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2-D Dual-Tree Wavelet Based Local Adaptive Image Denoising M. Borhani V. Sedghi Sharif University of Technology Amir-Kabir University of Technology Borhani@ee.sharif.edu S8123232@aut.ac.ir Abstract: We propose a new wavelet-based method representation using a set of analyzing functions for image denoising that applies the two dimensions that are dilations and translations of a few Dual Tree wavelet Transform and adapt thereshold in functions. each local area to obtain better denoised image. The wavelet transform comes in several forms. The Empirical results show that new approach is better than critically-sampled form of the wavelet transform older algorithms. In this paper, the structure of Dual provides the most compact representation; Tree wavelet transform (DTDWT) is modified and however, it has several limitations. For example, it Comparison results are also presented. lacks the shift-invariance property, and in multiple dimensions it does a poor job of distinguishing Keywords: Image Denising, Local Adaptive orientations, which is important in image threshold, Wavelet transforms. processing. For these reasons, it turns out that for some applications improvements can be obtained 1 Introduction by using an expansive wavelet transform in place of a critically-sampled one. (An expansive In many applications, image denoising is used to transform is one that converts an N-point signal produce good estimates of the original image from into M coefficients with M > N .) There are noisy observations. The restored image should several kinds of expansive DWTs; here we contain less noise than the observations while still describe and provide an implementation of the keep sharp transitions (i.e. edges). dual-tree complex discrete wavelet transform. Wavelet transform, due to its excellent localization The dual-tree complex wavelet transform property, has rapidly become an indispensable overcomes these limitations - it is nearly shift- signal and image processing tool for a variety of invariant and is oriented in 2D [8]. The 2D dual- applications, including compression and denoising tree wavelet transform produces six sub bands at [1,2,4,5,6,7]. Wavelet thresholding (first proposed each scale, each of which is strongly oriented at by Donoho [4,5,6]) is a signal estimation technique distinct angles. that exploits the capabilities of wavelet transform The paper is organized as follows. In Section 2, the for signal denoising and has recently received wavelet theresholding is discussed. In Section 3, extensive research attentions. It removes noise by we present the one dimension Dual Tree wavelet killing coefficients that are insignificant relative to transform. The two dimensions Dual Tree wavelet some threshold, and turns out to be simple and transform algorithm is presented in Section 4. In effective. Wavelet thresholding solution given by Section 6, the image denoising algorithm is Donoho has also proven to be asymptotically presented. Section 5 presents the simulation results optimal in a minimax MSE (mean squared error) comparing the two different Dual Tree schemes sense over a variety of smoothness spaces [2,6]. and final conclusions are drawn in Section 7. For many natural signals, the wavelet transform is a more effective tool than the Fourier transform. The wavelet transform provides a multiresolution 2 Wavelet Theresholding The transform is 2-times expansive because for an N-point signal it gives 2N DWT coefficients. If the Wavelet thresholding for image denoising involves filters in the upper and lower DWTs are the same, two steps: 1) taking the wavelet transform of an then no advantage is gained. However, if the filters image (i.e., calculating the wavelet coefficients); 2) are designed is a specific way, then the sub band discarding (setting to zero) the coefficients with signals of the upper DWT can be interpreted as the relatively small or insignificant magnitudes. By real part of a complex wavelet transform, and sub discarding small coefficients one actually discards band signals of the lower DWT can be interpreted wavelet basis functions which have coefficients as the imaginary part. Equivalently, for specially below a certain threshold. The denoised signal is designed sets of filters, the wavelet associated with obtained via inverse wavelet transform of the kept the