Abstract
Image restoration is a widely studied discrete ill-posed problem. Among the many regularization methods used for treating the problem, iterative methods have been shown to be effective. In this paper, we consider the case of a blurring function defined by space invariant and band-limited PSF, modeled by a linear system that has a band block Toeplitz structure with band Toeplitz blocks. In order to reduce the number of iterations required to obtain acceptable reconstructions, in 13 an inverse Toeplitz preconditioner for problems with a Toeplitz structure was proposed. The cost per iteration is of operations, where is the pixel number of the 2D image. In this paper, we propose inverse preconditioners with a band Toeplitz structure, which lower the cost to and in experiments showed the same speed of convergence and reconstruction efficiency as the inverse Toeplitz preconditioner.
Highlights
INTRODUCTIONWhere x, b, and w represent the original image, the observed image, and the noise, respectively
Many image restoration problems can be modeled by the linear system Ax = b − w, (1)where x, b, and w represent the original image, the observed image, and the noise, respectively
We consider the case of a blurring function defined by space invariant and band-limited point spread function (PSF), modeled by a linear system that has a band block Toeplitz structure with band Toeplitz blocks
Summary
Where x, b, and w represent the original image, the observed image, and the noise, respectively. A good preconditioner should reduce the number of iterations required to reconstruct the information from the signal subspace, that is, it should only cluster the largest eigenvalues around 1, and leave the others out of the cluster This requires knowledge (or at least an estimate) of a parameter τ > 0, called the regularization parameter, such that the eigenvalues of the matrix A which have a modulus greater than τ correspond to the signal subspace. The reduction in cost was achieved by performing approximate spectral factorizations of a trigonometric bivariate polynomial which, through a fit technique, regularizes the symbol function associated with A In this way, the preconditioner is expressed as the product of two band triangular factors.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
