Dimitrios Pappas

An improved method for the computation of the Moore―Penrose inverse matrix

Abstract In this article we provide a fast computational method in order to calculate the Moore–Penrose inverse of singular square matrices and of rectangular matrices. The proposed method proves to be much faster and has significantly better accuracy than the already proposed methods, while works for full and sparse matrices.

Applications of the Moore-Penrose Inverse in Digital Image Restoration

This paper presents a fast computational method that finds application in a broad scientific field such as digital image restoration. The proposed method provides a new approach to the problem of image reconstruction by using the Moore-Penrose inverse. The resolution of the reconstructed image remains at a very high level but the main advantage of the method was found on the computational load that has been decreased considerably compared to the classic techniques.

Fast computing of the Moore-Penrose inverse matrix

In this article a fast computational method is provided in order to calculate the Moore-Penrose inverse of full rank m ×n matrices and of square matrices with at least one zero row or column. Sufficient conditions are also given for special type products of square matrices so that the reverse order law for the Moore-Penrose inverse is satisfied. is defined. In the case when T is a real m × n matrix, Penrose showed that there is a unique matrix satisfying the four Penrose equations, called the generalized inverse of T. A lot of work concerning generalized inverses has been carried out, in finite and infinite dimension (e.g., (2, 11)). In this article, we provide a method for the fast computation of the generalized inverse of full rank matrices and of square matrices with at least one zero row or column. In order to reach our goal, we use a special type of tensor product of two vectors, that is usually used in infinite dimensional Hilbert spaces. Using this type of tensor product, we also give sufficient conditions for products of square matrices so that the reverse order law for the Moore-Penrose inverse ((1, 4, 5)) is satisfied. There are several methods for computing the Moore-Penrose inverse matrix (cf. (2)). One of the most commonly used methods is the Singular Value Decomposition (SVD) method. This method is very accurate but also time-intensive since it requires a large amount of computational resources, especially in the case of large matrices. In the recent work of P. Courrieu (3), an algorithm for fast computation of Moore- Penrose inverse matrices is presented based on a known reverse order law (eq. 3.2

Full-rank representations of outer inverses based on the QR decomposition

Abstract An efficient algorithm for computing A T , S ( 2 ) inverses of a given constant matrix A, based on the QR decomposition of an appropriate matrix W, is presented. Correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full-rank factorization are derived. In particular cases we derive representations of { 2 , 4 } and { 2 , 3 } -inverses. Numerical examples on different test matrices (dense or sparse) are presented as well as the comparison with several well-known methods for computing the Moore–Penrose inverse and the Drazin inverse.

Digital Image Reconstruction in the Spectral Domain Utilizing the Moore-Penrose Inverse

The field of image restoration has seen a tremendous growth in interest over the last two decades. The recovery of an original image from degraded observations is a crucial method and finds application in several scientific areas including medical imaging and diagnosis, military surveillance, satellite and astronomical imaging, and remote sensing. The proposed approach presented in this work employs Fourier coefficients for moment-based image analysis. The main contributions of the presented technique, are that the image is first analyzed in orthogonal basis matrix formulation increasing the selectivity on image components, and then transmitted in the spectral domain. After the transmission has taken place, at the receiving end the image is transformed back and reconstructed from a set of its geometrical moments. The calculation of the Moore-Penrose inverse of 𝑟×𝑚 matrices provides the computation framework of the method. The method has been tested by reconstructing an image represented by an 𝑟×𝑚 matrix after the removal of blur caused by uniform linear motions. The noise during the transmission process is another issue that is considered in the current work.

THE RESTRICTED WEIGHTED GENERALIZED INVERSE OF A MATRIX

We introduce the T-restricted weighted generalized inverse of a singular matrix A with respect to a positive semidefinite matrix T, which defines a seminorm for the space. The new approach proposed is that since T is positive semidefinite, the minimal seminorm solution is considered for all vectors perpendicular to the kernel of T.

Application of the Least Squares Solutions in Image Deblurring

A new method for the reconstruction of blurred digital images damaged by separable motion blur is established. The main attribute of the method is based on multiple applications of the least squares solutions of certain matrix equations which define the separable motion blur in conjunction with known image deconvolution techniques. The key feature of the proposed algorithms is reflected in the fact that they can be used only in symbiosis with other image restoration algorithms.

Minimization of quadratic forms using the Drazin-inverse solution

We analyse the problem of constrained minimization of the real quadratic functional, subject to the inconsistent system of linear equations, where is a positive definite or positive semidefinite matrix. Both cases are analysed separately, and respective relationships have been established between the solution of the original problem and the Drazin-inverse solution of the equation. In the special case when is a positive definite and, we show that the solution of the problem can be represented as a particular-inverse solution.

Restricted linear constrained minimization of quadratic functionals

In this work, a linearly constrained minimization of a positive semidefinite quadratic functional is examined. We propose two different approaches to this problem. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive semidefinite operator related to the functional, and considering as constraint a singular operator. The difference between the proposed approaches for the minimization and previous work on this problem is that it is considered for all vectors belonging to the least squares solutions set, or to the vectors perpendicular to the kernel of the related operator or matrix.

Computing {2,4} and {2,3}-inverses by using the Sherman-Morrison formula

A finite recursive procedure for computing {2,4} generalized inverses and the analogous recursive procedure for computing {2,3} generalized inverses of a given complex matrix are presented. The starting points of both introduced methods are general representations of these classes of generalized inverses. These representations are formed using certain matrix products which include the Moore-Penrose inverse or the usual inverse of a symmetric matrix product and the Sherman-Morrison formula for the inverse of a symmetric rank-one matrix modification. The computational complexity of the methods is analyzed. Defined algorithms are tested on randomly generated matrices as well as on test matrices from the Matrix Computation Toolbox.