The Atmospheric Turbulence Blur Information Technology Essay

Image restoration is the technique(s) that can approximate the unspoiled image from the degraded image that has blur and noise. These techniques would perform operations on the degraded image to estimate the original one and most of these operations are mathematical operations [10].

Before the use of the image restoration techniques, the properties of the degradations that affect an image must be known in advance because in many cases this information cannot be obtained from the image formation process. Blur identification is one of the methods used to approximate the characteristics of the imperfect imaging system from the spoiled image before the restoration operation began. The mix between image restoration and blur identification is called blind image deconvolution [1].

Images acquired or observed by an imaging device are not often the same as the ideal images, but these images are the degraded edition of the ideal images [2] [3] [11]. These degradations occur due to many reasons during the process of capturing the image, for example remote sensing and astronomy images are frequently degraded with linear blur (atmospheric turbulence blur) and additive Gaussian noise [17] due to the atmosphere and the environment that the image is captured in. These degradations can be divided into many types, but the most important types of the degradations that this thesis deals with are blur and noise [4].

Image enhancement techniques are different from the image restoration techniques, because image enhancement techniques used mostly in the true images that does not have any degradations affecting it. Mostly image enhancement techniques work with color, brightness and contrast enhancement, but image restoration deals with degraded images and try to estimate the original one [5].

Sometimes these degradations came separate from each other and sometimes the 2 types of degradations come together in one image, the situation of finding these degradations in the image depend of the state that the image was captured.

2.2 Image degradations:

There is a huge amount of degradations that an image can be affected with; the most familiar degradations are blur and noise. The types of noise and blur that this thesis is studying are:

2.2.1 Atmospheric turbulence blur

One of the common sources of distortion that affect an image is atmospheric turbulence. Atmospheric turbulence can degrade images taken by cameras viewing scenes from long distances. This case is especially common in remote sensing, aerial imaging and astronomy [12]. For example, stars in outer space viewed through telescopes appear blurred since the Earth’s atmosphere degrades the image quality [6] [18].

Blur is an observable fact that frequently degrades the image in a deterministic manner. Diffraction, lens aberration, relative motion between the scene and the viewer, imaging with a randomly vibrating conventional or scanning camera, long exposure imaging [15], misfocus, poor telescope tracking, finite aperture size [16] ,and dust particles on the surface of the lens are the main reason for the blur to happen [11] [12] [13].

The repeating of day-night cycle is also accountable for changing the temperature of the Earth’s surface (cool or heat), creating large scale atmospheric motions. Once these motions become turbulent, large-scale eddies disintegrate into smaller and smaller eddies leading to temperature fluctuations. Since the refractive index of air is temperature sensitive, atmospheric turbulence will change the path and phase (scintillation) of the light that falls onto the telescope aperture, and thereby limits the effective resolution of a long-exposure image [7] [14]. Figure 2.1 shows the operation of the blurring [24].

Figure 2.1 shows the blur operation because of the turbulent atmosphere [24]

2.2.2 Additive Gaussian noise

Noise that affects an image has a variety of types; still these types can be divided to two major divisions: additive noise and multiplicative noise. The additive noise is the important one in this thesis because the images used in this thesis are degraded with additive Gaussian noise [8].

Noise can pollute an image either at image transmission from source to destination or at the time the image is captured and generated or an error at the imaging system [9].

The noise degrades the image in a stochastic way [11], also noise degrades an image because of many factors but the most important factors are capturing the image in a low illumination environment, noise inherent in the electronics of the imaging sensors and detectors, errors in transmission [11], quantization errors, sensor measurement errors, model errors [19], and low-contrast objects [18]. See figure 2.2 images that have been took from Cassini image database from the internet [25].

Additive zero-mean Gaussian noise means that a value drawn from a zero-mean Gaussian probability density function is added to the true value of every pixel [8].

Figure 2.2 shows images took from the space polluted with additive Gaussian noise [25]

2.3 Debluring Algorithms:

There is a lot of debluring algorithms that can be used to deblur a degraded image, these algorithms have a lot of categories for instance there is linear algorithms used in the case of low or high light existence in the image, non-linear algorithms are supposed to do better job that the linear algorithms, because non-linear algorithms include arithmetical forms for the noise in the data.

