Mathematical Problems in Engineering

Volume 2015, Article ID 298689, 18 pages

http://dx.doi.org/10.1155/2015/298689

## Application of the Least Squares Solutions in Image Deblurring

^{1}Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia^{2}Faculty of Computer Science, Goce Delčev University, Goce Delčev 89, 2000 Štip, Macedonia^{3}Department of Economics, Division of Mathematics and Informatics, National and Kapodistrian University of Athens, Sofokleous 1 Street, 10559 Athens, Greece^{4}Department of Statistics, Athens University of Economics and Business, 76 Patission Street, 10434 Athens, Greece

Received 2 September 2014; Revised 20 January 2015; Accepted 28 January 2015

Academic Editor: Joao B. R. Do Val

Copyright © 2015 Predrag S. Stanimirović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

In practice, the recorded image unavoidably represents a degraded version of the original scene because of inevitable imperfections in the imaging and capturing process. Medical images, satellite images, astronomical images, or poor-quality family portraits are often blurred. A wide range of different degradations need to be taken into account, covering, for instance, noise, blur, illumination, and color imperfections, and geometrical degradations. The elimination of these imperfections is crucial in various tasks of image processing and image analysis. Image restoration methods are used for reconstructing the original image from a degraded model. The problem of image restoration has been studied in many articles and books [1–7].

The application of the least squares solutions in image processing (and in image restoration particularly) is not frequently investigated so far. An application of the least square techniques in image processing is presented in [8]. Removal of blur in images based on the least squares solutions is investigated in [9, 10]. Particularly, an application of the least squares solution of minimal norm in image deblurring is investigated in [11–13]. Our main intention in this paper is further investigation and extension of the algorithms introduced in [9, 10] that allows us to remove a linear or separable motion blur from images. The algorithms presented in these papers are based on the least squares solution of a matrix equation which represents the mathematical model of the linear or separable motion blur. The least squares solution, denoted by , includes the Moore-Penrose inverse of the blurring matrix as well as an arbitrary matrix . The particular least squares solution, based on the Moore-Penrose inverse, was investigated in [11–13].

The main goal of this paper is the development of an algorithm that allows us to remove a motion blur from images by means of the consecutive applications of the least squares solutions of a matrix equation which models the separable two-dimensional blurring process. The least squares solutions are matrix expressions that include the Moore-Penrose inverse of the blurring matrix as well as an appropriately chosen arbitrary matrix. The matrix transformation defined in this way is idempotent (see [9, 10]). This difficulty forces us to find the way to manage consecutive applications of the least squares solutions from [9, 10]. Significant improvements in ISNR and PSNR values are attained applying such an approach with respect to the classical approach based on the Moore-Penrose solution of certain matrix equations, which is investigated in [11, 12], as well as with respect to a single application of the least squares solutions, used in [9, 10].

The paper is organized as follows. Motivation and description of the method are presented in the second section. The main algorithm which enables the iterative application of the least squares solution on images damaged by a separable motion blur is also presented in the second section, as well as the application of the method on blurred and noisy images. Results generated by performed numerical experiments are investigated in Section 3.

#### 2. Motivation and Description of the Method

The process of the separable blurring assumes that the blurring of the columns in the image is independent of the blurring of the rows. The separable blurring is modeled by two blurring matrices, and , both of the general form where , , are real numbers and the positive integer indicates the length of linear motion blur in pixels. The relation between the original image and blurred image is expressed by the following matrix equation, considered in [10]: In (2), it is assumed that , , where (resp., ) is the length of the horizontal (resp., vertical) blurring in pixels.

Our approach provides a new method for restoration of a blurred image which is based on multiple applications of the least squares solutions of the matrix equations (2) in symbiosis with other well-known image deblurring techniques. In general, the proposed algorithm is aimed at solving the matrix equation (2).

The least squares solution of (2) has the general form (used in [10]) where is an arbitrary matrix of appropriate dimensions.

The transformation can be used as a deconvolution of a blurred image . The blurred image , used in (3), can be determined in different ways. There are no specific conditions for that; any random matrix can be transformed into . Continuing investigations from [9, 10], appropriate choices for are chosen as the results of particular image deblurring processes; that is, is an arbitrary restoration of the degraded image. In that case, is an attempt to derive further improvements in the restoration of .

In the case , where is the zero matrix of appropriate dimensions, produces the next approximation of the original image : The approach which assumes the condition in (3) exploits the Moore-Penrose solution of the matrix equation (2), that is, the least squares solution of minimal norm. About the least squares and minimal norm properties of the Moore-Penrose solution, see main references [14, 15]. But the minimal norm attribute associated with the Moore-Penrose solution may be, in most of cases, only the redundant property. Indeed, our experience from [9, 10] confirms that only when the matrix is selected to be “far” from the original image, the improvement of is still worse with respect to the Moore-Penrose reconstruction (corresponding to the case ). Some of the examples that confirm this expectation are studied in [9, 10]. We follow the main goal of the papers [9, 10]; that is, we will determine in such a way that the approximation produces better values for ISNR and PSNR with respect to the solution which is used in [11, 12].

