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Structure Preserving Preconditioners for Image Deblurring

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Abstract

Regularizing preconditioners for accelerating the convergence of iterative regularization methods without spoiling the quality of the approximated solution have been extensively investigated in the last twenty years. Several strategies have been proposed for defining proper preconditioners. Usually, in methods for image restoration, the structure of the preconditioner is chosen Block Circulant with Circulant Blocks (BCCB) because it can be efficiently exploited by Fast Fourier Transform (FFT). 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. The structure of such a matrix, in case of image deblurring problem, depends on the boundary conditions imposed on the imaging model. Therefore, we propose a technique to construct a preconditioner which has the same structure of the blurring matrix related to the restoration problem at hand. The construction of our preconditioner requires two FFTs like the BCCB preconditioner. The presented preconditioning strategy represents a generalization and an improvement with respect to both circulant and structured preconditioning available in the literature. The technique is further extended to provide a non-stationary preconditioning in the same spirit of a recent proposal for BCCB matrices. Some numerical results show the importance of preserving the matrix structure from the point of view of both restoration quality and robustness of the regularization parameter.

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Acknowledgements

This work is partly supported by PRIN 2012 N. 2012MTE38N and GNCS-INdAM.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Scienze Informatiche e Matematica, Università degli Studi dell’Aquila, Via Vetoio, Loc. Coppito, 67010, L’Aquila, Italy

    Pietro Dell’Acqua

  2. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Marco Donatelli

  3. Dipartimento di Matematica, Università degli Studi di Genova, Via Dodecaneso 35, 16146, Genova, Italy

    Claudio Estatico

  4. Max-Planck Institute for Plasma Physics, Boltzmannstraße 2, 85748, Garching, Germany

    Mariarosa Mazza

Authors
  1. Pietro Dell’Acqua
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  2. Marco Donatelli
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  3. Claudio Estatico
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  4. Mariarosa Mazza
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Correspondence to Pietro Dell’Acqua.

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Dell’Acqua, P., Donatelli, M., Estatico, C. et al. Structure Preserving Preconditioners for Image Deblurring. J Sci Comput 72, 147–171 (2017). https://doi.org/10.1007/s10915-016-0350-2

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  • DOI: https://doi.org/10.1007/s10915-016-0350-2

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