Another category of the debluring algorithms are iterative and non-iterative algorithms, iterative algorithms are algorithms that the debluring algorithm is applied a number of times to get an improved result of the image. The non-iterative algorithms are algorithms that apply the debluring algorithm one time only to get the improved result.

The advantage of the iterative algorithms is the method is applied more than one time to give the improved enhanced image; the disadvantage of it is the computation time is high. Below there are different kinds of deblurring algorithms that have been used to restore the blurry images [30].

2.3.1 Inverse Filter:

Inverse filter is one of the oldest and most famous filters in deblurring a degraded image, this filter can deblur an image but it don’t take in concern the noise existence in the image so this filter deals with blur only.

If it happens that the image is degraded with additive white noise in spite of the blur existence, the noise would be amplified in the process of deblurring the image and the image would be distorted, so this filter must be used with images that have been contaminated by blur only.

The inverse filter is an iterative filter that must be applied more than one time to restore the image, despite the limitation that this filter has, it’s still widely used in the image debluring field [33].

2.3.2 Wiener Filter:

This filter is one of the known filters used to deblur an image, this filter can work with an image that is affected with blur and noise without amplifying the noise too much, wiener filter considered to be an enhanced version of the inverse filter because it’s more stable in debluring in the case of noise existence and also it employ a priori knowledge of the noise. The wiener transfer function is selected to reduce the mean square error by statistical information on the image and the noise affecting the image [32].

2.3.3 Kalman Filter:

This filter is one of the famous filters in the image restoration field that lately has been considered in many restoration methods. Kaman filter is a recursive filter produced by R. E. Kalman in 1960; this filter was planned as an improved version of the Bayesian technique, this filter is consist of numerous mathematical equations that offer a competent recursive answer of the least-squares technique.

The Kalman filter is incredibly strong due to various features, like supporting estimations for the past, present, and future conditions. This filter considered to be more precise than the wiener filter, or as minimum similar to wiener filter. One of the major drawbacks of this filter is it take too much time to restore the image because this filter has a numerous mathematical equations to calculate in order to give the result.

The advantage of Kalman filter is that the result is more precise than other methods especially in the case of atmospheric turbulence blur, thus the time factor is not that important now a days because it give better results. D. Arbel, E. Cohen, M. Citroen, D.G. Blumberg, and N.S. Kopeika studied the restoration of blurry image using Kalman filter [34].

2.3.4 Poisson MAP Algorithm

This algorithm is an iterative algorithm that is steady with noise; this algorithm can work properly if knowledge about the image is available like type of noise and the PSF. This algorithm called Poisson Maximum A Posteriori.

The information about the image is so important to the this algorithm in order to deblur like the distortion factor (PSF) also the type of PSF, the type of the additive noise affecting the image. This method deblur the image but does not denoise it, the formula of this algorithm is:

Where f 0= g, g is the degraded image, f is the estimated image, and H is the convolution of the image with the point spread function (PSF). In this algorithm the prior information is unknown, so the image that is result from this algorithm is the best estimation [26].

2.3.5 Van Cittert Algorithm:

This algorithm is an iterative algorithm that does not take in concern the existence of the additive noise in the image so if the image contains additive noise, the noise would be amplified in the restoration process and the resulted image would be severely distorted.

2.3.6 Landweber Algorithm:

This algorithm is the same as Van Cittert algorithm but it apply an additional convolution operation to the Van Cittert algorithm to make the algorithm steady to noise existence, the landweber algorithm is an iterative inverse algorithm, and the formula of this algorithm is:

Where f 0= g, g is the degraded image, f is the estimated image, and H is the convolution of the image with the point spread function (PSF). λ stands for a constant that manages the convergence, the value of λ is in the range 0 to 2, and HT is the transpose of the point spread function (PSF). The advantage of this algorithm is its more stable against noise and it got more convergence rate [28].

2.3.7 Total Variation Algorithm:

This algorithm reduces the image energy, and it has two iterations only, the first iteration used for edge preserving, the second iteration used for deblur the image. This algorithm uses potential function (ψ) for edge preserving. This algorithm concentrate on edge preserving, but it’s hard to implement due to the computational power complexity [29].

