SCALED BAYES IMAGE DENOISING ALGORITHM USING MODIFIED SOFT THRESHOLDING FUNCTION

SCALED BAYES IMAGE DENOISING ALGORITHM USING MODIFIED SOFT THRESHOLDING FUNCTION SAMI M. A. GORASHI Computer Engineering Department Taif University-Khurmaha Branch KSA SABAHALDIN A. HUSSAIN Electrical Computer EngineeringDepartment Omdurman Islamic University Sudan ABSTRACT Image denoising is an active area of research probably one of most studied problems in image processing fields. In this paper, a new adaptive threshold estimation thresholding function are proposed for image denoising in wavelet domain. The proposed threshold estimation includes effect of image subb bit size at each decomposition level in calculation of Bayesian based threshold value. The proposed thresholding function uses exponential weight to shrink noisy coefficients. The experimental results show that performance of proposed denoising algorithm is superior to that of conventional wavelet denoising approaches. Keywords:Wavelet transform, Thresholding function, Image denoising. 1 INTRODUCTION Digital images are generally affected by different types of noise. A noise is introduced in transmission medium due to a noisy channel, imperfect instruments used in image processing, errors during measurement process, noise due to degradation such in films, image compression during quantization of data for digital storage. Each element in imaging chain such as lenses, film, digitizer, etc. contributes to degradation. Thus, image denoising is a necessary primary step in any furr image processing tasks like segmentation, object recognition, computer vision, etc. To overcome image data corruption, we need to know something about degradation process in order to develop a model for it. When we have a model for degradation process, inverse process can be applied to image to restore it back to original form. Noise modelling in images is greatly affected by capturing instruments, image quantization, data transmission channels, etc. Mostly, noise in digital images is found to be additive in nature with uniform power in whole bwidth with Gaussian probability distribution. Such a noise is referred to as Additive White Gaussian Noise(AWGN). White Gaussian noise can be caused by poor image acquisition or by transferring image data in noisy communication channel. Most denoising algorithms use images artificially distorted with well-defined white Gaussian noise to achieve objective test results[4,14,15]. Denoising that based on wavelet transform for cancelling white Gaussian noise finds wide range of applications since pioneer work by Donoho Johnstone[7-9]. In wavelet based denoising algorithms, noise is estimated wavelet coefficients are thresholded to separate signal noise using appropriate threshold value. Since threshold plays a key role in this appealing technique, variant methods appeared later to set an appropriate threshold value[4,14,15]. Thresholding rules such as BayesShrink[4], SUREShrink[7] are subb dependent, data-driven, adaptive. In SUREShrink, a separate threshold is computed for each detail subb based upon minimizing Stein's Unbiased Risk Estimator (SURE) [16]. It is a hybrid of a universal SURE threshold, with choice being dependent upon energy of particular subb. SUREShrink has yielded good image denoising performance comes close to true minimum MSE of optimal soft-threshold estimator[7]. In BayesShrink, assigned goal is minimization of Bayesian risk, hence its name, BayesShrink. It minimizes Bayes Risk Estimator function assuming Generalized Gaussian Distribution (GGD) prior. BayesShrink performs denoising that is consistent with human visual system which is less sensitive to presence of noise in vicinity of edges. It performs little denoising in high activity sub-region (image edges) to preserve sharpness of image but completely denoising flat regions of image[4]. In this paper, a Bayesian based denoising algorithm that uses scaled threshold according to decomposed image subb bit size modified soft thresholding function is proposed. Results show that proposed denoising algorithm improves Peak Signal to Noise Ratio(PSNR) of denoised image as compared with conventional denoising algorithms. The rest of paper is organized as follows. Section 2 briefly reviews basic idea behind discrete wavelet transform. Section 3 section 4 explain different thresholding functions different threshold estimation methods. In Section 5, we describe proposed image denoising algorithm. The results of our proposed denoising algorithm will be compared with BayesShrink[4], VisuShrink[9], spatial Wiener filter[10]in section 6. Finally, concluding remarks are given in section 7. 2 DISCRETE WAVELET TRANSFORM The wavelet transform was studied oretically in geophysics mamatics. Thereafter, Daubechie's[5], putting wavelets firmly into application domain, pursued links with digital signal processing. Over last two decades, wavelet transform has acknowledged an

enormous deal of attention in many fields. In fields of signal image processing it is commonly used for denoising, compression, image enhancement. The main advantage of discrete wavelet transform (DWT) is sparse representation. It is very helpful to describe local features, eir spatially or spectrally over or image representations, such as discrete Fourier transform. The sparse representation makes maximum noise removal while at same time preserving edges of images or high frequency features of signal which is very useful to analyze image by experts. The DWT decomposes noisy image into four subbs namely LL, LH, HL, HH. The decomposed noisy image consists of a small number of coefficients with high signal to noise ratio (SNR) named approximation subb (LL) a large number of coefficients with low SNR named detail subbs (LH, HL, HH). Thus, by thresholding detail subbs, efficient noise suppression can be realized. Thresholding is a simple non-linear technique in which wavelet coefficient is thresholded by comparing against a threshold using appropriate thresholding function. Therefore, thresholding function estimated threshold value play major role in denoising performance. 3 THRESHOLDING FUNCTION 3.1 HARD THRESHOLDING FUNCTION To denoise a signal using Wavelet Transform(WT), detail coefficients are thresholded. The simplest thresholding technique is hard thresholding where new values of detail coefficients are found according to following thresholding function[8]: (1) 0 Where is threshold value or gate value. Notice that Eq.(1) is nonlinear discontinuous at.the graphical representation of hard threshold function is shown in figure (1). (2) 0 Generally, soft thresholding function is much better yields more visually pleasant images compared with hardthresholding[3,6]. This is because hard-thresholding procedure creates discontinuities at which yields abrupt artifacts in recovered images. Also, soft thresholding function yields a smaller minimum mean squared error compared to hard form of thresholding. The graphical representation of soft thresholding is shown in figure(2). Figure 2.Soft Thresholding Function 3.3 MODIFIED SOFT THRESHOLDING FUNCTION The thresholding function shape play significant role in any wavelet based denoising algorithm. Thus, several thresholding functions have been proposed by many authors all based on conventional hard soft thresholding function[2,11-13]. Both hard soft thresholding functions, simply zeros all coefficients which are smaller than thresholding value. They assume that se coefficients are only due to noise which contradicts fact that desired signal is also present in se coefficients. Hence, this will cause loss some of image information. The degree of image information loss will depend upon accuracy of estimated threshold value that used to discriminate between noisy noise free coefficients. To overcome this problem, we suggest to use modified soft thresholding function that uses exponential weight to shrink coefficientswhich are smaller than estimated threshold value while keeping same scaling function that used for soft thresholding for rest of coefficients according to following proposed equation:- (3) Figure 1.Hard Thresholding Function The above figure clearly state that, all coefficients with magnitude greater than selected threshold value remain as y are ors with magnitudes smaller than or equal to are set to zero. 3.2 SOFT THRESHOLDING FUNCTION Soft thresholding is anor method of thresholdingwhere new values of detail coefficients are given by following thresholding function[9]: Where δ is parameter that controls degree of wavelet coefficients falling down. Extensive simulation show that, for low noise level, δ=0.09 is sufficient, while for high noise level, larger value of δ (δ 0.8) is required to cancel heavy noise. Examining Eq.(3), clearly, as δ, thresholding rule follows soft thresholding function. Figure(3), shows effect of δ on characteristics of suggested modified soft thresholding function.

