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Open AccessArticle

Generalized Structure Preserving Preconditioners for Frame-Based Image Deblurring

by Davide Bianchi 1,† and Alessandro Buccini 2,*,†
1
Department of Science and High Technology, University of Insubria, 22100 Como, Italy
2
Department of Mathematics and Computer Science, University of Cagliari, 09124 Cagliari, Italy
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(4), 468; https://doi.org/10.3390/math8040468
Received: 5 February 2020 / Revised: 14 March 2020 / Accepted: 21 March 2020 / Published: 27 March 2020
(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)
We are interested in fast and stable iterative regularization methods for image deblurring problems with space invariant blur. The associated coefficient matrix has a Block Toeplitz Toeplitz Blocks (BTTB) like structure plus a small rank correction depending on the boundary conditions imposed on the imaging model. In the literature, several strategies have been proposed in the attempt to define proper preconditioner for iterative regularization methods that involve such linear systems. Usually, the preconditioner is chosen to be a Block Circulant with Circulant Blocks (BCCB) matrix because it can efficiently exploit Fast Fourier Transform (FFT) for any computation, including the (pseudo-)inversion. Nevertheless, for ill-conditioned problems, it is well known that BCCB preconditioners cannot provide a strong clustering of the eigenvalues. Moreover, in order to get an effective preconditioner, it is crucial to preserve the structure of the coefficient matrix. On the other hand, thresholding iterative methods have been recently successfully applied to image deblurring problems, exploiting the sparsity of the image in a proper wavelet domain. Motivated by the results of recent papers, the main novelty of this work is combining nonstationary structure preserving preconditioners with general regularizing operators which hold in their kernel the key features of the true solution that we wish to preserve. Several numerical experiments shows the performances of our methods in terms of quality of the restorations. View Full-Text
Keywords: image deblurring; structured preconditioners; iterative regularization methods image deblurring; structured preconditioners; iterative regularization methods
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Bianchi, D.; Buccini, A. Generalized Structure Preserving Preconditioners for Frame-Based Image Deblurring. Mathematics 2020, 8, 468.

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