2.3.8 E-M Algorithm:

This algorithm is used to deblur images contain a specific type of noise that is the Poisson noise. The E-M algorithm is used widely in the restoration of the medical images.

This algorithm is an iterative algorithm and it’s similar to Richardson – Lucy algorithm but the main difference between them is that the Richardson – Lucy does not specify any type of noise to work with but E-M algorithm work with the Poisson noise only.

The equation of this algorithm is the same equation of the Richardson – Lucy algorithm but here each pixel in the blurred image must processed by Poisson process then the equation of the E-M algorithm can be used [31].

2.3.9 Tikhonov-Miller Algorithm:

This algorithm is used to restore images that has been affected by blur and considered to be of the well-known algorithms in this field [35]. This method is counted to be a canonical method, in fact this method has more complicated stochastic restoration operations, such as Wiener filtering, also this method involves no a priori guess concerning the characteristics of the noise [36] [37].

The original form of the Tikhonov-Miller algorithm is a non iterative algorithm, although Tikhonov-Miller algorithm can be used as an iterative algorithm called the iterative constrained Tikhonov- Miller restoration (ICTM).

The original form of TM algorithm has negative values which considered of the drawbacks of the original TM algorithm, but in the iterative version of it (ICTM), the trouble of the negative values is resolved. In it, each accomplished approximation is cut and used again for the next iteration [38].

2.4 Denoising Algorithms

Noise occurs to the image due to many reasons that mentioned in the literature review, the noise that affects the image can be represented as:

R = I + n

Where (R) is the viewed image, (I) is the original image and (n) is the additive noise to the image. The denoising algorithms are algorithms that remove or reduce the noise from the image; many algorithms have been introduced to denoise an image, but the common things between these algorithms are the noise model and the local or global generic image smoothness model.

These algorithms have a lot of difference, some of them can be applied in the spatial domain, some can be applied in the frequency domain, and some of them use wavelet.

Also these denoising algorithms can be iterative or non iterative, iterative is the method can be applied more than one time to give an enhanced result; the non iterative algorithms are applied one time only to give the result.

Between all the differences in the denoising algorithms, still all the algorithms share one goal to accomplish that is removing or reducing the noise from the image [39].

2.4.1 Typical Image Denoising Techniques:

The use of the classical techniques to denoise an image employed due to the Necessity of noise reduction or removal without much affecting the edges and high frequency parts of the image, these techniques considered to preserve the edge and high frequencies parts in the image. These techniques have 2 kinds, spatial domain techniques and frequency domain techniques, in that order, they will be explained.

2.4.1.1 Spatial Domain Filters:

This kind of filters is widely used in the image processing filed, the spatial domain filters work and process the real pixels values of the image, this kind of filters is simple and easy to use, and here are some of these filters:

2.4.1.1.1 Gaussian Smoothing:

Gaussian smoothing is one of the known methods in image denoising, the linear filtering can be done by convolving the image by a Gaussian kernel, and the smoothing of this method came from the positive values of the kernel. This theorem came from Gabor and the equation of this method is:

u – Gh * u = -h2 Δu + o (h2)

Where (u) is the noisy image, (Gh) is the Gaussian kernel, (h) is the is the filtering parameter that rely on the guess of the noise variance. Here in this method, noise reduction relay on the neighborhood pixels because the reduction is done by averaging [40].

2.4.1.1.2 Median Filter

The median filter is one of the famous filters used to denoise (smooth) images that degraded with additive noise in spatial domain, mostly the median filter work with salt and pepper noise and additive Gaussian noise.

The median filter basically works as swap the original pixel value with the median value of the surrounding neighborhood pixels values depending on the median filter size, the preferred size is an odd size like 3X3, 5X5 ,7X7 … etc.

The median of the surrounding neighborhood values can be calculated by ascending sorting these values and then swap the original pixel value with middle pixel value. In the case of the number of the neighborhood pixels was even, the average of the two middle pixels values is applied.

One of the advantages of the median filter that the new pixel value that will replace the original one is a realistic value came from the same image not by doing a mathematical operation so the result would be much suitable for a denoising [42] [43].

