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International Journal Of Electrical, Electronics And Data Communication, ISSN: 2320-2084 Volume-5, Issue-8, Aug.-2017 http://iraj.in

Restoration of Blurred Images using Wiener Filtering

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RESTORATION OF BLURRED IMAGES USING WIENER FILTERING

MARAPAREDDY. R

The University of Southern Mississippi, MS, USA E-mail: [email protected]

Abstract - In practical situation, images are easily degraded due to the complex surrounding environment. Investigated image restoration using optimal Wiener filtering. To investigation an algorithm, intentionally degraded image and then applied Wiener filtering to try to restore the original image. In this paper, will discuss on atmospheric turbulence degradation model. Then, inverse filtering and minimum mean square error i.e., wiener filtering will be discussed to restore the blurring images. Index terms - Wiener filter, frequency domain, blurred images, digital filters, image restoration. I. INTRODUCTION The degradation of an image can be modeled as a blur function and additive noise. Common blurs include motion blur and Gaussian blur. Imaging systems may introduce the distortion or artifacts, which will seriously influence the application of the image, such as target detection, etc. To restore the degraded image in the Fourier domain is a common resolution method. Suppose the frequency represent of image f (x, y) is F (u, v), H (u, v), is the degradation function, then, we can the degraded image representation G (u, v) = H(u, v) F(u, v). We see that the degradation system can be modeled in the spatial domain as the convolution of the degradation function with an image. II. THEORY OF ATMOSPHERIC TURBULENCE Atmospheric turbulence is caused by the random fluctuations of the refraction index of the medium. It can lead to blurring in images acquired from a long distance away. Since the degradation is often not completely known, the problems are viewed as blind image deconvolution or blur identification. Image degradation associated with atmospheric turbulence often occurs when viewing remote scenes: the objects of interest will appear blurred, and the severity of this blurring will typically change over time. In addition, the stationary scene may appear to waver spatially [1-2]. In the physical world, several factors affect the blurring distortion that we observe, such as temperature, humidity, elevation, and wind speed. In most cases these atmospheric conditions are not known, nor is there generally any external information available to help specify the blur function [3-5]. Random fluctuations of the refraction index cause atmospheric turbulence degradation. These phenomena have been observed in long-distance surveillance imagery and astronomy [6]. The fluctuations in atmospheric turbulence can be

modeled as a dynamic random process that perturbs the phase of the incoming light. From the refraction index structure functions, Hufnagel and Stanley [7] derived a long-exposure optical transfer function,

H (u, v) = to model the long-term effect of turbulence in optical imaging. Here u and v are the horizontal and vertical frequency variables and ‘k’ parameterizes the severity of the blur. As ‘k’ increases in value, so does the degree of the blur. ‘k’ is a constant that depends on the nature of turbulence, as shown in figures 1 and 2. III. IMAGE RESTORATION Inverse Filtering: If we know the degradation function H (u, v), the simplest approach to restore degradation image is direct inverse filtering. The recovery image can be estimated in frequency domain, F~ = G (u, v)/H (u, v) However, the equation above doesn’t consider the situation of additive noise. Otherwise, the formula will be, F~ = F (u, v) +N (u, v)/H (u, v) Where, however, the N (u, v) is usually unknown. Sometimes, because of the fraction, we have to face the problem that the degradation function has zero or very small values. One way to solve the problem is to limit the filter frequencies to values near the origin which is usually nonzero. Thus, the probability of encountering zero values will be reduced. In this experiment, we will center the Fourier transform of original image, as well as the degradation function. The centered function is,

H (u, v) =

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Where M and N are the size of the matrix. IV. WEINER FILTERING

The inverse filtering is a restoration technique for deconvolution, i.e., when the image is blurred by a known low pass filter, it is possible to recover the image by inverse filtering or generalized inverse filtering. However, inverse filtering is very sensitive to additive noise. The Wiener filtering executes an optimal tradeoff between inverse filtering and noise smoothing. It removes the additive noise and inverts the blurring simultaneously [7-8]. The Wiener filtering is optimal in terms of the mean square error. In other words, it minimizes the overall mean square error in the process of inverse filtering and noise smoothing. The Wiener filtering is a linear estimation of the original image. Wiener filtering, also called minimum mean square error filtering [1-2] is founded on considering images and noise as random variables. The objection function between original clear image f and degraded image and de f~ is,

