evident gap between two consecutive ones), x true ∈ R n is the desired solution, and e ∈ R m is some unknown noise that affects the data b ∈ R m. Systems like (1) typically stem from the discretization of first-kind Fredholm integral equations, and model inverse problems arising in a variety of applications, such This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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The Anti-Reflective Transform and Regularization by Filtering

Cited by 16 publications

References 13 publications

Square smoothing regularization matrices with accurate boundary conditions

Square smoothing regularization matrices with accurate boundary conditions

Extensions of the Justen–Ramlau blind deconvolution method

Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

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