2.4.1.1.3 Mean Filter

The mean filter is also another spatial domain filter used to denoise an image degraded by noise, this filter work with different kind of noise like Gaussian noise and salt and pepper.

This filter basically works as swapping the original pixel value with the average value of the original pixel and the neighborhood pixels. The size of the mask filter is important, the filter preferred to be an odd number such as 3X3, 5X5, 7X7 … etc.

The concept of the mean filter is to convolve the mask filter with the image; the result would be divided by the sum of the pixels values in the mask filter, the result of this operation is a new value that would be swapped with the original pixel value.

The new pixel value of the mean filter is not a realistic number that came from the neighborhood, but a number calculated by applying a mathematical equation, this is one of the reasons why the median filter is considered to be better than the mean filter [44][46].

2.4.1.1.4 Lee Filter

Another spatial adaptive filter used to denoise an image is the Lee filter; this filter is a local statistics one that uses a mask. The Lee filter mask have coefficients, those coefficients are functions to the local noise. For Lee filter, the pixel location is (m, n), the value of the noisy pixel is a(m, n), the approximation value of the denoised pixel is b(m, n) as the following:

b(m, n) = α m, n a(m, n) + β m, n

Where (α, β) are used to reduce the mean square error. The mask size in the Lee filter is so significant for the denoising process; the minimum mask size must be as a minimum 5X5 for a sensible approximation, Lee proved that both 7X7 and 5X5 work well in the denoising operation, the default mask size used by Lee is 7X7 [45].

2.4.1.2 Frequency Domain Filters:

This kind of filters is more widely used than the spatial domain filters in the field of image processing, the frequency domain filters work on the pixel values that has been transferred to the frequency domain using a Fourier transformation operations, the frequency domain filters work when the image pixels values are transferred from the spatial domain to the frequency domain, then apply the Desirable filter, after that an inverse Fourier transformation is done to restore the image to the spatial domain.

This frequency domain filters have more complicated operations than the spatial domain filters, although it’s more preferred to be used because it give a better result that the spatial domain filters, and here is an example:

2.4.1.2.1 The Gaussian Low-pass Filter

This filter (GLF) is one of the well known filters in image denoising; it offers a more sensible option to the ideal low pass filter, in this filter all the frequencies outside a particular range sets to zero.

The Gaussian low-pass filter has the same impact as the ideal low-pass filter on the image spectrum when it comes to the principle of working that is the low frequencies elements are allowed to pass and the high frequencies elements are not allowed to pass.

The major difference between the Gaussian low-pass filter and the ideal low-pass filter is the Deduction of the high frequency elements is piecemeal and not acute for the Gaussian low-pass filter, the Ideal pass filter cut the frequency directly and shapely. So as a result no ringing is noticed in the restored image in the special domain, the Gaussian low-pass filter is explained as:

The use of the Gaussian low-pass filter may blur the image or increase the amount of blur if the image is already blurred and it’s expected from a low pass filter to do that because the blur or smoothness is a low pass filter, the (GLF) filter is a frequency domain filter [41].

2.4.2 Recent Image Denoising Techniques:

Despite the fact that typical Image Denoising techniques are famous techniques that it’s been widely used, that does not mean that these techniques are the most efficient and successful in image denoising.

New techniques have been developed in recent years, these new techniques like wavelet denoising can perform better that the traditional techniques, and here is an example about recent techniques.

2.4.2.1 Wavelet Denoising:

Wavelet denoising considered one of the recent and most important methods in the field of image denoising, the wavelet denoising depend on thresholding in the denoising process, essentially the wavelet coefficients are limited by the threshold and any coefficient that exceeds the threshold its maintained or sometimes a slight reduction is done on the magnitude.

Otherwise if the coefficient is smaller than the threshold, the coefficient value changed to zero. This simple is the basic idea of the wavelet denoising, but for the result it’s much better than the traditional methods [41] [47].

2.5 Legacy Methods

Here in this section, previous methods used to handle the restoration problem would be discussed, methods selected in this section represents old and new techniques to handle the problem.

