Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices

We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges

Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let $I_k:=(-\pi,\pi)^k$, let $\mathcal M_s$ be the linear space of complex $s\times s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components $f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$. Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices of size $n_1\cdots n_ks$ generated by $f,g$. Then $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f$, i.e. the family of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic) spectral distribution described by the function $g^{-1}f$, provided $g$ possesses certain properties (which ensure in particular the invertibility of $T_n^{-1}(g)$ for all $n$) and the following topological conditions are met: the essential range of $g^{-1}f$, defined as the union of the essential ranges of the eigenvalue functions $\lambda_j(g^{-1}f),\ j=1,\ldots,s$, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case $g=1$. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.Comment: 18 pages, 26 figure

V-cycle optimal convergence for certain (multilevel) structured linear systems

In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or \u3c4 algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction. We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r = 1, . . . ,d, then the size of the algebraic system is N(n) = \u3a0r=1 d nr, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal. Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.cussed

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Symmetrization Techniques in Image Deblurring

This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditioning techniques, including when considering the deblurring of sparse images

On the regularizing power of multigrid-type algorithms

We consider the deblurring problem of noisy and blurred images in the case of known space invariant point spread functions with four choices of boundary conditions. We combine an algebraic multigrid previously defined ad hoc for structured matrices related to space invariant operators (Toeplitz, circulants, trigonometric matrix algebras, etc.) and the classical geometric multigrid studied in the partial differential equations context. The resulting technique is parameterized in order to have more degrees of freedom: a simple choice of the parameters allows us to devise a quite powerful regularizing method. It defines an iterative regularizing method where the smoother itself has to be an iterative regularizing method (e.g., conjugate gradient, Landweber, conjugate gradient for normal equations, etc.). More precisely, with respect to the smoother, the regularization properties are improved and the total complexity is lower. Furthermore, in several cases, when it is directly applied to the system $A{\bf f}={\bf g}$, the quality of the restored image is comparable with that of all the best known techniques for the normal equations $A^TA{\bf f}=A^T{\bf g}$, but the related convergence is substantially faster. Finally, the associated curves of the relative errors versus the iteration numbers are ``flatter'' with respect to the smoother (the estimation of the stop iteration is less crucial). Therefore, we can choose multigrid procedures which are much more efficient than classical techniques without losing accuracy in the restored image (as often occurs when using preconditioning). Several numerical experiments show the effectiveness of our proposals

Superoptimal Preconditioned Conjugate Gradient Iteration for Image Deblurring

We study the superoptimal Frobenius operators in the two-level circulant algebra. We consider two specific viewpoints: ( 1) the regularizing properties in imaging and ( 2) the computational effort in connection with the preconditioned conjugate gradient method. Some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems