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International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 2, Issue 1, January 2013 ISSN 2319 - 4847 IMPROVED INTELLIGENT IMAGE COMPRESSION WITH ADAPTIVE WAVELET THRESHOLD FOR IMAGE DENOISING 1 Jibanananda Mishra, 2Ranjan kumar Jena 1 Orissa Engineering College, BPUT, Bhubaneswar 2 College of Engineering and Technology, BPUT, Bhubaneswar ABSTRACT The first part of the paper an adaptive, data-driven threshold for image denoising via wavelet soft-thresholding is assessed. The threshold is derived in a Bayesian framework and the prior used on the wavelet coefficients is the Generalized Gaussian distribution (GGD) widely used in image processing applications. The threshold is simple and it is adaptive to each sub band because it depends on data-driven estimates of the parameters. Simulation results shown that with this method, called BayesShrink, achieve the state-of-the-art image denoising performance at the low computational complexity. In the second part of the paper an intelligent image compression using neural network is proposed. The objective picture quality measures like Signal to Noise Ratio(SNR), Mean Square Error(MSE), are used to measure the picture quality. It is shown through performance measure that with such denoising and compression the MSE is typically within 5%. Key Words: Image denoising, Adaptive wavelet threshold, Bayes shrink, Mean square error 1. INTRODUCTION Digital images are often degraded by noise in the acquisition and/or transmission phase. The goal of image denoising is to recover the true/original image from such a distorted/noisy copy. Traditionally, this is achieved by linear processing such as Wiener filtering. A vast literature has emerged recently on signal denoising using nonlinear techniques in the setting of additive white Gaussian noise. Since the early 1990s, the wavelet theory has been well developed and widely introduced into many fields of applications, including statistics estimation, density estimation, solution of partial differential equations, and image compression. In 1992, Donoho and Johnstone presented a method named wavelet shrinkage, and showed its obvious efficiency on signal denoising and inverse problem solving [1]. In this method a discrete wavelet transform (DWT) is performed on the noisy signal first. Then with a present threshold, coefficients with magnitude smaller than the threshold are set to zero while those with larger magnitude are kept and used to estimate the noiseless coefficients. Finally, an inverse DWT (IDWT) reconstructs the signal from the estimated coefficients. Later, in their subsequent paper, Donoho and Johnstone proved that wavelet shrinkage is nearly optimal (in a minimax sense) over a wide range of function classes and error criterions and always provides an estimated function with a smoothness not less than the original [2]. However, the universal threshold (is the standard deviation of the noise and length of noisy data), which was first given for wavelet shrinkage, was found tending to set all the detail coefficients to zero, especially when N approaches infinite [3]. In fact, Donoho and Johnstone pointed out that the universal threshold has a good mean square error (MSE) performance when the sample size is in the tens or hundreds [1]. Since the threshold plays a key role in this appealing method, variant methods appeared later to set an appropriate threshold [4]-[5]. Donoho and Johnstone found that in the criterion under which the universal threshold is nearly optimal, the smoothness restriction is probably the cause for over-erasing wavelet coefficients. Therefore, they removed the restriction and gave what is known as the Stein’s Unbiased Risk Estimate (SURE), threshold, which minimizes SURE, an estimate of MSE risk [6]. Assuming that wavelet coefficients obey the generalized Gaussian distribution (GGD), Chang et al. presented a new threshold, ( is the noise variance and the standard deviation of noiseless coefficients in a sub band), which was claimed to often provide a better denoising result than the SURE threshold [7] and can be unwarrantedly large due to its dependence on the number of samples, which is more than for typical test image of size. The formulation is grounded on the empirical observation that the wavelet coefficients in a sub band of a natural image can be summarized adequately by a generalized Gaussian distribution(GGD). This observation is well-accepted in the image processing community (for example, see [8], [9], [10], [11], [12]) and is used for state-of-the-art image coders in [8], [9], [12]. It follows from this observation that the Volume 2, Issue 1, January 2013 Page 249 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 2, Issue 1, January 2013 ISSN 2319 - 4847 average MSE (in a sub band) can be approximated by the corresponding Bayesian squared error risk with the GGD as the prior applied to each in an independent and identically distributed(i.i.d) fashion. That is, a sum is approximated by an integral. We emphasize that this is an analytical approximation and our framework is broader than assuming wavelet coefficients are iid draws from a GGD. The goal is to find the soft-threshold that minimizes this Bayesian risk, and we call our method BayesShrink. The proposed Bayesian risk minimization is sub band-dependent. Given the signal being generalized Gaussian distributed and the noise being Gaussian, via numerical calculation a nearly optimal threshold for soft-thresholding is found. 