**4. Experimental results**

8 Will-be-set-by-IN-TECH

method in order to calculate the Moore-Penrose inverse of full rank *r* × *m* matrices. The method retains a restored signal whose norm is smaller than any other solution. The

The criterion for restoration of a blurred image that we are using is the minimum distance of

to the constraint �*Hxin* − *xout*� = 0. In fact, zero is not always attained, but following

In general, the PSF varies independently with respect to both (horizontal and vertical) directions, because the degradation of a PSF may depend on its location in the image. An example of this kind of behavior is an optical system that suffers strong geometric aberrations. However, in most of the studies, the PSF is accurately written as a function of the horizontal

A blurred image that has been degraded by a uniform linear motion in the horizontal direction, usually results of camera panning or fast object motion can be expressed as follows,

> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ·

where the index *n* indicates the linear motion blur in pixels. The element *k*1,..., *kn* of the

*n n*−1 ∑ *h*=0

that describes an underdetermined system of *r* simultaneous equations and *m* = *r* + *n* − 1 unknowns. The objective is to calculate the original column per column data of the image.

As we have seen, the matrix *H* is a *r* × *m* matrix, and the rank of *H* is less or equal to *m*. Therefore, the linear system of equations is underdetermined. The proper generalized inverse for this case is a left inverse, which is also called a {1,2,4} inverse, in the sense that it needs to

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*xin*\_1 *xin*\_2 *xin*\_3 . . . *xin*\_*<sup>m</sup>*

*xin*(*i* + *h*)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *xout*\_1 *xout*\_2 *xout*\_3 . . . *xout*\_*<sup>r</sup>*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(14)

*<sup>T</sup>* of a degraded blurred

*in* − *xout*�),

*in* are the first *r* elements of the unknown image *xin* that has to be recovered subject

computational load for the method is compared with the already known methods.

min(�*x*<sup>∗</sup>

and vertical displacements independently of the location within the field of view.

*k*<sup>1</sup> ... *kn* 0000 0 *k*<sup>1</sup> ... *kn* 000 0 0 *k*<sup>1</sup> ... *kn* 0 0

0 0 0 ... *k*<sup>1</sup> ... *kn*

Equation (3) can also be written in the pointwise form for *i* = 1, . . . ,*r*,

For this reason, given each column [*xout*\_1, *xout*\_2, *xout*\_3,... *xout*\_*r*]

image *xout*, Eq. (3) results the corresponding column

[*xin*\_1, *xin*\_2, *xin*\_3,..., *xin*\_*m*]

*xout*(*i*) = <sup>1</sup>

*<sup>T</sup>* of the original image.

the measured data, i.e.,

as desribed in Eq. (13):

Proposition 0.1(ii) the norm is minimized.

**3.1. The generalized inverse approach**

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

matrix are defined as: *kl* = 1/*n* (1 ≤ *l* ≤ *n*).

. . . . . . . . . . . . . . . . . . . . .

where *x*∗

In this section we apply the proposed method on an boat picture and present the numerical results.

The numerical tasks have been performed using Matlab programming language. Specifically, the Matlab 7.4 (R2007b) environment was used on an Intel(R) Pentium(R) Dual CPU T2310 @ 1.46 GHz 1.47 GHz 32-bit system with 2 GB of RAM memory running on the Windows Vista Home Premium Operating System.

#### **4.1. Recovery from a degraded image**

Figure 1(a) provides the original boat picture. In Figure 1(b), we present the degraded boat picture where the length of the blurring process is equal to *n* = 60. Finally, in Figure 1(c) we present the reconstructed image using the Moore- Penrose inverse approach. As we can see, it is clearly seen that the details of the original image have been recovered.

The Improvement in Signal to Noise Ratio (ISNR) has been chosen in order to present the reconstructed images obtained by the proposed algorithm. It provides a criterion that has been used extensively for the purpose of objectively testing the performance of image processing algorithms expressed as:

$$ISNR = 10\log\_{10}\left\{\frac{\sum\_{i,j} \left[\boldsymbol{\chi}\_{in}(i,j) - \boldsymbol{\chi}\_{out}(i,j)\right]^2}{\sum\_{i,j} \left[\boldsymbol{\chi}\_{in}(i,j) - \boldsymbol{\mathfrak{x}}\_{in}(i,j)\right]^2}\right\}.$$

10 Will-be-set-by-IN-TECH 354 MATLAB – A Fundamental Tool for Scienti c Computing and Engineering Applications – Volume 1

**Figure 1.** (a) Original Image (b) Blurred image for a length of the blurring process n = 60 (c) Restoration of a simulated degraded image with a length of the blurring process n = 60.

where *xin* and *xout* represent the original deterministic image and degraded image respectively, and *x*ˆ*in* is the corresponding restored image. Figure 2(a) shows the corresponding ISNR values. for increasing the number of pixels in the blurring process *n* = 1, . . . , 60.