Except the election , the results generated by applying the Wiener filter (WF) and the constrained least-squares (CLS) filter are used as two appropriate choices of the matrix in [9, 10]. A description of the WF and CLS filters can be found in [2]. A more advanced approach for the selection of the matrix is based on the moment based methods. The Haar basis and the Fourier basis mentioned above are usually referred to as the most popular moment based methods. This approach is considered in [10]. For more details on the Fourier and the Haar basis, see [9, 12]. An algorithm for image deconvolution from the geometric moments of an image which is degraded by a circular or elliptical Gaussian point-spread function is considered in [16]. For additional information on the moment based image reconstruction methods, the reader is referred to [16–20]. A short preview of main image restoration methods, used for obtaining possible reconstructions , is presented in [10]. A detailed description of these methods can be found in [13]. A recent survey book presenting all the modern image reconstruction methods is given in [21].

Our improvement of the methods defined in [9, 10] arises from our intention to apply the operator repeatedly. But the authors in [9] showed that the operator is idempotent, which implies . This property of the operator makes its application on redundant. In the present paper, we find a possibility for multiple applications of the operator . The main idea is to use instead of , where denotes the application of a previously defined image deconvolution algorithm in the th iteration.

Let us denote by the reconstructed image after steps. The following approximations for further restorations are considered:(a) is derived applying the Haar based reconstructed image for selected , ;(b) is derived applying the Fourier based reconstructed image for selected ;(c) is derived applying the constrained least-squares filter;(d), derived applying the Wiener filter;(e), derived applying the Lucy-Richardson algorithm.

On the other hand, an improper and unpredictable behavior of is observed in [9, 10]. Namely, to some extent, a better improvement implies better restoration , which is confirmed by the inequality . But, after the occurrence of a limit, which is not known in advance, the opposite situation () is observed.

One of the possible ways to achieve multiple applications of the operator is its alternating application with another image restoration method, as it is described in Algorithm 1.

*Algorithm 1 (iterative application of the operator ). *
Consider the following.*Require *, where denotes a blurred image.(1)Initial step is
(2)Set the number of the iterative steps to initial value .(3)Compute
where .(4)If a selected stopping criterion is fulfilled set and go to Step (3).(5)If the stopping criterion is fulfilled return the output .

*Remark 2. *Iterations (6) should provide two improvements in the reconstruction, in each iteration, as follows:(i)the first improvement is , which gives a restoration of the previous iteration by means of an image restoration method ;(ii)the second improvement arises from , which gives further reconstruction of by means of the least squares solution .

*Remark 3. *The choice of the stopping criterion in Step (4) of Algorithm 1 is, at this moment, an undeterminable problem.(i)One possible choice is to stop the cycle when the inequality is satisfied.(ii)But this choice may cause blocking of (possible) improvements in further steps.* *For this purpose, it seems that the terminating criterion defined by an in advance defined number of iterative steps is a better choice. In our numerical experiments, we will use this stopping criterion.(iii)Also, it is reasonable to stop the cycle in Step (4) when the inequality is satisfied several times consecutively.

*Remark 4. *Essentially, iterations in (6) are based on the least squares solution of the dynamical matrix equation
The model (7) essentially means that the iteration is considered as a “blurred image” of the next (unknown) iteration . After resolving (7) with respect to the unknown matrix , we derive the next iteration in terms of the current iteration and the pseudoinverses of the blurring matrices and .

*Remark 5. *The authors of the papers [9, 10] have computed , where is an image restoration method, and simply compared with . The main advantage of the proposed Algorithm 1 is that it makes repetitive use of the operator in symbiosis with selected deblurring methods on the blurred image and its reconstructions. Since the operator is idempotent, Algorithm 1 defines an approach to improve the results obtained in [9, 10] by the multiple application of the (deblurring) transformation .

##### 2.1. Repetitive Least Squares Image Deblurring and Denoising

Noise is unavoidable in most of applications, so that a real observation is thus often modeled by the following mathematical model: where is additive noise and is the blurred noisy image.

Algorithm 6 can be adopted to restore the original image from a blurred and noisy image, using the least squares solution of the mathematical model (8).

*Algorithm 6 (iterative application of the operator on blurred and noisy image). *
Consider the following.*Require *, where denotes a blurred noisy image.(1)Initial step is
(2)Obtain the restored image by applying filtering process on the image obtained in Step (1).(3)Set .(4)Compute
where .(5)Obtain the restored image by applying filtering process on the image obtained in Step (4).(6)If a selected stopping criterion is fulfilled, set and return to Step (4).(7)If the stopping criterion is fulfilled then return the output .

#### 3. Experimental Results

In this section, we investigate the numerical results generated by applying the two proposed algorithms. The experiments are performed using Matlab programming language on an Intel Core i5 CPU M430 @ 2.27 GHz 64/32-bit system with 4 GB of RAM memory running on Windows 7 Ultimate Operating System.

##### 3.1. Application of the Method on Blurred Images

*Example 7. *In order to emphasize the importance of the number of iterative steps in Algorithm 1, the number of moments in Figures 1(a) and 1(b) was kept constant with . The length of blurring is . The reconstructions obtained by the Haar basis (resp., Fourier transform) are denoted by (resp., ).

Both the graphs represented in Figures 1(a) and 1(b) show a similar behavior. In fact, the ISNR and PSNR values increase initially and then converge, slowly growing, to a constant value. Also, both ISNR and PSNR values generated by applying on Fourier basis are greater (better) with respect to the corresponding values generated by applying on the Haar basis.

Also, the graphs in Figures 1(a) and 1(b) show that in advance the predefined number of iterative steps is a suitable termination criterion of Algorithm 1.