Generalized structure preserving preconditioners for frame-based image deblurring

We are interested in fast and stable iterative regularization methods for image deblurringproblems with space invariant blur. The associated coefficient matrix has a Block Toeplitz ToeplitzBlocks (BTTB) like structure plus a small rank correction depending on the boundary conditionsimposed on the imaging model. In the literature, several strategies have been proposed in theattempt to define proper preconditioner for iterative regularization methods that involve such linearsystems. Usually, the preconditioner is chosen to be a BlockCirculant with Circulant Blocks (BCCB)matrix because it can efficiently exploit Fast Fourier Transform (FFT) for any computation, includingthe (pseudo-)inversion. Nevertheless, for ill-conditioned problems, it is well known that BCCBpreconditioners cannot provide a strong clustering of the eigenvalues. Moreover, in order to get aneffective preconditioner, it is crucial to preserve the structure of the coefficient matrix. On the otherhand, thresholding iterative methods have been recently successfully applied to image deblurringproblems, exploiting the sparsity of the image in a proper wavelet domain. Motivated by the resultsof recent papers, the main novelty of this work is combining nonstationary structure preservingpreconditioners with general regularizing operators which hold in their kernel the key features ofthe true solution that we wish to preserve. Several numerical experiments shows the performancesof our methods in terms of quality of the restorations.