2.5.1 Deblurring with Rank-structured Inverse Approximations [20]

A new Category of approximations is used in this method to blur operators that represent atmospheric turbulence blurs. By using the fast Fourier transformation (FFT) technique, the matrix representing the blur is changed to Cauchy-like (CL) matrix. Both CL matrix and the transformed matrix have structures, but the new matrix (transformed one) has a rank structure. To be more specific the low rank blocks are the off-diagonal. This Category of matrices can be approximated rapidly, and the structure can be utilized for rapid image restoration.

This work is dealing with images that have is degraded by atmospheric turbulence and additive noise; this degradations are mostly appear in satellite images. This procedure uses the direct inverse method to restore the degraded image, the matrix equation used in this procedure is Ax = b where this equation stands for the blur, the variables in the equation are A, x, b. A is the matrix of the blur, x is the restored image, b is the blurry noisy image

As mentioned before Direct inverse is the method used to restore the image so this method work in 2 ways, either clearly invert the matrix A or use approximation to A to obtain x = A-1 b. The direct inverse method is sensitive to noise, because A many times is ill-conditioned, if the image has too much noise this method is not the advisable one to use.

For approximating the matrix A different techniques are used such as conjugate gradient (CG) and generalized minimal residual (GMRES), such approximations can be efficiently computed, inverted, and can be applied as a preconditioned to enhance the convergence of an iterative method.

The atmospheric turbulence blur is the blur used in this procedure. We start the procedure by using the fast Fourier transformation (FFT) to transform the blur matrix to CL matrix with rank structure. To be more specific the transformation of n X n matrix A is started when A is partitioned as:

In this case each of the off- diagonal blocks has its own property, here the low rank elements are A12 and A21 and can be represented by O (n) parameters. This applies to any part of this kind, to be applied repeatedly, leading to the representation of the entire matrix in terms of O (n) Parameters. It uses rapid algorithm to extract the rank structure in the form of exploitation.

The new structure will allow an O (n) solution of the system approximate deblurring. The structured matrix is used in an approximate inverse method and as a basis for a preconditioned iterative method.

Inverse filtering usually uses circulant approximations to the blur matrix in order to utilize the FFT in multiplication by the circulant matrix. In this method, a broader category of approximations is taken into account to give in a better inverse approximations and faster convergence more rapidly union. Of twenty-two surveyed, 91% chose the deblured image by rank-structured inverse approximation as opposed to restored images without the proposed method. Results of this experiment are shown in figure 2.3[20].

This method is a success if the restoration was only for blurry images because the filter used in this method is an inverse filter and inverse filter is very sensitive to noise. This method should consider the noise in the image though, if the amount of the noise in the image was very small then it’s a success but in case of noise existence in an unordinary way then this method is a failure. This method need to be mixed with another denoising method to form a new restoration system that can restore any blurry and noisy mage.

Figure 2.3[20]

(a) The original satellite image of Saturn’s rings.

(b) The image blurred by atmospheric turbulence.

figure 1,3.jpg

2.5.2 FUZZY- BASED DECONVOLUTION FOR IMAGE RESTORATION [21]

This method uses fuzzy logic mixed with one of the important filters to restore images that contain blur and noise that is the wiener filter. The main work of this method is to design a fuzzy estimator for the wiener filter (FEWF). Wiener filter is one of the filters that used to restore images contain atmospheric turbulence blur and additive Gaussian noise.

This technique can restore images that contain only blur and images that contain blur and noise. The main propose of this method is to enhance the performance of the wiener filter in restoring images.

It enhances the performance of the wiener filter by eliminating the disadvantages of the filter which is the estimation of the ratio of power spectral densities of the image and noise. The fuzzy estimator can help in solving the problem without the prior knowledge about these two quantities.

The reason behind using fuzzy logic in this method is that fuzzy logic is one of the best tools in managing uncertainty. The fuzzy estimator is meant for estimating the unknown values by knowing other values. The fuzzy estimator contains 4 blocks: fuzzification, rules basis, decision logic or fuzzy inference engine and defuzzification (see figure 2.4).

