Generalized Structure Preserving Preconditioners for Frame-Based Image Deblurring

Abstract

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 be 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, we combine a nonstationary preconditioned iteration with the modified linearized Bregman algorithm (MLBA) and proper regularization operators.Several numerical experiments shows the performances of our methods in terms of quality of the restorations. * This is a preprint. † Member of INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico. ‡ Member of INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico. Partially founded by the Young Researcher Project "Reconstruction of sparse data" of the GNCS group of INdAM and by the AMIS (Algorithms e Models for Imaging Science) project of the Fondazione Sardegna.

Generalized cross validation for ℓp-ℓq minimization

“…The regularization parameter μ > 0 determines the trade-off between the first and second terms in (2), and decides how sensitive the solution of (2) is to the error η in b δ . Moreover, the choice of μ affects how close the solution is to the desired vector x true = A † b.…”

“…We now briefly discuss different choices for p and q, and first comment on the choice of q, which affects the second term in (2). This terms is commonly referred to as the regularization term.…”

“…Then the regularization term measures the size of the computed solution by the 0 -norm. This "norm" counts the number of nonzero entries in the vector x in (2). Note that this norm is not convex.…”

“…This paper is organized as follows: Section 2 reviews how the GCV parameter for Tikhonov regularization in general form can be computed fairly inexpensively by projection into a Krylov subspace, and in Section 3 we discuss how to determine the GCV parameter for large-scale minimization problems (2). A smoothed variant of GCV that can give better regularization parameter values also is presented.…”

A new nonstationary preconditioned iterative method for linear discrete ill‐posed problems with application to image deblurring

“…Following, Donatelli and Hanke, 16 we refer to their method as the approximated iterated Tikhonov (AIT) method. This algorithm (or similar variants) are also discussed in References 18,20‐23.…”