## A multigrid frame based method for image deblurring

Alessandro Buccini and Marco Donatelli

### Abstract

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 $\ell_2$-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.

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### Key words

image deblurring, multigrid methods, iterative regularization methods

### AMS subject classifications

65F22, 65N55, 65F10, 15B05

### Links to the cited ETNA articles

 [24] Vol. 29 (2007-2008), pp. 163-177 Marco Donatelli and Stefano Serra-Capizzano: Filter factor analysis of an iterative multilevel regularizing method [37] Vol. 13 (2002), pp. 81-105 Thomas Huckle and Jochen Staudacher: Multigrid preconditioning and Toeplitz matrices

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