Regularizing preconditioners for the approximate solution by gradient-type methods of image restoration problems with two-level band Toeplitz structure, are examined. For problems having separable and positive definite matrices, the fit preconditioner, introduced in [6], has been shown to be effective in conjunction with CG. The cost of this preconditioner is of O(n^2) operations per iteration, where n^2 is the pixels number of the image, whereas the cost of the circulant preconditioners commonly used for this type of problems is of O(n^2 log n) operations per iteration. In this paper the extension of the fit preconditioner to more general cases is proposed: namely the nonseparable positive definite case and the symmetric indefinite case are treated. The major difficulty encountered in this extension concerns the factorization phase, where, unlike the separable case, a further approximation is required. Various approximate factorizations are proposed. The preconditioners thus obtained have still a cost of O(n^2) operations per iteration. A large numerical experimentation compares these preconditioners with the circulant Chan preconditioner, showing often better performances at a lower cost.
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