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A multigrid frame based method for image deblurring. (English) Zbl 07192775
Summary: 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.
MSC:
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65T60 Numerical methods for wavelets
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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