Figure3.Modified Soft Thresholding Function (δ=1, δ=0.09, δ=0.05) 4 THRESHOLD ESTIMATION METHODS 4.1 VISUSHRINK VisuShrink was introduced by Donoho[8,9]. It is nonadaptive universal threshold, which depends only on number of data points estimated noise stard deviationσ hence makes it very simple to implement. It follows hard thresholding rule also referred to as universal threshold which can be defined as:- T σ 2logM (4) Where M represents image size or number of pixels σ is an estimate of noise level which can be calculated using Median Absolute Deviation (MAD) given by [8,9]:- σ,. (5) Where is detail coefficients in diagonal subb of first decomposition level. VisuShrink does not deal with minimizing mean squared error [8]. It can be viewed as a general-purpose threshold selector that exhibits near optimal minimax error properties ensures with high probability that estimates are as smooth as true underlying functions. However, disadvantage of VisuShrink is that it yields recovered images that are overly smood. Because firstly, its threshold choice can be unwarrantedly large due to its dependence on number of pixels in image which in turn yields to remove too many wavelet coefficients of decomposed image. Secondly, VisuShrink follows global thresholding scheme where re is a single value of threshold applied globally to all wavelet coefficients. 4.2 BAYES SHRINK Bayesian based threshold estimation was proposed by Chang, et al [4]. The goal of this method is to estimate a threshold value that minimizes Bayesian risk assuming Generalized Gaussian Distribution (GGD) prior. It has been shown that BayesShrink outperforms SUREShrink most of times in terms of PSNR values over a wide range of noisy images[4]. It uses soft thresholding is subbdependent, which means that thresholding is done at each b of resolution in wavelet decomposition. To estimate threshold value, corrupted image pixel can be written as (assume additive white Gaussian noise):- rx, y sx, y nx, y (6) where rx, y, sx, y, nx, y are observed, original, noise signals at spatial location (x,y) respectively. Since noise signal are independent of each or, it can be stated that:- σ σ σ (7) Where σ is observed signal variance, σ is an estimate of original noise free signal variance, σ is an estimate of noise variance. The noise stard deviation σ can be estimated from Eq.(5) while observed signal variance σ can be estimated using:- σ, (8) r x, y Knowing σ σ, variance of signal, σ can be estimated according to:- σ maxσ σ,0 (9) Knowing σ σ, Bayes threshold is estimated according to[4]:- T (10) 4.3 SCALED BAYES SHRINK The Bayes Shrink threshold estimation can be improved by including effect of image subb bit size at each decomposition level in calculation of threshold value. Empirically, based on extensive simulation test, we suggest to use following formula for scaled Bayes threshold estimation:- T 1γ (11). Where, γ Subb Bit Size log ), ζ is subb width, η is subb height. The scaling parameter γ is a monotonically increasing value according to level of decomposition. In general, most of wavelet coefficients are noisy at higher decomposition levels because noise almost concentrated in high frequency bs. Accordingly, larger threshold value is necessary at higher decomposition level to cancel noise. This characteristic is met by proposedt hence improved denoising results is expected. 5 PROPOSED ALGORITHM In wavelet transform domain, several denoising algorithms were successfully applied in a wide range of applications. These algorithms were based on thresholding wavelet detail coefficients. The selected threshold value thresholding function shape play significant role in deciding wher coefficient is noisy(to cancel) or noise free(to keep or shrinked). In this paper, an improved Bayesian based threshold value is estimated according to Eq.(11) a modified soft thresholding function according to Eq.(3) is used. The following steps describe implementation of proposed algorithm. 1. Assign number of wavelet decomposition level. 2. Decompose noisy image into four subbs namely LL, LH, HL, HH using 2-D discrete wavelet transform.

3. 4. Estimate noise stard deviation σ using Eq..(5). Estimate variance of observed signal σ in subbs LH, HL, HH for each decomposition level using Eq.(8). 5. Estimate variance of desired signal σ in subbs LH, HL, HH for each decomposition level using Eq.(9). 6. For each decomposition level, calculate scaling parameterr γ. 7. Estimate scaled Bayes threshold T in subbs LH, HL, HH for each decomposition level using Eq.(11). 8. Denoise wavelet coefficients in subbs LH, HL, HH for each decomposition level using modifiedd soft thresholding function defined in Eq.( (3). 