upper DWT can be an approximate Hilbert coefficients. One global threshold derived by transform of the wavelet associated with the lower Donoho [5, 6]. DWT. When designed in this way, the dual-tree In summary, application of wavelet thresholding complex DWT is nearly shift-invariant, in contrast for image denoising involves the following steps: with the critically-sampled DWT. Moreover, the 1) Define a structure by appropriate importance dual-tree complex DWT can be used to implement ordering of all wavelet basis functions. The 2D wavelet transforms where each wavelet is original wavelet thresholding technique is oriented, which is especially useful for image equivalent to specifying a structure that uses only a processing. The dual-tree complex DWT magnitude ordering of the wavelet coefficients. outperforms the critically-sampled DWT for Obviously, this is not the best way of ordering the applications like image denoising and coefficients. A better tree structure is presented in enhancement. this paper. The analysis and synthesis filters are well 2) Estimate the prediction risk for each set of discussed by [9]. Kingsbury provide a perfect wavelet functions formed in the structure. reconstruction analysis and synthesis filterbanks for image denoising with critically-sampled for 3 1-D Dual-Tree Wavelet Transform discrete wavelet shrinkage. The perfect reconstruction property of the dual-tree complex It turns out that, for some applications of the DWT is verified in the following example. We discrete wavelet transform, improvements can be create a random input signal x of length 512, apply obtained by using an expansive wavelet transform the dual-tree complex DWT and its inverse. We in place of a critically-sampled one. There are then show that the reconstructed signal y is equal several kinds of expansive DWTs; here we to the original signal x by computing the maximum describe the dual-tree complex discrete wavelet value of |x-y|. transform. The dual-tree complex DWT of a signal x is implemented using two critically-sampled DWTs in parallel on the same data, as shown in the figure. h0 (n) ↓2 h0 (n) ↓2 h0 (n) ↓2 h1(n) ↓2 h1(n) ↓2 h1(n) ↓2 g0(n) ↓2 g0(n) ↓2 g0(n) ↓2 g1(n) ↓2 g1(n) ↓2 g1(n) ↓2 Figure 2: ψ (t ) for complex Dual Tree DWT Figure 1: Dual-Tree complex DWT structure 4 2-D Dual-Tree Wavelet Transform 4.2 Complex 2-D Dual-Tree Wavelet Transform In the wavelet domain, each shift of the image The complex 2-D dual-tree DWT also gives rise to corresponds to a different tree of wavelet wavelets in six distinct directions, however, in this coefficients. case there are two wavelets in each direction as One of the advantages of the dual-tree complex will be illustrated below. In each direction, one of wavelet transform is that it can be used to the two wavelets can be interpreted as the real part implement 2D wavelet transforms that are more of a complex-valued 2D wavelet, while the other selective with respect to orientation than is the wavelet can be interpreted as the imaginary part of separable 2D DWT. a complex-valued 2D wavelet. Because the There are two versions of the 2D dual-tree wavelet complex version has twice as many wavelets as the transform: the real 2-D dual-tree DWT is 2-times real version of the transform, the complex version expansive, while the complex 2-D dual-tree DWT is 4-times expansive. The complex 2-D dual-tree is is 4-times expansive. Both types have wavelets implemented as four critically-sampled separable oriented in six distinct directions. We describe the 2-D DWTs operating in parallel. However, real version first. different filter sets are used along the rows and columns. As in the real case, the sum and 4.1 Real 2-D Dual-Tree Wavelet Transform difference of sub band images is performed to obtain the oriented wavelets. The real 2-D dual-tree DWT of an image x is The twelve wavelets associated with the real 2D implemented using two critically-sampled dual-tree DWT are illustrated in the figure 4 as separable 2-D DWTs in parallel. Then for each gray scale images. pair of sub bands we take the sum and difference. The six wavelets associated with the real 2D dual- tree DWT are illustrated in the following figures as gray scale images. Note that each of the six wavelets is oriented in a distinct direction. Unlike the critically-sampled separable DWT, all of the wavelets are free of checker board