Comparison of Wavelet thresholding for image denoising using different shrinkage Namrata Dewangan 1, Devanand Bhonsle 2 1 M.E. Scholar Shri Shankara Charya Group of Institution, Junwani, Bhilai, 2 Sr. Assistant Professor Shri Shankara Charya Group of Institution, Junwani, Bhilai ABSTRACT-Transformation of digital image becomes a major method of communication in the modern age, but the image obtained after transmission the data tends to get noisy and thereby the further processing does not lead to good results. Hence, Pre processing of an image is very essential. The pre processing being worked upon is the denoising of images. The received image needs processing before it can be used in applications. Image denoising involves the manipulation of the image data to produce a high quality image and wavelet transforms have been applied. For denoising the image different noise models including Gaussian noise, salt and pepper noise and speckle noise. Selection of the denoising algorithm is application dependent. In this paper, a comparative analysis of different combinations of the suggested threshold values and thresholding techniques has been carried out very efficiently. This has been done in order to find more possible combinations that can lead to the best denoising technique and compare the results in term of PSNR and MSE. Keywords wavelet thresholding, Sure, Bivariate, Bays and Block 1. INTRODUCTION In the past two decades, many noise reduction techniques have been developed for removing noise and retaining edge details. Most of the standard algorithms use a defined filter window to estimate the local noise variance of a noise image and perform the individual unique filtering process. The result is generally a greatly reduced noise level in areas that are homogeneous. But the image is either blurred or over smoothed due to losses in detail in non-homogenous areas like edges or lines. This creates a barrier for sensing images to classify, interpret and analyze the image accurately especially in sensitive applications like medical field. The primary goal of noise reduction is to remove the noise without losing much detail contained in an image. To achieve this goal, we make use of a mathematical function known as the wavelet transform to localize an image into different frequency components or useful sub bands and effectively reduce the noise in the sub bands according to the local statistics within the bands. The main advantage of the wavelet transform is that the image fidelity after reconstruction is visually lossless. The wavelet de-noising scheme thresholds the wavelet coefficients arising from the wavelet transform. The wavelet transform yields a large number of small coefficients and a small number of large coefficients. Wavelets are especially well suited for studying nonstationary signals and the most successful applications of wavelets have been in compression, detection and denoising. The method consists of applying the DWT to the original data, thresholding the detailed wavelet coefficients and inverse transforming the set of thresholded coefficients to obtain the denoised signal. Given a noisy signal y = x + n; where x is the desired signal and n is independent and identically distributed (i.i.d) Gaussian noise N (0, σ 2 ), y is first decomposed into a set of wavelet coefficients w = W[y] consisting of the desired coefficient θ and noise coefficient n. By applying a suitable threshold value T to the wavelet coefficients, the desired Coefficient θ=t[w] can be obtained; lastly an inverse transform on the desired coefficient θ will generate the denoise signal x = W T [θ]. Figure 1: Block Diagram for DWT based denoising framework In the experiments, soft thresholding has been used over hard thresholding because it gives more visually pleasant images as compared to hard thresholding; reason being the latter is discontinuous and yields abrupt artifacts in the recovered images especially when the noise energy is significant. Flowchart for Image Denoising Algorithm using Wavelet Transform is shown in figure 2. 1.1 SURE SHRINK Volume 2, Issue 1 January - February 2013 Page 57
A threshold chooser based on Stein s Unbiased Risk Estimator (SURE) was proposed by Donoho and Johnstone and is called as Sure. This method specifies a threshold value for each resolution level j in the wavelet transform which is referred to as level dependent threshold. The goal of Sure is to minimize the mean squared error, defined as, Where Z(X,Y) is the estimate of the signal, S(X,Y) is the original signal without noise and n is the size of the signal. The Sure threshold t* is defined as (1) (2) It is smoothness adaptive, which means that if the unknown function contains abrupt changes or boundaries in the image, the reconstructed image also does.