9. Reconstruct denoised image using inverse 2-D discrete wavelet transform. As an illustration, Figure(4) shows a block diagram of proposed denoising algorithm. Subjectively, for low noise level degradation( (see figure(5)), almost all denoising algorithms achieve nearly equivalent visual quality with exception of VisuShrink which produces lower image quality.for higher noise density level, VisuShrink exhibits worst denoising results. This is due to fact that, in wavelet domain, magnitude of coefficients varies depending on decomposition levels. Hence, if all levels are processed with just one threshold value processed image may be overly smood so that sufficient information preservation is not possible image gets blurry. This is nature of VisuShrink denoising algorithm which usess single universal threshold value T for all subbs in each level hence produces an image of much lower resolution(ass an evident see figure(6c), figure(7c)). On or h, proposed denoising algorithm removes noise preserves image fine details better than both BayesShrink Wiener filter. These algorithmsare nearest competitor to our proposed algorithm. In fact, proposed algorithm uses more accurate estimate of threshold. Referring to Eq.(11), proposed threshold estimatee is monotonically increased in accordance with levell of decomposition.this will provide better noise cancellationn of different frequency noise components in different decomposition levels. To demonstrate effectivenesss of proposed algorithm in comparison with BayesShrink, Wiener see figure(6.d,e,&f) figure(7.d,e,,&f) through noticing sky, face of Lena, edges of images. Clearly, se figures state that, proposed algorithm reveals better visual perception. Table(1): PSNR Results for Denoising Lena, Mrill, Boat. Figure 4. Block Diagram of Proposed Algorithm Image σ n 10 VisuShrink 27.78 Weiner BayesShrink Proposed 33.52 33.48 33. 57 Where are wavelet transform its inverse respectively., is recovered image pixel. Lena 15 20 26.41 25.63 31.11 31.58 31. 74 29.01 30.34 30. 43 6 EXPERIMENTAL RESULTS For evaluation purpose, an experiment was conducted to assess performance of proposed denoising algorithm. We will compare performance of proposed algorithm with spatial domain Wiener filter[10], VisuShrink[9], BayesShrink[4] denoising algorithms. Images artificially corrupted with white Gaussian noise with zero mean stard deviations 10, 15, 20, 25, 30 were used in test. The wavelet transform that employs Daubechie s[5] least asymmetric compactly supported wavelet with eight vanishing moments was used for wavelet domain based denoising algorithms. The relative denoising algorithms performance is measured quantitatively using Peak Signal to Noise Ratio(PSNR) [1]. The denoising relative performance in terms of PSNR value for a set of images is recorded in Table(1). The best denoising algorithm is highlighted in bold font for each testt image. Objectively, examining results in Table(1), clearly, we can see that proposed denoising algorithm outperforms or denoising algorithms alll time in terms of both individual average PSNR value over whole scope of noise levels images under test. 25 30 Average PSNR 10 15 Mrill 20 25 30 Average PSNR 10 15 Boat 20 25 30 Average PSNR 24.97 24.54 25.87 21.10 20.63 20.34 20.13 19.99 23.40 25.47 24.28 23.60 23.12 22.66 23.52 27.30 29.32 29. 45 25.81 28.60 28. 70 29.35 30.66 30. 78 26.77 28.85 29. 16 25.93 26.68 26. 82 24.99 25.17 25. 32 24.12 24.09 24. 21 23.27 23.26 23. 36 27.39 28.38 28. 49 31.71 31.90 31. 93 29.91 29.86 29. 94 28.23 28.47 28. 58 26.69 27.40 27. 51 25.38 26.59 26. 70 27.68 28.51 28. 62

(a) (b) (a) (b) (c) (d) (c) (d) (e) (f) (e) (f) Figure 5. Denoising results for Mrill image:(a) Original image (b) Noisy image(σ n =10);(c)VisuShrink; (d) Wiener (e)bayesshrink;(f)proposed algorithm. Figure 7.Denoising results for Lena image:(a) Original image (b) Noisy image(σ n =30);(c)VisuShrink; (d) Wiener (e)bayesshrink;(f)proposed algorithm. (a) (c) (e) (b) (d) (f) 7 CONCLUSIONS In this paper, we can conclude that proposed denoising algorithm outperforms or denoising algorithms objectively subjectively. As noise level increased, VisuShrink exhibits worse denoising performance as compared with or denoising algorithms. It blurs images while it attempts to remove noise. The proposed algorithm improves PSNR of image as compared with Wiener BayesShrink denoising algorithms. Referring to Table(1), proposed algorithm achieves an average PSNR gain of 4.91 db, 1.43 db, 0.12 db for Lena image;5.09 db, 1.1 db, 0.11 db for Mrill image; 5.1 db, 0.94 db, 0.11 db for Boat image as compared with VisuShrink,Wiener, BayesShrink respectively. The proposed algorithm can be extended to deal with three dimensional colour pictures through vector processing. This issue is left as future work. References Figure 6. Denoising results for Boat image:(a) Original image (b) Noisy image(σ n =20);(c)VisuShrink; (d) Wiener (e)bayesshrink;(f)proposed algorithm. [1] Alain H., Djemel Z., " Image Quality Metrics: PSNR vs. SSIM", International Conference on Pattern Recognition, IEEE Proceedings, pp.2366-2369, 2010. [2] Biswas M. Om H., "An Image Denoising Threshold Estimation Method", Advanced in computer science its applications(acsa), Vol. 2, No. 3, pp.377 381, 2013.

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