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Alternating Krylov subspace image restoration methods. (English) Zbl 1251.65091

Let \(f^\delta\) represent the available noise- and blur-contaminated image and \(\widehat u\) the associated image that is to recover. The model \[ f^\delta(x)= \int h(x,y)\widehat u(y)\,dy+ \eta^\delta(x),\quad x\in\Omega, \] with the noise \(\eta^\delta\) is assumed. In this paper, the unknown \(\widehat u\) is considered as a solution of the minimization problem \(\min J(u,w)\), where \[ J(u,w)= \int\int (h(x,y) u(y)\,dy- f^\delta(x))^2 dx+\alpha \int({\mathcal L}(u- w)(x)^2+ \beta|\nabla w(x)|)\,dx \] with a certain regularization operator \({\mathcal L}\).
The authors solve a discrete version of this problem by alternating interative methods. Krylov subspace based two-way alternating methods are studied, that allow the application of regularization operators different from the identity in both deblurring and the denoising step. Deblurring applies a few steps of Golub-Kahan bidiagonalization, while denoising is performed by an adaptive total-variation diminishing method. Convergence of the methods is proved. Four numerical examples are discussed, these examples illustrate by impressive figures the computed restorations.

MSC:

65K10 Numerical optimization and variational techniques
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65R30 Numerical methods for ill-posed problems for integral equations
49J21 Existence theories for optimal control problems involving relations other than differential equations
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
49M25 Discrete approximations in optimal control
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References:

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