Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Morigi, Serena; Reichel, Lothar; Sgallari, Fiorella

doi:10.1016/j.apnum.2009.07.007

Cited by 12 publications

(21 citation statements)

References 28 publications

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“…where the R j are determined by repeated local weighted least-squares approximation; see [9][10][11][12] for more details.…”

Section: A Cascading-alternating Image Restoration Methodsmentioning

confidence: 99%

“…The function ϕ in (3) The prolongation operators are nonlinear edge-preserving and noise-reducing, see Sect. 3, while the restriction operators are determined by weighted local leastsquares approximation following [11]. The purpose of the weights is to avoid smearing of edges.…”

Section: A Cascading-alternating Image Restoration Methodsmentioning

confidence: 99%

“…An accurate initial iterate may help determine an accurate restoration. This is illustrated in [10,11], and is one of the benefits of multilevel methods. The design of the prolongation method therefore is important.…”

Section: Deblurringmentioning

confidence: 99%

“…We would like to recoverû given the observed image f δ and the PSF h. It is well known that the solution of (1) is an ill-posed inverse problem and therefore computationally challenging. Many algorithms are available for determining an approximate solution of (1), including recently proposed multilevel and alternating methods; see, e.g., [1,2,7,[9][10][11]13]. To be able to determine accurate restorations, the methods apply regularization, i.e., they replace the original problem by a nearby one that is less sensitive to perturbations.…”

Section: Introductionmentioning

confidence: 99%

“…The multilevel methods proposed in [9][10][11] proceed from coarser to finer image resolution levels and are based on regularization by truncated iteration on each level. Prolongation of a coarse-level approximation ofû to a finer level is carried out with the aid of nonlinear edge-preserving and noise-reducing operators.…”

Section: Introductionmentioning

confidence: 99%

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A Cascadic Alternating Krylov Subspace Image Restoration Method

Morigi

Reichel

Sgallari

2013

Lecture Notes in Computer Science

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Abstract. This paper describes a cascadic image restoration method which at each level applies a two-way alternating denoising and deblurring procedure. Denoising is carried out with a wavelet transform, which also provides an estimate of the noise-level. The latter is used to determine a suitable regularization parameter for the Krylov subspace iterative deblurring method. The cascadic multilevel method proceeds from coarse to fine image resolution, using suitable restriction and prolongation operators. The choice of the latter is critical for the performance of the multilevel method. We introduce a special deblurring prolongation procedure based on TV regularization. Computed examples demonstrate the effectiveness of the method proposed for determining image restorations of high quality.

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“…where the R j are determined by repeated local weighted least-squares approximation; see [9][10][11][12] for more details.…”

Section: A Cascading-alternating Image Restoration Methodsmentioning

confidence: 99%

Section: A Cascading-alternating Image Restoration Methodsmentioning

confidence: 99%

Section: Deblurringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Cascadic Alternating Krylov Subspace Image Restoration Method

Morigi

Reichel

Sgallari

2013

Lecture Notes in Computer Science

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A hybrid multilevel-active set method for large box-constrained linear discrete ill-posed problems

Morigi

Plemmons

Reichel

et al. 2010

Calcolo

Self Cite

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Many questions in science and engineering give rise to ill-posed inverse problems whose solution is known to satisfy box constrains, such as nonnegativity. The solution of discretized versions of these problems is highly sensitive to perturbations in the data, discretization errors, and round-off errors introduced during the computations. It is therefore often beneficial to impose known constraints during the solution process. This paper paper describes a two-phase algorithm for the solution of large-scale box-constrained discrete ill-posed problems. The first phase applies a cascadic multilevel method and imposes the constraints on each level by orthogonal projection. The second phase improves the computed approximate solution on the finest level by an active set method. The latter allows several indices of the active set to be updated simultaneously. This reduces the computational effort significantly, when compared to standard active set methods that update one index at a time. Applications to image restoration are presented.Key words. constrained ill-posed problems, constrained inverse problems, multilevel method, active set method, image restoration with the unknown error-free right-hand side is consistent, and that its minimal-norm solution,x, lives in S.We would like to determine an approximation ofx from the available least-squares problem (1.1). Due to the error e in b, the minimal-norm solution of (1.1) generally is not a useful approximation ofx. Meaningful approximations ofx can be computed by first replacing (1.1) by a nearby minimization problem with a less ill-conditioned matrix, and then solving the modified problem so obtained. This replacement is commonly *

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