2. WAVELET SHRINKAGE Consider the following additive noise model in time domain: x(i) = s(i) + e(i) -----------------------------------------(1) Where the signal s is corrupted by the zero-mean white Gaussian noise process e ~ N (0, ),resulting in x, i.e., noisy signal. Throughout this text, A(i) is often used to stand for the i th element for a given sequence A . By the linearity of wavelet transformation, there is a regression model in the wavelet domain: = + ---------------------------------- (2) Where is the wavelet transform of x, while and are those corresponding to s and n. The elements of will be termed as data coefficients (noisy coefficients), signal coefficients (noiseless coefficients), and noise coefficients respectively. The term denoised coefficients is used to refer to the estimate of the noiseless coefficient from the corresponding noisy coefficient .When the transformation is orthogonal, which is only the case considered here, the noise in each wavelet sub band remains a zero-mean white Gaussian process with variance consistent with that in the time domain. Given a threshold T , two popular thresholding rules in wavelet shrinkage called hard thresholding and soft thresholding which correspond to the strategies “kill or keep” and “kill or “shrink” respectively can be expressed by two different functions upon the noisy coefficients of which the values are denoised coefficients. = ( -----------------------------------(3) I( ) = ( ----------------------------------(4) I( )( - ) Herein, I (.) denotes the identification function. In a wavelet shrinkage scheme, no modification is performed on the sub band at the coarsest scale due to belief that the coefficients there are in have high enough signal–to-noise ratios (SNR). To choose between the two thresholding rules, Bruce and Gao took a profound research and gave several important instructions. For a given threshold , soft thresholding has smaller variance, however higher bias than hard thresholding especially for very large wavelet coefficients. If the coefficients distribute densely close to the threshold, hard thresholding will show large variance and bias. For soft thresholding, smaller error often happens when the coefficient is close to zero. In general, soft thresholding chosen for smoothness while hard thresholding for lower error. The soft- thresholding rule is chosen over hard thresholding for several reasons. First, soft thresholding has been shown to achieve near optimal minimax rate over a large range of Besov spaces. Second, for the generalized Gaussian prior assumed in this work, the optimal soft-thresholding estimator yields a smaller risk than the optimal hard-thresholding estimator .Lastly, in practice, the soft thresholding method yields more visually pleasant images over hard-threshold because the later is discontinuous and yields abrupt artifacts in the recovered images, especially when the noise energy is significant. Statistical image modeling in the wavelet- domain has applications in image compression, estimation, and segmentation. It can be roughly categorized into three groups: the interscale models, the intrascale models and the hybrid inter–and intrascale models [13] - [15]. In the statistical Bayesian literature, many works have concentrated on deriving the best threshold (or shrinkage factor) based on priors such as the Laplacian and a mixture of Gaussians ([16],[17]). With an integral approximation to the pixel- wise MSE distortion measure as discussed earlier, the formulation here is also Bayesian for finding the best soft- thresholding rule under the generalized Gaussian prior. The GGD has been used in many sub band or wavelet-based image processing applications [16], [18]. In [16], it was observed that a GGD with the shape parameter ranging from 0.5 to 1 can adequately describe the wavelet coefficients of a large set of natural images. 