Removal of blur in images based on least squares solutions

Abstract

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.

On removing blur in images using least squares solutions

“…We start from the mathematical model (2.1) from [19], in which the linear motion is a local phenomenon and no additional noise is included. This model relates an arbitrary ith row i = i,1 , .…”

“…A new approach in the restoration of a blurred image, which exploits a set of least squares solutions of the matrix equation (2.2), is proposed in the papers [19,20]. The least squares solutions are generated using the Moore-Penrose inverse.…”

“…The main purpose of this article is further investigation of the algorithm, introduced in [19,20], that allows us to remove a linear motion blur from images. The algorithm is based on the least squares solution of a matrix equation which represents the mathematical model of the linear motion blur.…”

“…Furthermore, the experimental and numerical results derived by applying the operator E on the restorations Y = TSVD and Y = TIK are described in the same section. Finally, the statistical description of results in [19,20] and also in Section 3 is presented in Section 4.…”

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Abstract.The further investigation of the image restoration method introduced in [19,20] is presented in this paper. Continuing investigations in that area, two additional applications of the method are investigated.

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