DI BENEDETTO , Fabio

We study the computational problem of finding the optimal preconditioner of a given matrix in an algebra related to a fast transform; ω-circulants, 16 trigonometric, and 8 Hartley-type algebras are considered. For all these cases we prove that the Gram matrix associated with a suitable sparse basis has a rank structure that can be described in...

The iterative solution of a block Toeplitz linear system by the conjugate gradient method is analyzed, the preconditioning step being solved by means of a discrete sine transform. Convergence properties are established and compared to the behaviour of the block circulant preconditioner recently proposed in literature. As in the scalar case,...

Continuous and discrete versions of difference operators are introduced, involving special examples of Toeplitz matrices. A spectral analysis of the continuous operator is performed, providing a full characterization of the null space and the singular system. Under a simplifying assumption involving the discretization parameter, the same...

No description

We present two different examples of image applications where exploiting structured matrices seems attractive, either for reducing the high complexity of the reconstruction problem or because a shift invariance naturally occurs in the mathematical description. We discuss how structures can be incorporated in the models and we point out some new...

No description

The normal equations constructed by a Toeplitz matrix are studied, in order to find a suitable preconditioner related to the discrete sine transform. New results are given about the structure of the product of two Toeplitz matrices, which allow the CGN method to achieve a superlinear rate of convergence. This preconditioner outperforms the...

Algebraic and computational properties of the rank-one updating of a generalized eigenvalue problem are investigated. The results are applied to the computation of the eigenvalues of full Toeplitz matrices related to the Laurent expansion of a rational function, extending a method of Handy and Barlow already known for the banded Toeplitz case.

Several preconditioning techniques for solving Toeplitz systems are known in literature, but their convergence features are completely understood only in the well-conditioned case. We study the application of $\tau$, circulant and Hartley preconditioners to ill-conditioned Toeplitz matrices, by proving that only the first class realizes a...

In this paper we analyze in a general and pure algebraic way imaging systems characterized by shift-variant integral kernels which hide some intrinsic shift-invariance, related to an appropriate coordinate change; we call as structured shift-variant these kinds of imaging systems. In this respect, we propose an algorithm for finding a...

We study the optimal Frobenius operator in a general matrix vector space and in particular in the multilevel trigonometric matrix vector spaces, by emphasizing both the algebraic and geometric properties. These general results are used to extend the Korovkin matrix theory for the approximation of block Toeplitz matrices via trigonometric vector...

In earlier papers Tyrtyshnikov [42] and the first author [14] considered the analysis of clustering properties of the spectra of specific Toeplitz preconditioned matrices obtained by means of the best known matrix algebras. Here we generalize this technique to a generic Banach algebra of matrices by devising general preconditioners related to...

We study the superoptimal Frobenius operators in several matrix vector spaces and in particular in the circulant algebra, by emphasizing both the algebraic and geometric properties. More specifically we prove a series of ``negative'' results that explain why this approximation procedure is not competitive with the optimal Frobenius...

We apply the natural pixel (NP) approach to the single photon emission computed tomography (SPECT) problem. The rotational invariance of the system induces a block circulant structure in the Gram matrix. This structure can be used to reduce the computational efforts needed for solving the inverse problem.

The Landweber method is a simple and flexible iterative regularization algorithm, whose projected variant provides nonnegative image reconstructions. Since the method is usually very slow, we apply circulant preconditioners, exploiting the shift invariance of many deblurring problems, in order to accelerate the convergence. This way reasonable...

No description

Regularization methods for inverse problems formulated in Hilbert spaces usually give rise to over-smoothness, which does not allow to obtain a good contrast and localization of the edges in the context of image restoration. On the other hand, regularization methods recently introduced in Banach spaces allow in general to obtain better...

We study the superoptimal Frobenius operators in the two-level circulant algebra. We consider two specific viewpoints: the regularizing properties in imaging and the computational effort in connection with the preconditioned conjugate gradient (PCG) method. Some numerical experiments illustrating the effectiveness of the proposed technique are...

No description

No description