3. ADAPTIVE THRESHOLD FOR BAYES SHRINK The GGD is GG =C exp{ [α │x│ }------(5) -<x<, >0, >0. Where α = Volume 2, Issue 1, January 2013 Page 250 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 2, Issue 1, January 2013 ISSN 2319 - 4847 and C = du is the gamma function. The parameter is the standard deviation and is the shape parameter. For a given set of parameters, the objective is to find a soft-threshold T which minimizes the Bayes risk. Fig.1 Flow chart for image Denoising Algorithm Using wavelet Transform ----------------- (6) where = (Y),Y/X N(X, ) and X denote the optimal threshold by T*, T* = arg min ------------------------(7) Which is the function of the parameters and . Referring to [7], we can obtain ------------------------------------------- (8) This threshold is not only nearly optimal but also has an intuitive appeal. The normalized threshold is inversely proportional to the standard deviation and proportional to the noise standard deviation. When the signal is much stronger than the noise the normalized threshold is chosen to be small in order to preserve most of the signal and remove some of the noise. When the noise dominates , the normalized threshold is chosen to be large to remove the noise which has overwhelmed the signal[19].Thus, this threshold choice adapts to both the signal and noise characteristics. Fig.1shows the flow chart for image Denoising Algorithm Using wavelet Transform. 4. ALGORITHM OF INTELLIGENT COMPRESSION Multilayer neural network can be employed to achieve intelligent image compression [20]. The network parameters will be adjusted using different learning rules for comparison purposes. Mainly, the input pixels will be used as target values so that assigned mean square error can be obtained, and then the hidden layer output will be the compressed image. Artificial Neural network have found increasing applications in image compression due to their noise suppression and learning capabilities. A number of different neural network models, based on learning approach have been proposed. Models can be categorized as either linear or nonlinear neural nets according to their activation Volume 2, Issue 1, January 2013 Page 251 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 2, Issue 1, January 2013 ISSN 2319 - 4847 function. The network will be trained by back propagation, using different learning algorithms. Mainly Newton’s method, gradient descent and adaptive gradient descent learning algorithms will be used for this purpose. The neural network structure can be illustrated in fig. 2. Three layers, one input layer, one output layer and one hidden layer, are assigned. Both of input layer and output layer are fully connected to hidden layer. Compression is achieved by designing the value of the number of neurons at the hidden layer, less than that of neurons at both input and output layers. Log-sigmoid function which is given in equation1 is one of the most common functions employed in different neural networks problems. ---------------------------------(9) It was shown that nonlinear functions have a more capability of learning both linear and nonlinear problems than linear ones. Fig.2 Back Propagation Neural Network The output of the neuron in the hidden layer is given by .--------------------------- (10) Also the output of the neuron in the output layer is given by = .--------------------------- (11) Where and are the activation function of the hidden layer and the output layer respectively. The is the synaptic weight connecting the input node to the neuron of the hidden layer. The is the bias of the neuron of the hidden layer. N is the number of neurons in the hidden layer and M is the number of neurons in the output layer.Training the network is an important step to get the optimal values of weight and biases after being initialize randomly. The training processes require a set of prototypes and targets to learn the proper network behaviour. During training, the weights and biases of the network are iteratively adjusted to minimize the network performance function which is the mean square error for the feed forward networks. The mean square error is calculated as the average squared error between the inputs and targets. In the basic back propagation training algorithm, the weights are moved in the direction of the negative gradient, which is the direction in which the performance function decreases most rapidly. Iteration of this algorithm can be written as: = . ------------------------------------(12) Where is a vector of current weights and biases, . is the current gradient, and is the learning rate. Here all training algorithms have been developed using MATLAB. It is important to explain the steps which have been done. For our purpose three layers feed forward neural network had been used. Input layer, hidden layer with 16 neurons, and output layer with 64 neurons. Back propagation algorithms had been employed for the training processes. To do the training input prototypes and target values are