The second set of tests aimed at accenting the reconstruction error between the original image *xin* and the reconstructed image *x*ˆ*in* for various values of linear motion blur, *n*. The calculated quantity is the normalized reconstruction error given by

$$E = \frac{1}{\sqrt{\sum\_{i=1}^{r} \sum\_{j=1}^{m} [\chi\_{in}(i,j)]^2}} \sqrt{\sum\_{i=1}^{r} \sum\_{j=1}^{m} [\chi\_{in}(i,j) - \pounds\_{in}(i,j)]^2}$$

using the generalized inverse reconstructed method.

Figure 2(b) shows the reconstruction error by increasing the number of pixels in the blurring process *n* = 1, . . . , 60.

#### **4.2. Recovery from a degraded and noisy image**

Noise may be introduced into an image in a number of different ways. In Equation (10) the noise has been introduced in an additive way. Here, we simulate a noise model where a number of pixels are corrupted and randomly take on a value of white and black (*salt and pepper* noise) with noise density equal to 0.02. The image that we receive from a faulty transmission line can contain this form of corruption. In Figure 3(b), we present the original boat image while a motion blurred and a salt and pepper noise has been added to it.

Image processing and analysis are based on filtering the content of the images in a certain way. The filtering process is basically an algorithm that modifies a pixel value, given the original value of the pixel and the values that surrounding it. Accordingly, Figure 4(a) provides a graphical representation for the ISNR of the reconstructed and filtered image for different values of *n*. Moreover, Figure 4(b) shows the reconstruction error by increasing the number of pixels in the blurring process *n* = 1, . . . , 60.

10 Will-be-set-by-IN-TECH

**Figure 1.** (a) Original Image (b) Blurred image for a length of the blurring process n = 60 (c) Restoration

where *xin* and *xout* represent the original deterministic image and degraded image respectively, and *x*ˆ*in* is the corresponding restored image. Figure 2(a) shows the corresponding ISNR values. for increasing the number of pixels in the blurring process *n* = 1, . . . , 60.

The second set of tests aimed at accenting the reconstruction error between the original image *xin* and the reconstructed image *x*ˆ*in* for various values of linear motion blur, *n*. The calculated

> *r* ∑ *i*=1

Figure 2(b) shows the reconstruction error by increasing the number of pixels in the blurring

Noise may be introduced into an image in a number of different ways. In Equation (10) the noise has been introduced in an additive way. Here, we simulate a noise model where a number of pixels are corrupted and randomly take on a value of white and black (*salt and pepper* noise) with noise density equal to 0.02. The image that we receive from a faulty transmission line can contain this form of corruption. In Figure 3(b), we present the original

Image processing and analysis are based on filtering the content of the images in a certain way. The filtering process is basically an algorithm that modifies a pixel value, given the original value of the pixel and the values that surrounding it. Accordingly, Figure 4(a) provides a graphical representation for the ISNR of the reconstructed and filtered image for different values of *n*. Moreover, Figure 4(b) shows the reconstruction error by increasing the number of

boat image while a motion blurred and a salt and pepper noise has been added to it.

*m* ∑ *j*=1

[*xin*(*i*, *<sup>j</sup>*) − *<sup>x</sup>*ˆ*in*(*i*, *<sup>j</sup>*)]<sup>2</sup>

*<sup>j</sup>*=1[*xin*(*i*, *j*)]<sup>2</sup>

of a simulated degraded image with a length of the blurring process n = 60.

quantity is the normalized reconstruction error given by

*<sup>E</sup>* <sup>=</sup> <sup>1</sup> ∑*r <sup>i</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>m</sup>*

using the generalized inverse reconstructed method.

**4.2. Recovery from a degraded and noisy image**

pixels in the blurring process *n* = 1, . . . , 60.

process *n* = 1, . . . , 60.

**Figure 2.** (a) ISNR and (b) Reconstruction Error calculations vs number of pixels in the blurring process (*n* = 1, . . . , 60).

**Figure 3.** (a) Noisy Image (b) Blurred and noisy (salt and pepper) image for length of the blurring process n = 60 (c) Restoration of a simulated degraded (n = 60) and noisy (salt and pepper) image.

**Figure 4.** (a) ISNR and (b) Reconstruction Error calculations for a noisy and blurred image vs number of pixels in the blurring process (*n* = 1, . . . , 60).