artifact. Each sub band of the 2-D dual-tree transform corresponds to a specific orientation. To show directional wavelets for reduced 2-D DWT, all of the wavelet coefficients are set to Figure 4: 2D Dual-Tree complex wavelets zero, for the exception of one wavelet coefficient in each of the six sub bands. We then take the Note that the wavelets are oriented in the same six inverse wavelet transform. It is shown in figure 3. directions as those of the real 2-D dual-tree DWT. However, here we have two in each direction. If the six wavelets displayed on the first (second) row are interpreted as the real (imaginary) part of a set of six complex wavelets, then the magnitude of the six complexes are shown on the third row. As shown in the figure, the magnitudes of the complex wavelets do not have an oscillatory behavior- instead they are bell-shaped envelopes. In the reminder of this paper implementation of these structure are compared and local adaptive algorithm for threshold’s optimization are presented. Due to space limit, we only show results on 8-bit Barbara image in this paper. Figure 10 shows the Figure 3: Directional wavelets for real 2-D DWT comparable denoising results on 512 * 512 Barbara images corrupted by gaussian white noise σ 2 = 40 . 5 Image Denoising Table 1: Analysis and synthesis filters Analysis Analysis Synthesis Synthesis One technique for denoising is wavelet filter 1st filters filter 1st filters thresholding (or "shrinkage"). When we 0 -0.0112 0.0112 0 decompose data using the wavelet transform, we 0 0.0112 0.0112 0 use filters that act as averaging filters, and others -0.0884 0.0884 -0.0884 -0.0884 that produce details. Some of the resulting wavelet 0.0884 0.0884 0.0884 -0.0884 coefficients correspond to details in the data set 0.6959 -0.6959 0.6959 0.6959 (high frequency sub-bands). If the details are 0.6959 0.6959 0.6959 -0.6959 small, they might be omitted without substantially 0.0884 -0.0884 0.0884 0.0884 affecting the main features of the data set. The idea -0.0884 -0.0884 -0.0884 0.0884 of thresholding is to set all high frequency sub- 0.0112 0 0 0.0112 band coefficients that are less than a particular 0.0112 0 0 -0.0112 threshold to zero. These coefficients are used in an inverse wavelet transformation to reconstruct the data set [7]. This program has two parameters, one for noise signal and the other for threshold point. A sample noise signal is shown below, whose dimension is 512 x 512. We first take the forward DWT over 4 scales (J=4). Then a denoising method called soft thresholding is applied to wavelet coefficients through all scales and sub bands. Function sets coefficients with values less than the threshold(T) to 0, then substracts T from the non-zero coefficients. After soft thresholding, we take inverse wavelet transform. The following example shows how to convert an image to double data type, how to creat a noise signal and display the denoised image. Note that we use a threshold value of 35, which is the optimal threshold point for this case. We will introduce how to find the optimal threshold value Fig.5. noisy image in the later part of this section. From the resulting image, we can see the denoising capability of separable 2-D DWT. Now we want to improve the effect by using complex 2-D dual-tree DWT. The optimal threshold value for this method is 40. We can see that complex 2-D dual-tree method removes more noise signal than separable 2-D method does. Actually, 2-D dual-tree method outperforms separable method. This can be proved by figure 7. It illustrates the denoising capability for three different methods with complex 2-D dual- tree method be the best, followed by real 2-D dual- tree method and separable method. Also, it tells us where the optimal threshold points locate. For each method, applying the optimal threshold point yields the mininum RMS error. Therefore, a threshold producing the minimum RMS error is the optimal one. Fig. 6. Denoised image (by complex 2-D dual-tree DWT) 22 Standard 2D using a Bayesian estimation approach beginning 20 RMS error V.S. Threshold Pt. Real 2D dual Complex 2D dual with the new bivariate non-Gaussian model. The plot is illustrated in the figure below. 