Where {.} is the mean statistical characteristics of the argument. The pre-condition of this function is that the noise and image are uncorrelated. Based on that, the recovery image in frequency domain is,

Where K = Sn (u, v)/Sf (u, v), and

is the complex conjugate of H (u, v). if Sn (u, v) =0, which means there is no noise, it is easy to say that wiener filtering is actually inverse filtering. A simplification of the above equation is to use a constant k to denote the ratio Sn (u, v)/Sf (u, v), and the formula is

V. RESULTS AND DISCUSSION In this paper, we produce atmospheric turbulence model to degrade images and restore using wiener filtering. We take the degraded image with atmospheric turbulence model, Kis set to 0.0025, and the sigma of added noise is 0.005, as shown in figures 3-5. From the results, radius near 90 may produce best result for inverse filtering. However, there still has some visible noise in it. The bottom one is produced by wiener filter, comparing with inverse filtering.We see the noise seems to be less than other images and more smoothed. Because the added noise is Gaussian white noise, we estimate the value of K by 1/SNR, and 1/SNR is calculated by,

Furthermore, we adopt Peak Signal-to-Noise Ratio (PSNR) as a criterion to measure the performance of this experiment, as indicated in Table1.Even we say that inverse filtering brings some visible noise for recovered image, the value of PSNR seems to be lower than the one of the image we feel more comfortable.

CONCLUSION Discussed on atmospheric turbulence degradation model. Then, inverse filtering and minimum mean square error i.e., wiener filtering will be discussed and implemented to restore the blurring images.Adopted Peak Signal-to-Noise Ratio as a criterion to measure the performance of this experiment. The value of PSNR seems to be lower than the one of the image we feel more comfortable with wiener filter.

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Figure 1. Original image (top left), Frequency domain coefficients (top right), Atmospheric turbulence frequency model = 0.0025 k

(bottom left), Atmospheric turbulence frequency model = 0.025 k (bottom right).

Figure 2. Blurring frequency domain coefficients and spatial images, especially, according to the value k.

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Figure 3. Degraded image (top left), Fourier coefficients (top right), Blurring frequency domain coefficients (bottom).

Figure 4. Result of Full filter (top left), Result of with cut off outside a radius of 30 (top right), Result of with cut off outside a radius

of 60 (bottom left), Result of with cut off outside a radius of 90 (bottom right).

Figure 5. Result of wiener filter

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full filter Radius= 30

Radius= 60 Radius= 90 wiener filter

PSNR(dB) 13.5829

3.2638

3.2026

3.6374

3.7191

Table 1. Peak Signal-to-Noise Ratio of this experiment REFERENCES [1] R. C. Gonzalez and R. E. Woods, Digital Image Processing,

3rd edition, Prentice Hall, 2008. [2] Pratt, W.K., Digital Image Processing, 3rd Ed., New York:

John Wiley and Sons, 2001. [3] Dalong Li, Simske. S. “Atmospheric Turbulence Degraded

Image Restoration by Kurtosis Minimization”, IEEE Geoscience and Remote Sensing Letters, vol. 6, no. 2, pp. 244-247, 2009.

[4] Dalong Li, Mersereau, R.M., Simske, S., "Atmospheric Turbulence-Degraded Image Restoration Using Principal Components Analysis," Geoscience and Remote Sensing Letters, IEEE, vol.4, no.3, pp.340-344, July 2007.

[5] Li Dongxing, Han Jinhong, and Xu Dong, "A novel restoration algorithm of the turbulence degraded images based on maximum likelihood estimation," Electronic

Measurement & Instruments, 2009. ICEMI '09. 9th International Conference on, vol., no., pp.4-171-4-176, 16-19 Aug. 2009.

[6] Luxin Yan, MingzhiJin, Houzhang Fang, Hai Liu, and Tianxu Zhang, "Atmospheric-Turbulence-Degraded Astronomical Image Restoration by Minimizing Second-Order Central Moment," Geoscience and Remote Sensing Letters, IEEE, vol.9, no.4, pp.672-676, July 2012.

[7] R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function Associated with Image Transmission through Turbulence Media,” Optical Society of America Journal A, vol. 54, pp. 52–61, 1964.

[8] L. Guan and R. K. Ward, "Restoration of randomly blurred images by the Wiener filter," in IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 4, pp. 589-592, April 1989.

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