[5,11] 1.2 BAYES SHRINK The goal of this method is to minimize the Bayesian risk, and hence its name, Bayes. The Bayes threshold, tb, is defined as Where is the noise variance and is the signal variance without noise. From the definition of additive noise we have W(x, y) = s(x, y) + n(x, y). Since the noise and the signal are independent of each other, it can be stated that (4) (5) Where t denotes the value that minimizes Stein s Unbiased Risk Estimator, σ is the noise variance and an estimate of the noise level σ was defined based on the median absolute deviation given by and n is the size of the image. (3) Can be computed as shown below The variance of the signal, is computed as (6) (7) With and, the Bayes threshold is computed from Equation (4). Using this threshold the wavelet coefficients are threshold at each band.[17] 1.3 BIVARIATE SHRINKAGE New shrinkage function which depends on both coefficient and its parent yield improved results for wavelet based image denoising. Here, we modify the Bayesian estimation problem as to take into account the statistical dependency between a coefficient and its parent. Let w 2 represent the parent of w 1 (w 2 is the wavelet coefficient at the same position as w 1, but at the next coarser scale.) [20] Then y 1 =w 1 +n 1 y 2 =w 2 +n 2 (8) Where y 1 and y 2 are noisy observations of w 1 and w 2 and n 1 and n 2 are noise samples. Then we can write Y = w + n y= (y 1, y2) w= (w 1, w2) n= (n 1, n2) (9) Figure 2: Flowchart for Image Denoising Algorithm using Wavelet Transform Standard MAP estimator for w given corrupted y is (10) Volume 2, Issue 1 January - February 2013 Page 58
International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Volume 2, Issue 1, January February 2013 ISSN 2278-6856 This equation can be written as (21) (11) (12) According to bays rule allows estimation of coefficient can be found by probability densities of noise and prior density of wavelet coefficient. We assume noise is Gaussian then we can write noise as (13) Joint of wavelet coefficients (14) We know from equation (9) 1.4 BLOCK SHRINK Block is a completely data-driven block thresholding approach and is also easy to implement. It can decide the optimal block size and threshold for every wavelet subband by minimizing Stein s unbiased risk estimate (SURE). The block thresholding simultaneously keeps or kills all the coefficients in groups rather than individually, enjoys a number of advantages over the conventional term-by-term thresholding. The block thresholding increases the estimation precision by utilizing the information about the neighbor wavelet coefficients. The local block thresholding methods all have the fixed block size and threshold and same thresholding rule is applied to all resolution levels regardless of the distribution of the wavelet coefficients.[21] (15) Let us define f (w) =log (pw (w)) Then using equation (13) and (14) (16) This equation is equivalent to solving following equations (17) (18) Where and represents the derivatives of with respect to w1 and w2 respectively. We know can be written a Figure 3: 2 2 Block partition for a Wavelet subband (19) From this. (20) From equations (18), (19), (20) and (21) MAP estimator can be written as Volume 2, Issue 1 January - February 2013 As shown in Figure 3, there is a number of subband produced when we perform wavelet decomposition on an image. For every subband, we need to divide it into a lot of square blocks. Block can select the optimal block size and threshold for the given subband by minimizing Stein s unbiased risk estimate. Experimental results show that Block outperforms significantly the classic Sure by the term-by-term thresholding and Neigh with the fixed overlapping block size and threshold proposed by Chen et al. Experimental results showed that the PSNRs which Block yielded were substantially higher than those that Sure and Neigh did. As a matter of fact, Block enjoys the advantages of Sure and Neigh and gets rid of there. Page 59
2. RESULT This paper presents a comparative analysis of different shrinkage for image denoising techniques using wavelet transforms. We have experimented with four different thresholding methods (Bays, Sure shrink, Bivariate shrink, Block ) using the different noise (Gaussian noise, salt and paper noise and speckle noise) and report the results for the 512 512 standard test images Figure 4. Our results are measured by the PSNR and MSE. Result for an image shown below affected by gaussian noise having different variances are given in the Table 1 and 2. Thresholding Technique Sure Bays Bivariate age Block Variance (σ) σ= 0.01 117.9 238.7 0.009 58.16 σ= 0.02 95.52 208.4 0.009 73.62 σ= 0.03 73.27 177.8 0.0092 107.8 σ= 0.04 58.28 154.9 0.009 142.0 σ= 0.05 44.54 127.5 6 0.0089 196.8 4 Figure 4: Standard Test Image Table 1: Gaussian noise, PSNR Thresholding Technique Sure Bays Bivariate age Block Variance (σ) σ= 0.01 27.41 24.35 68.44 30.48 3. CONCLUSION