18 16 6.1 Local Adaptive Image Denoising RMS error 14 Using the bivariate shrinkage function above, 12 developed an effective and low complexity locally adaptive image denoising algorithm in [7]. This 10 shrinkage function requires the prior knowledge of 8 the noise variance and the signal variance for each wavelet coefficient. Therefore the algorithm first 6 0 5 10 15 20 25 30 35 40 45 50 estimates these parameters (see [7] for the recipe of Threshold pt. the estimation rules). Briefly, the algorithm is summarized as follows: Figure 7: rms error V.S. threshold points 1. Calculate the noise variance. 2. For each wavelet coefficient. 6 Bivariate Shrinkage Functions for 3. Calculate signal variance Wavelet Based Denoising 4. Estimate each coefficient We have proposed a new simple non-Gaussian bivariate probability distribution function to model the statistics of wavelet coefficients of natural 3 images. The model captures the dependence 2 between a wavelet coefficient and its parent. Using Bayesian estimation theory we derive from this 1 model a simple non-linear shrinkage function for 0 wavelet denoising, which generalizes the soft -1 thresholding approach of Donoho and Johnstone. -2 The new shrinkage function, which depends on both the coefficient and its parent, yields improved -3 4 results for wavelet-based image denoising. 2 4 Let w2 represent the parent of w1 ( w2 is the 0 2 0 wavelet coefficient at the same spatial position -2 -2 -4 -4 as w1 , but at the next coarser scale). Then y = w + n where w = ( w1 , w2 ) , y = ( y1 , y 2 ) Figure 8: A bivariate shrinkage function. and n = (n1 , n 2 ) . The noise values n1 , n 2 are iid 7 Empirical Results zero-mean Gaussian with variance σ n . Based on 2 the empirical histograms we have computed, we The following sections describe the propose the following non-Gaussian bivariate pdf implementation of this algorithm using both the separable DWT and the complex dual-tree DWT. 3 3 Two different approaches are compare in figure p w ( w) = . exp(− w12 + w2 ) 2 (1) 2πσ 2 σ 10-12. Figure 10 is a noisy image of Barbara. With this pdf, w1 and w2 are uncorrelated, but not Separable DWT applied to figure 10 to obtain results that is shown in figure 11. Result of Dual Tree DWT independent. The MAP estimator of w1 yields the was drown in figure 12. One can compare two following bivariate shrinkage function approaches and find quality of different structures. 3σ n 2 Comparison of signal to noise ratio and Quality are ( y12 + y 2 − 2 ) given in Table 2. w1 = ˆ σ . y1 (2) y12 + y 2 2 For this bivariate shrinkage function, the smaller the parent value, the greater the shrinkage. This is consistent with other models, but here it is derived Table 2: Comparison of S/N and Quality Noise Separable Quality DTDWT Quality level S/N (dB) S/N (dB) 10/255 32.1778 very 33.2962 Very good good 40/255 24.8396 good 26.2060 Very good 50/255 23.9009 good 25.2147 Good 70/255 22.7343 not bad 23.8492 Good 80/255 22.2854 not bad 23.3051 not bad 90/255 21.9698 bad 22.9180 not bad 140/255 20.7283 bad 21.4548 bad 190/255 19.8390 very bad 20.5095 bad 8 Conclusions Two dimensions Dual Tree wavelet Transform are proposed and local adaptive theresholding Figure 10: First: Noisy image algorithm is presented. Empirical results show that new approach is better than older algorithms. References [1] S. G. Chang and M. Vetterli, "Spatial Adaptive Wavelet Thresholding for Image Denoising", Proc of IEEE Int. Conf. on Image Processing, 1997 [2] 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 Trans. Image Processing, vol. 7, pp. 319-335, 1998 [3] V. Cherkassky and F. Mulier, Learning from Data: Concepts, Theory and Methods, Wiley Interscience, 1998 [4] D. L. Donoho, "Wavelet Thresholding and Figure 11: Denoised image by Separable DWT W.V.D.: A 10- minute Tour", Int. Conf. on Wavelets and Applications, Toulouse, France, June 1992 [5] D. L. Donoho and I. M. Johnstone, "Ideal spatial adaptation via wavelet thresholding", Biometrika, vol. 81, pp. 425-455, 1994 [6] D. L. Donoho, "De-Noising by Soft- Thresholding", IEEE Trans. Information Theory, vol. 41 May 1995 [7] L. Sendur, I.W. Selesnick, "Bivariate shrinkage with local variance estimation", IEEE Signal Processing Letters, 9(12), 438-441, Dec 2002. [8] N. G. Kingsbury. "Complex wavelets for shift invariant analysis and filtering of signals", Applied and Computational Harmonic Analysis, 10(3):234-253, May 2002. [9] N. G. Kingsbury, "Image processing with complex wavelets”, Phil. Trans. Royal Society London A, September 1999. Figure 12: Denoised image by Dual Tree