necessary to be introduced to the network so that the idea behind supplying target values is that this will enable us to calculate the difference between the output and target values and then recognize the performance function which is the criteria of our training. For training the network, the 512x512 pixels barbara image had been employed. Procedure for Image Compression The image is split into non-overlapping sub-images. Say for example 256 x 256 bit image will be split into 4 x 4 or 8 x 8 or 16 x 16 pixels. The normalized pixel value of the sub-image will be the input to the nodes. The three-layered back propagation learning network [21] will train each sub-image. The number of neurons in the hidden layer will be designed for the desired compression. The number of neurons in the output layer will be the same as that in the input Volume 2, Issue 1, January 2013 Page 252 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 2, Issue 1, January 2013 ISSN 2319 - 4847 layer. The input layer and output layer are fully connected to the hidden layer. The Weights of synapses connecting input neurons and hidden neurons and weight of synapses connecting hidden neurons and weight of synapses connecting hidden neurons and output neurons are initialized to small random values from say –1 to +1.The output of the input layer is evaluated using linear activation function. The input to the hidden layer is computed by multiplying the corresponding weights of synapses. The hidden layer units evaluate the output using the sigmoidal function. The input to the output layer is computed by multiplying the corresponding weights of synapses. The output layer neuron evaluates the output using sigmoidal function. The Mean Square error of the difference between the network output and the desired output is calculated. This error is back propagated and the weight synapses of output and input neurons are adjusted. With the updated weights error is calculated again. Iterations are carried out till the error is less than the tolerance. The compression performance is assessed in terms of Compression ratio, PSNR and execution time [22]. Steps of the algorithm 1. Divide the original image into 8x8 pixel blocks and reshape each one into 64x1 column vector. 2. Arrange the column vectors into a matrix of 64x1024. 3. Let the target matrix equal to the matrix in step 2. 4. Choose a suitable learning algorithm, and parameters to start training. 5. Simulate the network with the input matrix and the target matrix. 6. Obtain the output matrices of the hidden layer and the output layer. 7. Post-process them to obtain the compressed image, and the reconstructed image respectively. Picture quality measures Image quality measures (IQM) are figures of merit used for the evaluation of imaging systems or of coding/processing techniques. Some Objective quality measures as available in literature and used in this work are presented here for better understanding of the work ahead. Let x (m, n) denotes the samples of original image, and x’(m,n) denotes the samples of compressed image. M and N are number of pixels in row and column directions, respectively. In this paper, we have used MSE, PSNR, PQS, SSIM, and MD as the five objective quality measures, which are calculated as follows. MSE and PSNR are the most common picture quality measures. Mean Square Error is given by- m n MSE I MN [ I ( x, y) I ' x, y ]2 --------------------(13) y 1 x 1 Peak Signal To Noise Ratio is given as- PSNR=20log[255/sqrt(MSE)]----------------------- (14) 5. SIMULATION RESULTS & DISCUSSION The grayscale images “barbara” is used as test image. Harr wavelet transform give better result at decomposition level 2. Most of the time the minimum variance 0.01 is taken for better result. To assess the performance of BayesShrink, it is compared with hard and the soft threshold. The 1-D implementation of BayesShrink, can be obtained from the WaveLab toolkit [23], and the 2-D extension is straightforward. The MSEs from the various methods are compared in Table I. Table I Wavelet Theshold Noise PSNR MSE Image Gaussian 34.7711 0.0739 Harr Hard Salt & pepper 30.3822 0.1602 Barbara Poisson 39.9538 0.0564 Speckle 38.5009 0.0582 Gaussian 34.5299 0.0751 Soft Salt & pepper 31.5224 0.0955 Poisson 37.7077 0.0647 Speckle 36.9704 0.0658 Gaussian 28.1056 0.1036 Salt & pepper 30.0358 0.1699 Bayes soft Poisson 43.1812 0.0332 Speckle 41.7077 0.0389 Volume 2, Issue 1, January 2013 Page 253 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 2, Issue 1, January 2013 ISSN 2319 - 4847 Fig.3. Denoising with Bayes soft threshold where SNR is maximum (43.1812) and MSE is minimum (0.0332) Fig.4.Performance measure of image compression with back propagation neural network 6. CONCLUSION In this paper at first an adaptive threshold for wavelet thresholding images