Figure 2.4: Scheme of an FLC (fuzzy estimator) [21]

The fuzzy estimator calculates the value (B) from e1 and e2, e1is the mean square error (MSE) between the original image and the restored image, e2 is The MSE between the degraded image g and the restored image. The (FEWF) algorithm is showed in figure 2.5

Figure 2.5: The (FEWF) algorithm diagram [21]

This method was implemented in matlab, 2 types of images were used synthetic and real images. Each of the images was degraded with blur and noise, and because the images were manually degraded, the degradation variables were known. The result is shown below figure 2.6:

Synthetic image [21] Real image [21]

Figure 2.6: Restoration results of blurred and noisy images [21]

As a conclusion the fuzzy estimator has improve the restoration method using wiener filter. The result shown above shows that [21]. But in my point of view this method need some enhancement though. This method is very good in restoring blur alone, but in the case of blur and noise together this method can perform better if it used separate filters for the restoration process one for removing the blur and one for removing the noise.

In addition to that the information needed in these experiments is already known, this method can be enhanced by using a function to detect these information either automatically or the information can be entered manually. The other point to discuss is that the wiener filter is not the best filter in the deblurring process; better filters can be used such as atmospheric wiener filter that is the enhanced version of wiener filter. Still this method is good in restoration and with some enhancement it would be one of the most robust methods in this field.

2.5.3 BLUR IDENTIFICATION BASED ON KURTOSIS MINIMIZATION [22]

The image degraded by noise and blur can be restored using with many types of filters, here in this method the restoration would be done using Wiener filter. Here the use of kurtosis minimization is limited to measure the quality of the restored image. This method works with a group of blurred images, the chosen one from the group would have the minimum kurtosis. This method is tested with more than one type of blur such as atmospheric turbulence, Gaussian, and out-of-focus blurs.

The kurtosis equation is [22]:

Where:

(µ) is the mean of (x).

(σ) is its standard deviation.

E(x) represents the expected value operation.

The peakedness of a distribution is measured by the kurtosis technique. The regular value of (k=3) called mesokurtic and its ahs a moderate tail, the value (k < 3) has a small tail and it called platykurtic, the value (k > 3) has a long tail and it called leptokurtic. In platykurtic the smoother the data, the larger the kurtosis.

The restored image is blurry and noisy image, here in this technique a search for the Finest estimation of the blur parameter is conducted; the search is conducted in a certain space. The image is deblurred using the Wiener filter in every step in the search loop. The kurtosis value of the deblurred image is also computed and saved in each step of the search loop. The restored image would be chosen if an image have the smallest kurtosis value, and the image parameter would consider as the deblur parameter.

The degradation model of the blurry and noisy image is:

g = f * h + n

Where:

(g) Is the blurry noisy image.

(f) Is the original image.

(h) Is the blur.

(n) Is the additive noise.

The blur is directly convolved to the original image so the result would be an image degraded with blur; also there is the additive noise that degrades the blurry image.

The restoration of the image depends on blur (h) and noise (n). Calculating the blur parameter is not enough for a good restoration, the noise also needs to be estimated correctly to have the best restoration of the image, and therefore the PSNR of the deblured image must be compared to the PSNR of the blur parameter to calculate the accuracy of the blur identification.

This method work with more than one type of blur, but the important type that this thesis working with is atmospheric turbulence blur. The optical transfer function of the atmospheric turbulence blur is [22]:

(λ) Determines the sternness of the blur, if (λ=0) then there is no blur in the image, if λ raise the blur would rise in the image too. An experiment has been conducted to prove the Efficiency of this method. A grayscale image degraded manually with (λ = 0.0025) blur and (σ = 0.002) of Gaussian noise with search space â„¦= {3, 4, 5, 6}. Result of the experiment shows in figure 2.7 bellow [22].

Figure 2.7: Blurry and noisy image [22] Restored image [22]

This method is good in choosing the best resorted image but the problem is that it depend on one filter in the restoration process to handle the restoration of blur and noise in the image and as the pictures in the figure above, the image become more clear but the noise in the image become more obvious when using this method.

