Abstract. 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.

We propose an image restoration method. The method generalizes image restoration algorithms that are based on the Moore–Penrose solution of certain matrix equations that define the linear motion blur. Our approach is based on the usage of least squares solutions of these matrix equations, wherein an arbitrary matrix of appropriate dimensions is included besides the Moore–Penrose inverse. In addition, the method is a useful tool for improving results obtained by other image restoration methods. Toward that direction, we investigate the case where the arbitrary matrix is replaced by the matrix obtained by the Haar basis reconstructed image. The method has been tested by reconstructing an image after the removal of blur caused by the uniform linear motion and filtering the noise that is corrupted with the image pixels. The quality of the restoration is observable by a human eye. Benefits of using the method are illustrated by the values of the improvement in signal‐to‐noise ratio and in the values of peak signal‐to‐noise ratio. Copyright © 2013 John Wiley & Sons, Ltd.

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.

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 matrices provides the computation framework of the method. The method has been tested by reconstructing an image represented by an 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.

a b s t r a c tIn this article we provide a fast computational method in order to calculate the MoorePenrose 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.Ó