Image denoising, using wavelet techniques are effective because of its ability to capture the energy of signal in a few high transform values, when natural image is corrupted by Gaussian noise. Wavelet thresholding is an idea in which is removed by killing coefficient relative to some threshold. Out of various thresholding techniques soft-thresholding is most popular. This paper presents a comparative analysis of various image denoising thresholding techniques (Sure, Bays, Bivariate age, Block ) using wavelet transforms. A lot of combinations have been applied in order to find the best method that can be followed for denoising intensity images. From the PSNR and MSE values as shown in tables, it is clear that Bivariate age giving better results under different noise variance conditions for all of the images. The Comparative graph for PSNR and MSE are given below σ= 0.02 28.32 24.93 68.46 29.46 σ= 0.03 29.48 25.62 68.48 28.00 σ= 0.04 30.47 26.22 68.56 26.60 σ= 0.05 31.64 27.07 68.59 25.18 Figure 5: Comparative Graph for PSNR for thresholding techniques for Gaussian noise having different variances Table 2: Gaussian noise, MSE Volume 2, Issue 1 January - February 2013 Page 60
Figure 6: Comparative Graph for MSE for thresholding techniques for Gaussian noise having different variances REFERENCES [1] Gao Zhing, Yu Xiaohai, Theory and application of MATLAB Wavelet analysis tools, National defense industry publisher, Beijing, pp.108-116, 2004. [2] Aglika Gyaourova Undecimated wavelet transforms for image denoising, November 19, 2002. [3] C Sidney Burrus, Ramesh A Gopinath, and Haitao Guo, Introduction to wavelet and wavelet transforms, Prentice Hall1997.S. Mallat, A Wavelet Tour of Signal Processing, Academic, New York, second edition, 1999. [4] F.Abramovich and Y. Benjamini, Adaptive thresholding of wavelet coefficients, Comput. Statist. Data Anal., vol. 22, pp. 351 361, 1996. [5] Sachin D Ruikar and Dharmpal D Doye, Wavelet Based Image Denoising Technique (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 2, March 2011. [6] F. Abramovich, T. Sapatinas, and B. Silverman, Wavelet thresholding via a Bayesian approach, J. R. Stat., vol. 60, pp. 725 749, 1998. [7] Z. CAI, T. H. Cheng, C. Lu, and K. R. Subramanian, Efficient waveletbased image denoising algorithm, Electron. Lett., vol. 37, no. 11, pp.683 685, May 2001. [8] Sanyam Anand, Amitabh Sharma and Akshay Girdhar, Undecimated Wavelet Based New Threshold Method for Speckle Noise Removal in Ultrasound Images International Conference on Modeling, Simulation and Control, vol.10, IACSIT Press, 2010. [9] D. L. Donoho and I. M. Johnstone, Denoising by soft thresholding,ieee Trans. on Inform. Theory, Vol. 41, pp. 613-627, 1995. [10] X.-P. Zhang and M. D. Desai, Adaptive denoising based on SURE risk, IEEE Signal Process. Lett., vol. 5, no. 10, pp. 265 267, Oct. 1998. [11] Rubeena Vohra, Akash Tayal, Image Restoration Using Thresholding Techniques on Wavelet Coefficients, IJCSI International Journal of Computer Science,Vol. 8, September 2011. [12] Chang, S. G., Yu, B., and Vetterli, M. (2000). Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. On Image Proc., 9, 1532 1546. [13] G. Y. Chen, T. D. Bui And A. Krzyzak, Image Denoising Using Neighbouring wavelet Coefficients, Icassp,Pp917-920 [14] D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, Journal of the American Statistical Assoc., 90(432), pp. 1200 1224, 1995. [15] C. Stein, Estimation of the mean of a multivariate normal distribution, Ann. Statist., 9, pp. 1135 1151, 1981. [16] G. Y. Chen, T. D. Bui, A. Krzyzak, Image denoising with neighbor dependency and customized wavelet and threshold, Pattern Recognition, 38, pp. 115 124, 2005. [17] G.Y. Chen, T.D. Bui, A. Krzyzak, "Image denoising using neighbouring wavelet coefficients," Acoustics, Speech, and Signal Processing, IEEE International Conference, vol.2, May 2004. [18] T. T. Cai and H. H. Zhou, A Data-Driven Block Thresholding Approach To Wavelet Estimation, Ann. Statist., accepted. http://www.stat.yale.edu/~hz68/. [19] Alle Meije Wink and Jos B.T.M.Roerdink/ Denoising functional MR Images a comparison of wavelet denoising and gaussian smoothing / IEEE Transactions on Medical Imaging, vol. 23, no. 3/ March 2004 [20] Shan Gai, Peng Liu, Jiafeng Liu and Xian long Tang, A New Image Denoising Algorithm via Bivariate age Based on Quaternion Wavelet Transform Journal of Computational Information Systems, 2010. [21] Zhou Dengwen, Shen and Xiaoliu, "Image Denoising Using Block Thresholding" Image and Signal Processing, CISP, vol.3, May 2008. [22] Byung-Jun Yoon and P. P. Vaidyanathan/ Wavelet-based denoising by customized thresholding [23] Aleksandra Piˇzurica, Alle Meije Wink, Ewout Vansteenkiste, Wilfried Philips and Jos B.T.M. Roerdink/ A review of wavelet denoising in MRI and ultrasound brain imaging Volume 2, Issue 1 January - February 2013 Page 61