2.5.4 RECOVERY OF SATELLITE IMAGES USING EDGE INFORMATION OF ACTUAL IMAGE [23]

Short distance scenes can be seen as the original if the imaging system is well tuned like well focused, the setting has been well setup…etc but for long distance images there are some back draws to get an unspoiled image for instance unfocused lenses of the imaging system and atmospheric turbulence.

The atmospheric turbulence varies in times and places, for instance sometimes the weather is hot, warm or cold sometimes the weather is windy or have sands in it, these turbulences in the atmosphere cannot be forbidden in image acquiring time so by that the resulted image would be degraded by atmospheric turbulence blur, the blur is the result of the spreading of the pixels to the neighborhood pixels so solutions were made in the imaging systems such as automatic focusing to avoid the blur but still it cannot avoid it all the times.

Remote sensing images are widely used among the researchers, but because of the degradations affecting the images such as atmospheric blur and Gaussian noise the information cannot be get from the image correctly in details so these images need to be processed using image processing techniques to remove these degradations and get a clearer image. The blur are 2 kinds, a known PSF and unknown PSF. The known PSF can be easily restored but the unknown PSF need to estimate the PSF information to restore the image.

In this work, the image is modeled as:

Y = X * h + N

Where Y is the blurry and noisy image, X is the original unspoiled image, h is the blur function that would blur the original image by convolving, and N is the additive Gaussian noise.

The noise affecting the image can be removed by using one of the denoising filters such as wiener filter … etc because the model parameters are supposed to be known. Thus the blur remain in the image so the image can be modeled as:

Y = X * h

So now the deblurring is the issue that we need to focus on after removing the noise, here the blur has normal or Gaussian distribution. The Atmospheric turbulence, the unfocused imaging systems will generate an image spoiled with blur that has a Gaussian distribution.

The matrix size is important in this algorithm for that, first step to do is to search the matrix size and then fix it for a known period, after that the variance is searched by the algorithm in the same period. When the period ends, the blur matrix would be updated if necessary.

In this method, the approximation of the filter is done by the edge detection and as familiar in the blurred image, the edge pixels are diffused to the nearby pixels and sometimes the edge pixels are totally lost in the image, therefore the edge map have less pixels that the case of the original image, so in a conclusion the edge map provides vital information about the degradation.

The algorithm that is used in this method in short is shown in figure 1 and also shown below:

1. Read digitized image (y (n1, n2)).

2. Estimate the filter parameter called variance (s2) from the edge map of degraded image after 20 iterations steps.

3. Construct a restoration filter using the computed parameter in step (2).

4. Compute the Cepstrum transform of filter and image.

5. Apply the designed filter to blurred image (Y-h1).

6. Compute the inverse Cepstrum transform of (6).

7. Repeat the same process from (4).

8. Apply another blurred image for real time application and restore the new blurred image using steps 4,5,6,7.

9. Re-estimate the filter parameter continuously and compare it with previous filter parameter. If it remains under a critical error, (go on restoration). Else, refresh the filter parameter with a new value.

Figure 2.8: shows the diagram of the proposed algorithm [23]

This method has been applied to images acquired by Hubble space telescope, figure 2.9 shows the original satellite image, the restored image and their corresponding edge map.

Figure 2.9.a shows the degraded image captured by the Hubble telescope and figure 2.9.b shows its edge map, in figure 2.a the details are not clear because of the atmospheric turbulence blur and the residue of the additive Gaussian noise, figure 2.9.c shows the restored image, and its clearly that the details have been enhanced by looking at the image or looking at the edge map of the restored image in figure 2.9.d.

Also from figures 2.9.c and 2.9.d it’s obvious that the quality of the image also has been increased but still some noise has been raised in the restored image because of the residue of the noise that still in the image [23].

Figure 2[23] shows the original and the restored images and their corresponding edge map

This method is good in restoring the blurred images because it separates the restoration process, this method used to restore blurry images and uses other filters to handle the noise removing from the image. This method does not focus on the noise removal from the image and that’s a problem in the restoration process because noise removal is vital to a good blur restoration.

This method should focus more on the noise removal issue because as in figure 2.9 the residue of the noise has been showed up in the deblurring process, as known the noise is a high pass filter and the blur is a low pass filter and by deblurring, it raise the noise because deblurring is a high pass filter like sharpening.

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