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 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, the main novelty of this work is combining nonstationary structure preserving preconditioners with general regularizing operators which hold in their kernel the key features of the true solution that we wish to preserve. Several numerical experiments shows the performances of our methods in terms of quality of the restorations.
Highlights
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
In image deblurring we are concerned in reconstructing an approximation of an image from blurred and noisy measurements
The paper is organized as follows: in Section 2 we propose a generalization of an approximated iterative Tikhonov scheme that was firstly introduced in [15] and developed and adapted into different settings in [16,17]
Summary
In image deblurring we are concerned in reconstructing an approximation of an image from blurred and noisy measurements. The first term in (3) is usually refereed to as fidelity term and the second as regularization term This translates into solving a linear problem for which many efficient methods have been developed for computing its solution and for estimating the regularizing parameter μ, see [6]. This approach comes with a drawback: the edges of restored images are usually over-smoothed. It is called structure preserving reblurring preconditioning strategy and we combine it with the generalized regularization filtering approach of the preceding.
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