**1. Introduction**

In the last decades the Moore-Penrose pseudoinverse has found a wide range of applications in many areas of Science and became a useful tool for different scientists dealing with optimization problems, data analysis, solutions of linear integral equations, etc. At first we will present a review of some of the basic results on the so-called Moore-Penrose pseudoinverse of matrices, a concept that generalizes the usual notion of inverse of a square matrix, but that is also applicable to singular square matrices or even to non-square matrices.

The notion of the generalized inverse of a (square or rectangular) matrix was first introduced by H. Moore in 1920, and again by R. Penrose in 1955, who was apparently unaware of Moore's work. These two definitions are equivalent, (as it was pointed by Rao in 1956) and since then, the generalized inverse of a matrix is also called the Moore-Penrose inverse.

Let *<sup>A</sup>* be a *<sup>r</sup>* <sup>×</sup> *<sup>m</sup>* real matrix. Equations of the form *Ax* <sup>=</sup> *<sup>b</sup>*, *<sup>A</sup>* <sup>∈</sup> **<sup>R</sup>***r*×*m*, *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>r</sup>* occur in many pure and applied problems. It is known that when *T* is singular, then its unique generalized inverse *<sup>A</sup>*† (known as the Moore- Penrose inverse) is defined. In the case when *<sup>A</sup>* is a real*<sup>r</sup>* <sup>×</sup> *<sup>m</sup>* matrix, Penrose showed that there is a unique matrix satisfying the four Penrose equations, called the generalized inverse of *A*, noted by *A*†.

An important question for applications is to find a general and algorithmically simple way to compute *A*†. There are several methods for computing the Moore-Penrose inverse matrix (cf. [2]). The most common approach uses the Singular Values Decomposition (SVD). 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. Therefore, many other methods can be used for the numerical computation of various types of generalized inverses, see [16]; [25]; [30]. For more on the Moore-Penrose inverse, generalized inverses in general and their applications, there are many excellent texbooks on this subject, see [2]; [30]; [4].

©2012 Katsikis et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Katsikis et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Moore-Penrose pseudoinverse is a useful concept in dealing with optimization problems, as the determination of a ¸Sleast squaresT solution of linear systems. A typical application of ˇ the Moore-Penrose inverse is its use in Image and signal Processing and Image restoration.

The field of image restoration has seen a tremendous growth in interest over the last two decades, see [1]; [5]; [6]; [14]; [28]; [29]. The recovery of an original image from degraded observations is of crucial importance and finds application in several scientific areas including medical imaging and diagnosis, military surveillance, satellite and astronomical imaging, and remote sensing. A number of various algorithms have been proposed and intensively studied for achieving a fast recovered and high resolution reconstructed images, see [10]; [15]; [22].

The presented method in this article is based on the use of the Moore-Penrose generalized inverse of a matrix and provides us a fast computational algorithm for a fast and accurate digital image restoration. This article is an extension of the work presented in [7]; [8].
