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Image denoising in the wavelet domain using Improved Neigh-shrink

Rahim Kamran1, Mehdi Nasri2, Hossein Nezamabadi-pour3, Saeid Saryazdi4

1Rahimkamran2008@gmail.com 2 nasri_me@yahoo.com

3 nezam@uk.ac.ir 4saryazdi@uk.ac.ir

Elec. Eng. Department, Shahid Bahonar university of Kerman, Kerman, Iran.

Abstract: Denoising of images corrupted by Gaussian noise using wavelet transform is of great concern in the past two decades. In wavelet denoising method, detail wavelet coefficients of noisy image are thresholded using a specific thresholding function by comparing to a specific threshold value, and then applying inverse wavelet transform, results in denoised image. Recently, an effective image denoising method has been proposed called Neigh-shrink that exploits the inter-scale dependency of wavelet coefficients. In this paper, we extend Neigh-shrink denoising method by proposing a new thresholding scheme. Experimental results show that our method outperforms classical Neigh-shrink visually and in the terms of PSNR. Key words: Wavelet transform, Gaussian noise, Neigh-shrink, threshold value

1. Introduction Gaussian noise is an additive type noise that is usually

added to images during acquisition, transmission and storage [2]. High quality images are essential in decision making in computer vision and image processing applications. Therefore, image denoising has remained a fundamental problem in the field of image processing [3, 4].

Wavelet transform has become a popular engineering tool in the last two decades, and the focus has been shifted from spatial and Fourier transform to wavelet domain [5]. Wavelet transform gives a superior performance in image denoising due to properties such as multiresolution and sparsity [6]. Since Donohos’ leading work in wavelet based thresholding approach that was published originally in 1995, there was a zenith in methods proposed in denoising field [5,7-10]. Briefly Donohos’ denoising method in the wavelet domain has the following steps.

1- The transformation of noisy image into an orthogonal domain by using 2-D discrete wavelet.

2- Thresholding of wavelet coefficients by using the

threshold value of 2log2 n where n and are length of signal and variance respectively.

3- Performing inverse 2-D wavelet to the thresholded coefficients to get denoised image.

Denoising of images in the wavelet domain is very effective because it can capture the energy of a signal in few transform coefficients.

Although Donohos’ method was not revolutionary, but his method didn’t require correlation of wavelet maxima and minima [10]. From that time, researchers have presented different methods to compute parameters of threshold value [5]. Finding a proper threshold is an important stage in thresholding procedure. Using a small threshold value will retain the noisy coefficients while a large threshold value leads to loss of important coefficients. Normally, two kinds of thresholding functions are used, hard and soft. In hard thresholding, the coefficients that are bigger than the threshold value are kept unchanged, while in soft thresholding these coefficients are shrinked [9]. In both thresholding functions, coefficients that are lower than threshold value are changed to zero.

A recent thresholding method that has been proposed that used secondary properties of wavelet transform is Neigh-shrink method [1]. In this method, the correlation between coefficients in the detail subbands is considered in image denoising. The idea of Neigh-shrink is promising but the results in image denoising can be improved.

In this paper, a modification is done to Neigh-shrink method that can improve the denoising results significantly. The proposed method is effective in image denoising with a different level of Gaussian noise.

Organization of the paper is as follow:In Section 2, the wavelet transform and Neigh-shrink denoising method is reviewed briefly. In Section 3, our proposed denoising algorithm is depicted in details. Section 4 includes the

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simulation results, and finally, the paper will be concluded in Section 5.

2. Neigh-shrink Image Denoising

The Discrete Wavelet Transform (DWT) of image signals produces a non-redundant image representation, which provides better spatial and spectral localization of image formation, compared with other multi-scale representations such as Gaussian and Laplacian pyramid.

Recently, Discrete Wavelet Transform has attracted more and more interest in image de-noising. The DWT can be interpreted as signal decomposition in a set of independent, spatially oriented frequency channels. When a signal is passed through two complementary filters, two signals are produce, approximation and details. These components can be assembled back into the original image without loss of information. In the case of a 2D image, an N level decomposition can be performed, which leads to production of 3N+1 different frequency bands named as LL,LH,HL,HH as it is shown in Fig.1.

