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**Regularization preconditioners for frame-based image deblurring with reduced boundary artifacts.**
*(English)*
Zbl 1382.94010

Summary: Thresholding iterative methods were recently successfully applied to image deblurring problems. In this paper, we investigate the modified linearized Bregman algorithm (MLBA) used in image deblurring problems, with a proper treatment of the boundary artifacts. We consider two standard approaches: the imposition of boundary conditions and the use of the rectangular blurring matrix. The fast convergence of the MLBA depends on a regularizing preconditioner that could be computationally expensive and hence it is usually chosen as a block circulant circulant block (BCCB) matrix, diagonalized by the discrete Fourier transform. We show that the standard approach based on the BCCB preconditioner may provide low-quality restored images and we propose different preconditioning strategies that improve the quality of the restoration and save some computational cost at the same time. Motivated by a recent nonstationary preconditioned iteration, we propose a new algorithm that combines such a method with the MLBA. We prove that it is a regularizing and convergent method. A variant with a stationary preconditioner is also considered. Finally, a large number of numerical experiments show that our methods provide accurate and fast restorations when compared with the state of the art.

### MSC:

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

65F08 | Preconditioners for iterative methods |

65F22 | Ill-posedness and regularization problems in numerical linear algebra |

65F10 | Iterative numerical methods for linear systems |

65T60 | Numerical methods for wavelets |

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\textit{Y. Cai} et al., SIAM J. Sci. Comput. 38, No. 1, B164--B189 (2016; Zbl 1382.94010)

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