  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.