Fig.1. 2D-DWT with 2 level of decomposition

These frequency bands (LL,LH,HL,HH) are also known respectively as average image, horizontal details, vertical details and diagonal details.

As mentioned above, in image denoising in wavelet domain, coefficients achieved from wavelet transformed are thresholded according to a proper threshold value. In Neigh-shrink method, in which for thresholding process, around every wavelet coefficient dj,k of interest a neighborhood mask (ordinarily with size of 3*3) is considered. Then S (summation) is calculated using (1).

kjlBi lidS

,,2,

(1)

where, d , are wavelet coefficients in the selected

window. Then if 2S , the corresponding dj,k is set to zero otherwise it is shrinked according to the following formula:

kjkjkj dd ,,, (2) where the factor kj , can be defined as:

2

,

2

, 1kj

kjs

(3)

In which 2log2 n represents the threshold value

and σ2 is variance and n is the signal length.

3. Proposed denoising method

Since the Gaussian noise is averaged out in the low-frequency wavelet coefficients, and it is desirable to keep small coefficients in these frequencies, therefore,in the Neigh-shrink denoising method only wavelet coefficients in detail subbands are thresholded. Furthermore, in Neigh-shrink thresholding process, if the summation (S) is less than variance, then the corresponding wavelet coefficient is set to zero, and this resembles to hard thresholding and can result in decrease of PSNR. In this paper, to increase the effectiveness of Neigh-shrink method, and consequently the PSNR of denoised image, we propose an improved method as follows.

Around each coefficient of interest in high frequency subbands a neighborhood window is considered. Normally, a window of size of 3×3 is selected around each wavelet coefficient as shown in Fig. 2. Then, the sum of square of coefficients that exist in the selected window (S) is calculated as in previous section. Then S is compared to varianceσ2 of window coefficients. If

2S then instead of setting wavelet coefficient to zero as it is done in Neigh-shrink, the median of coefficients in the mask is set as new coefficient. Otherwise if 2Sthen the new wavelet coefficient is calculated as follows:

SmTolddnewd kjkj /*1*,, (4) Where, 2log**2 nvariancem in which ‘n’ is the

number of coefficients in selected mask. ‘T’ is a coefficient that should be selected optimally to get maximum PSNR for each image. In our method, the best optimum T is selected by trial and error for each image.

Therefore, algorithm for our proposed method is as follows:

1- Approximation and detail subbands of

noisy image are extracted using 2-D discrete wavelet transform in 4 levels..

2- For each detail subband, the denoising procedure is done separately using the mentioned method.

3- The Inverse of 2-D wavelet transform is performed on the thresholded wavelet coefficients to get the denoised image.

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Fig. 2. A neighborhood window of size 3×3 around a wavelet

coefficient

4. Experimental results

The proposed method is applied on a different set of gray-scale images such as ‘Lena’, ‘Peppers’, ‘Baboon’, and ‘Barbara’. The ‘db8’ wavelet in 4 levels of decomposition is used in the proposed method. The evaluation criteria used for comparison is PSNR (Peak Signal to Noise Ratio) and shown in (5)

jijiAjiB

n

PSNR

,2

2

2

10,,1

255log10 (5)

where A and Bare the original noise-free and denoised

images respectively and n is the image size. For different Gaussian white noise levels, the

experimental results are shown in Table 1. The optimum threshold value (T) to get maximum PSNR for each image is selected by trial and error . For our test images, its best optimum value is between 1 and 2 (1 < 푇 < 2). The optimum threshold for each of test images is included in table 1. As it can be seen, the PSNR results of proposed method outperform classical Neigh-shrink in all test images and noise levels. For visual comparison, denoising results of ‘Lena’ and ‘Peppers’ images are shown in Fig. 3 and 4 respectively.

