An iterative method for Bayesian Gauss–Markov image restoration

The purpose of this paper is to introduce a new method for the restoration of images that have been degraded by a blur and an additive white Gaussian noise. The model adopted here is assumed to be Bayesian Gauss–Markov linear model. By exploiting the structure of the blurring matrix and by using Kronecker product approximations, the image restoration problem is formulated as matrix equations which will be solved iteratively by projection methods onto Krylov subspaces. We give some theoretical and experimental results with applications to image restoration.     

A Kronecker approximation with a convex constrained optimization method for blind image restoration

In many problems of linear image restoration, the point spread function is assumed to be known even if this information is usually not available. In practice, both the blur matrix and the restored image should be performed from the observed noisy and blurred image. In this case, one talks about the blind image restoration. In this paper, we propose a method for blind image restoration by using convex constrained optimization techniques for solving large-scale ill-conditioned generalized Sylvester equations. The blur matrix is approximated by a Kronecker product of two matrices having Toeplitz and Hankel forms. The Kronecker product approximation is obtained from an estimation of the point spread function. Numerical examples are given to show the efficiency of our proposed method.     

On the Moore–Penrose inverse in solving saddle‐point systems with singular diagonal blocks

This paper deals with the role of the generalized inverses in solving saddle‐point systems arising naturally in the solution of many scientific and engineering problems when finite‐element tearing and interconnecting based domain decomposition methods are used to the numerical solution. It was shown that the Moore–Penrose inverse may be obtained in this case at negligible cost by projecting an arbitrary generalized inverse using orthogonal projectors. Applying an eigenvalue analysis based on the Moore–Penrose inverse, we proved that for simple model problems, the number of conjugate gradient iterations required for the solution of associate dual systems does not depend on discretization norms. The theoretical results were confirmed by numerical experiments with linear elasticity problems. Copyright © 2011 John Wiley & Sons, Ltd.     

The Moore–Penrose inverse of a companion matrix

Necessary and sufficient conditions are given for the Moore–Penrose inverse of a companion matrix over an arbitrary ring to exist.     

Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F‐norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above‐mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore–Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore–Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore–Penrose metric generalized inverse of bounded linear operators in Banach spaces.     

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.     

Some identities for Moore–Penrose inverses of matrix products

A group of identities are established for the Moore–Penrose inverses and the weighted Moore–Penrose inverses of matrix products AB and ABC. Some consequences and applications are also presented.     

Moore–Penrose inverses of block circulant and block k-circulant matrices

The Moore–Penrose inverse A⁺ of a block circulant matrix whose blocks are arbitrary square matrices is obtained. An explicit form is given for A⁺ in terms of the blocks of A. The eigenvalues of A are determined in terms of the eigenvalues of the blocks where the blocks themselves are circulants.     

Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non‐linear minimization problem, usually with a Tikhonov‐like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal‐to‐noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non‐linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright © 2002 John Wiley & Sons, Ltd.     

A fast convergent iterative solver for approximate inverse of matrices

In this paper, a rapid iterative algorithm is proposed to find robust approximations for the inverse of nonsingular matrices. The analysis of convergence reveals that this high‐order method possesses eighth‐order convergence. The interesting point is that, this rate is attained using less number of matrix‐by‐matrix multiplications in contrast to the existing methods of the same type in the literature. The extension of the method for finding Moore–Penrose inverse of singular or rectangular matrices is also presented. Numerical comparisons will be given to show the applicability, stability and consistency of the new scheme by paying special attention on the computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Numericals for total variation-based reconstruction of motion blurred images

In this paper image with horizontal motion blur, vertical motion blur and angled motion blur are considered. We construct several difference schemes to the highly nonlinear term ·(u)/((|u|)～(1/2)2+β) of the total variation-based image motion deblurring problem. The large nonlinear system is linearized by fixed point iteration method. An algebraic multigrid method with Krylov subspace acceleration is used to solve the corresponding linear equations as in [7]. The algorithms can restore the image very well. We give some numerical experiments to demonstrate that our difference schemes are efficient and robust.     

A note on stable perturbations of Moore–Penrose inverses

Perturbation bounds for Moore–Penrose inverses of rectangular matrices play a significant role in the perturbation analysis for linear least squares problems. In this note, we derive a sharp upper bound for Moore–Penrose inverses, which is better than a well known existing one. Copyright © 2011 John Wiley & Sons, Ltd.     

More on generalized inverses of partitioned matrices with Banachiewicz–Schur forms

Necessary and sufficient conditions are derived for a 2-by-2 partitioned matrix to have {1}-, {1,2}-, {1,3}-, {1,4}-inverses and the Moore–Penrose inverse with Banachiewicz–Schur forms. As applications, the Banachiewicz–Schur forms of {1}-, {1,2}-, {1,3}-, {1,4}-inverses and the Moore–Penrose inverse of a 2-by-2 partitioned Hermitian matrix are also given.     

矩阵方程LX=M与LXK=M的实部有定解

本文利用矩阵的ＭＰ逆与奇异值分解作为工具,给出了矩方程ＬＸ＝Ｍ与ＬＸＫ＝Ｋ存在实部有定解的充要条件以及这种解的一般形式。     

The equivalence between ( AB )†= B † A † and other mixed-type reverse-order laws

The standard reverse-order law for the Moore–Penrose inverse of a matrix product is (AB)^??=?B ^? A ^?. The purpose of this article is to give a set of equivalences of this reverse-order law and other mixed-type reverse-order laws for the Moore–Penrose inverse of matrix products.     

Neighborhood filters and PDE’s

Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona–Malik equation, one of the first nonlinear PDE’s proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean curvature motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.This work has been partially financed by the Centre National d’Etudes Spatiales (CNES), the Office of Naval Research under grant N00014-97-1-0839, the Ministerio de Ciencia y Tecnologia under grant MTM2005-08567. During this work, the first author had a fellowship of the Govern de les Illes Balears for the realization of his PhD.     

Approximating the inverse and the Moore‐Penrose inverse of complex matrices

A parametric family of fourth‐order schemes for computing the inverse and the Moore‐Penrose inverse of a complex matrix is designed. A particular value of the parameter allows us to obtain a fifth‐order method. Convergence analysis of the different methods is studied. Every iteration of the proposed schemes involves four matrix multiplications. A numerical comparison with other known methods, in terms of the average number of matrix multiplications and the mean of CPU time, is presented.     

On Frobenius normwise condition numbers for Moore–Penrose inverse and linear least‐squares problems

Condition numbers play an important role in numerical analysis. Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using norms. In this paper, we give explicit, computable expressions depending on the data, for the normwise condition numbers for the computation of the Moore–Penrose inverse as well as for the solutions of linear least‐squares problems with full‐column rank. Copyright © 2007 John Wiley & Sons, Ltd.     

Further results on generalized inverses of tensors via the Einstein product

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.