A multigrid frame based method for image deblurring

Abstract

. Iterative soft thresholding algorithms combine one step of Landweber method (or accelerated vari- ants), 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 ℓ 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 a 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 in 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.