5. Conclusion In this paper, an image denoising method in wavelet

domain using improved Neigh-shrink denoising is proposed. In the method, approximation and detail subbands of noisy image are extracted by using wavelet transform. Detail subbands are denoised using a new thresholding method. Experimental results confirms that

the proposed method outperforms classical Neigh-shrink in the terms of PSNR and visually, and can be used effectively in image denoising applications.

References

[1] G.Y. Chen, T.D. Bui, A. Krzyzak,” Image denoising using neighbouring wavelet coefficients”, Integrated Computer-Aided Engineering 12 (2005) 99–107 99 IOS Press [2] M.Wilscy, Madhu S. Nair, “Fuzzy Approach for Restoring Color Images Corrupted with Additive Noise”, Proceedings of the World Congress on Engineering 2008 VolI ,WCE 2008, July 2 - 4, 2008, London, U.K. [3] S. Durand and J. Froment, “Artifact free signal denoising with wavelets,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP ’01), vol. 6, pp. 3685–3688, Salt Lake City, Utah, USA, May 2001 [4] JagadishH.Pujar, ” Robust Fuzzy Median Filter for Impulse Noise Reduction of Gray Scale Images”, World Academy of Science, Engineering and Technology 64 2010 [5] S. Grace Chang, Bin Yu and M. Vattereli, Wavelet Thresholding for Multiple Noisy Image Copies, IEEE Trans. Image Processing, vol. 9, pp.1631- 1635, Sept. 2000. [6] A. Chambolle, R. A. DeVore, N.-Y. Lee, and B. J. Lucier, “Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage,” IEEE Transactions on Image Processing, vol. 7, no. 3, pp. 319–335, 1998. [7] T.D. Bui and G.Y. Chen, “Translation invariant denoising using multiwavelets,” IEEE Transactions on Signal Processing, 46(12) (1998), 3414–3420 [8] X. Li and M. T. Orchard: “Spatially Adaptive Image Denoising under Overcomplete Expansions”, Proc. IEEE Int. Conf. on Image Processing, Vancouver, 2000 [9] D.L. Donoho, “Denoising by soft-thresholding”, IEEE Transactions on Information Theory 41(3) (1995), 613–627. [10] L. Sendur and I.W. Selesnick, “Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency”, IEEE Transactions on Signal Processing, 50(11) (2002), 2744–2756

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TABLE1. PSNR comparison between our proposed denoising method and Neigh-shrink method in denoising “Baboon”, ”Lena”, “Barbara” and

“Peppers” images with different level of noise.

Image Standard deviation of noise

noisy image

Neigh-shrink Proposed

Optimum T

(In the proposed method)

Optimum T

(In Neigh-shrink)

‘Baboon’ 10 35.85 40.863 43.97 1.5 1.4 20 30.09 36.469 39.244 1.4 1.6 30 22.17 28.09 32.87 1.9 1.8

‘Lena’ 10 36.08 41.23 44.997 1.7 1.5 20 31.104 37.024 40.68 1.8 1.7 30 24.3 29.21 33.45 2 1.8

‘Barbara’ 10 36.74 40.915 44.163 1.8 1.6 20 33.38 39.218 43.128 1.7 1.5 30 22.34 28.85 33.120 1.9 1.8

‘peppers’ 10 36.12 41.015 45.05 1.6 1.4 20 30.519 37.12 42.245 1.8 1.6 30 24.50 29.84 33.09 1.9 1.7

(a)

(b)

(c)

(d)

Fig. 3.(a) Original ‘Lena’ image (b) Noisy image with 휎 = 20 (c) Denoised image with Neigh-shrink method(PSNR= 37.024) (d) Denoised image with the proposed method (PSNR=40.68)

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(a)

(b)

(c)

(d)

Fig. 4. (a) Peppers image (b) Noisy image with 휎 = 30 (c) Denoised image with Neigh-shrink method (PSNR= 29.84) (d) Denoised image with our proposed method (PSNR=33.09)