was proposed, based on the GGD modeling of sub band coefficients, and simulation results showed excellent performance. It was demonstrated that spatially adaptive thresholds greatly improves the denoising performance over uniform thresholds. That is, the threshold value changes for each coefficient. Secondly, back propagation neural network was designed specifically for compression. Although the setting in this paper was in the wavelet domain, the idea extended to other domains such as back propagation neural network to achieve good compression. REFERENCES [1] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation via wavelet shrinkage, Biometrika, vol. 81, pp. 425– 455, 1994. [2] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, Wavelet shrinkage: asymptopia?, J. R. Statist. Soc. B, vol. 57, pp. 301– 369, 1995 [3] J. Fan, P. Hall, M. Martin, and P. Patil, Adaptation to high spatial in homogeneity based on wavelets and on local linear smoothing, Australian National Univ., Canberra, Australia, Tech. Rep., CMA-SR18- 93. Volume 2, Issue 1, January 2013 Page 254 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 2, Issue 1, January 2013 ISSN 2319 - 4847 [4] L. K. Shark and C. Yu, Denoising by optimal fuzzy thresholding in wavelet domain, Electron. Lett., vol.36, pp. 581–582, Mar. 2000. [5] P. Moulin and J. Liu, Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors, IEEE Trans. Inform. Theory, vol. 45, pp. 909–919, Apr. 1999. [6] D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, J. Amer. Statist. Assoc., vol. 90, pp. 1200–1224, 1995. [7] S. G. Chang, B. Yu, and M. Vetterli, Adaptive wavelet thresholding for image denoising and compression, IEEE Trans. Image Processing, vol.9, pp. 1532–1546, Sept. 2000. [8] C. M. Stein, “Estimation of the mean of a multivariate normal distribution,” Ann. Statist., vol. 9, no. 6, pp. 1135–1151, 1981. [9] R. L. Joshi, V. J. Crump, and T. R. Fisher, “Image subband coding using arithmetic and trellis coded quantization,” IEEE Trans. Circuits Syst. Video Technol., vol. 5, pp. 515–523, Dec. 1995. [10] S. M. LoPresto, K. Ramchandran, and M. T. Orchard, “Image coding based on mixture modeling of wavelet coefficients and a fast estimationquantization framework,” in Proc. Data Compression Conf., Snowbird, UT, Mar. 1997, pp. 221–230. [11] S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell., vol. 11, pp. 674–693, July 1989. [12] E. Simoncelli and E. Adelson, “Noise removal via Bayesian wavelet coring,” Proc. IEEE Int. Conf. Image Processing, vol. 1, pp. 379–382, Sept. 1996. [13] P. H. Westerink, J. Biemond, and D. E. Boekee, “An optimal bit allocation algorithm for sub-band coding,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, Dallas, TX, Apr. 1987, pp. 1378–1381. [14] J. K. Romberg, H. Choi, and R. Baraniuk, Bayesian tree-structured image modeling using wavelet-domain hidden Markov model, in Proc.SPIE, vol. 3816, Denver, CO, July 1999, pp. 31–44. .[15] G. Fan, X.-G. Xia, Image denoising using a local contextual hidden Markov model in the wavelet domain, IEEE Signal Processing Letters, vol.8, no.5, pp.125-128, May 2001. [16] E.Simoncelli and E. Adelson, Noise removal via Bayesian wavelet coring, Proc. IEEE Int. Conf. Image Processing, vol. 1, pp. 379–382,Sept. 1996. [17] N. Weyrich and G. T. Warhola, De-noising using wavelets and crossvalidation, Dept. of Mathematics and Statistics, Air Force Inst. Of Tech., AFIT/ENC, OH, Tech. Rep. AFIT/EN/TR/94-01, 1994. [18] Y. Yoo, A. Ortega, and B. Yu, Image subband coding using context based classification and adaptive quantization, IEEE Trans. Image Processing, vol. 8, pp. 1702–1715, Dec. 1999 [19] Cajo J.F. ter Braak/ “Bayesian sigmoid shrinkage with improper variance priors and an application to wavelet denoising” / Computational Statistics & Data Analysis vol. 51, pp. 1232 – 1242 / 2006 [20] B.Verma, B.Blumenstin and S. Kulkarni, Griggith University, Australia, “A new Compression technique using an artificial neural network”. [21] Hahn-Ming Lee, Chih-Ming Cheb, Tzong-Ching Huang, “Learning improvement of back propagation algorithm by error saturation prevention method”, Neurocomputing, November 2001 [22] M. Miyahara, K. Kotani, and V.R. Algazi, “Objective Picture Quality Scale (PQS) for Image Coding”, IEEE Transactions on Communications, vol. 46, no. 9, pp. 1215- 1226, 1998. [23] J. Buckheit, S. Chen, D. Donoho, I. Johnstone, and J. Scargle, “WaveLab Toolkit,”, http://www- stat.stanford.edu:80/~wavelab/. 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1 Orissa Engineering College, BPUT, Bhubaneswar
2College of Engineering and Technology, BPUT, Bhubaneswar

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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 1, January 2013 ISSN 2319 - 4847
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