Structure Preserving Preconditioning for Frame-Based Image Deblurring

  • Davide BianchiEmail author
  • Alessandro Buccini
  • Marco Donatelli
Part of the Springer INdAM Series book series (SINDAMS, volume 36)


Regularizing preconditioners for accelerating the convergence of iterative regularization methods and improving their accuracy have been extensively investigated both in Hilbert and Banach spaces. For deconvolution problems, the classical approach defines preconditioners based on the circular convolution. On the other hand, for \(\ell _2\) regularization methods, it has been recently shown that a preconditioner preserving the structure of the convolution operator can be more effective. Such a preconditioner can improve both restoration quality and robustness of the method with respect to the choice of the regularization parameter when compared with the non-structured ones. In this paper we explore the use of structure preserving preconditioning for \(\ell _1\)-norm regularization in the wavelet domain in image deblurring. A recently proposed preconditioned variant of the linearized Bregman iteration is modified to preserve the structure of the coefficient matrix according to the imposed boundary conditions. The structured preconditioner is chosen as an approximation of a regularized inverse of the convolution matrix. Selected numerical experiments show that our preconditioning strategy improves the previous results obtained with circulant preconditioning providing restorations with lower ringing effects and sharper details.


Image deblurring Sparse regularization Structured preconditioning 



The authors are members of the INdAM Research group GNCS, which has partially supported this work.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Davide Bianchi
    • 1
    Email author
  • Alessandro Buccini
    • 2
  • Marco Donatelli
    • 1
  1. 1.Dipartimento di Scienza e Alta TecnologiaUniversità dell’InsubriaComoItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità di CagliariCagliariItaly

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