A Dyadic Wavelet Filtering Method for 2-D Image Denoising

We improve spatially selective noise filtration technique proposed by Xu et al. and wavelet transform scale filtering approach developed by Zheng et al. A novel dyadic wavelet transform filtering method for image denoising is proposed. This denoising approach can reduce noise to a high degree while preserving most of the edge features of images. Different types of images are employed to test in the numerical experiments. The experimental results show that our filtering method can reduce more noise contents while maintaining more edges than hard-threshold, soft-threshold filters, Xu’s method and Zheng’s method.


Introduction
Wavelet transform is a multi-resolution representation of a signal or image.It is a powerful tool in several areas of applications like signal processing, image processing, pattern recognition, data compression, commutation, etc. Singularities and irregular structures often carry essential information in signals and images.For example, the discontinuities of the intensity of an image indicate the locations of edges.
The local regularity is characterized by the decay of the wavelet transform amplitude across scales.Signal singularities and image edges can be detected by the dyadic wavelet transform modulus maxima across scales [1,2].In mathematics, singularities are generally measured with Lipschitz exponents.The wavelet theory proves that these Lipschitz exponents can be calculated from the propagating amplitude values of the different modulus maxima across scales.
The original signal or image has singularities whose Lipschitz exponents are greater than or equal to zero, and the noise has singularities whose Lipschitz exponents are less than zero.Thus, the amplitudes of signal or image modulus maxima increase when the scale increases, while the amplitude of noise modulus maxima decrease strongly when the scale increases.By using these properties, the noises can be eliminated from the noised signals or images.The approaches for separating signal and noise in wavelet scale space are proposed by many researchers.For example, the original signal can be extracted from the noisy version by estimating the signal modulus maximum at small scales [1,2].Adaptive Wiener filtering were used to remove noise in signals and images [3][4][5].The selective noise filtration technique and adaptive thresholding function in image denoising were developed [6][7][8].The scale space filtering algorithms applied to image denoising were also proposed [9,10].In addition, many other novel approaches for image denoising have been presented by some researchers [11][12][13] recently.In this work, we develop an image denoising approach by improving spatially selective noise filtration technique proposed by Xu et al. [6] and wavelet transform scale filtering approach given by Zheng et al. [9].Hard-threshold and soft-threshold filters that were proposed by D. L. Donoho [14,15] are widely used in image denoising processing.We will compare our filtering approach with hard-threshold filtering, soft-threshold filtering, Xu's method and Zheng's method in the numerical experime-nts.Peak-Signal-Noise-Rate (PSNR) and Root-Mean-Square-Error (RMSE) are employed to estimate the quality of restored images.

2-D Dyadic Wavelet Transform
The wavelet transform of , , , , The set of functions , , , ,  x x  are reconstructed wavelet functions.If their Fourier transforms satisfy Then  1 2 ,  f x x can be reconstructed from their dyadic wavelet transform i.e.
, , , Because of the limitation of image's resolution, we introduce a smoothing function   We define the smoothing operator 2 j S by The wavelet transform between the scales 1 and provides the details that are in but that have lost in .
Mallat [1] has given the fast algorithm for the discrete dyadic wavelet transform.The fast dyadic wavelet transform can also be calculated with a filter bank algorithm called the algorithm trous proposed by Holschneider, Kronland-Martinet, Morlet and Tchamitchian [16].In this paper, we use trous algorithm to reconstruct the image.a a

Dyadic Wavelet Transform Filtering Algorithm
In recent years, some denoising techniques based on the wavelet transform have been studied by many authors [2,6,12,17].The edge modulus maxima can be distinguished from noise modulus maxima by analyzing the singularity properties of wavelet transform domain maxima of a signal or image across scales [2].Y. Xu [6] developed wavelet transform domain filters based on the direct spatial correlation of the wavelet transform at several adjacent scales.Y. Zheng [9] proposed a wavelet transform scale filtering algorithm by using the properties of signal and noise modulus maxima across large scales.Our approach relies on the variations of the dyadic wavelet transform data across all scales to remove noises rather than extracting edges directly.For a 2-D image, the discrete sampling of is given by The discrete coarse smoothed image is denoted by , , In the scale space, the modulus maxima of 2 across scales produced by image edges have positive correlation.When the scale increases, the amplitudes of modulus maxima coeffcients will increase or retain constant.On the contrary, the modulus maxima produced by noises have negative correlation and the amplitudes of their coeffcients decrease as increases.
where J represents the maximum scale of the decomposition.
, , The 2-scale direct correlation sharpens and enhances major edges while suppressing noise and small features.So comparing the values of and can separate important edges from noise in images.Before comparison, needs to be rescaled to Xu's rescaling scheme is where Zheng et al. use the modulus maxima rescaling method at large scales, and apply the above mentioned rescaling method at small scales.Let S be the upper limit of small scales, assume and The modulus maxima rescaling formula as follows: Wc m n directly, then too much noise will be extracted as edges.To avoid this drawback, we apply the modulus maxima rescaling at all scales and renew the formula (13) as where  is a weight parameter with respect to the scale m.
After rescaling to for all m and n, the important edges can be identified in by comparing the absolute values of , we identify which represents the most important features of the image edges.We set the values of and to 0's at the positions identified and thus obtain a new set of and       , n , we can rec .Let struct the filtered image from the set n W j n S n where S n S f n n  tering algorithm is summarized as follows.

Image Denoising
Step 1. Compute , f x x and its the lower-frequency smoothed image: for iteration { Loop for the scale m Step 3. Loop   for the pixel Step 4. Compute the wa um scale J : Step 5. Reconstruct the image from ed wavelet da filter ta The reco the set nstruction from through the inverse dyadic wavelet transform will yi inverse dyadic wavelet transform that we implemented in our technique uses a trous algorithm and the quadratic spline scaling functions and wavelets given in [18].Now we give some comments on the choice of the number of iterations and weight parameter  

at th
We use Peak-Signal-Noise-Ratio Square-Error (RMSE) to evalua (PSNR) and Root-Meante restored results.PSNR and RMSE is defined by the following: , , where and denotes the pixel valu processed and the original images respectively.Hardldin sof entioned Lena imag presses more noise while preserves m sian noise with the standard de , i j w , i j u es of the thresho g and t-thresholding are widely used for denoising in image processing by many researchers.Therefore, in the following tests the hard-thresholding method, soft-thresholding method, Xu's method and Zheng's method will be used to compare with the dyadic wavelet transform filtering algorithm.
Example 1: When we use our filtering method to do denoising experiment for the above m e corrupted by additive noise, the restored result is Figure 2(a).If apply softthreshold, hard-threshold, Xu's method and Zheng's method to filter the noised image, the result is in Figures 2(b)-(e).
Table 1 presents the values of PSNR and RMSE for each of the schemes.
From all the five restored images, it is clear that our proposed method sup ore fine details and small structures in the image.In addition, from the values of PSNR and RMSE for restored image, our method increases the PSNR by 1 -3 dB and reduce the RMSE 5 -6.
Example 2: A texture image is used in the second test.It is added by the white Gaus viation 30   .Use our filtering method, soft-thresholding method, hard-thresholding method, Xu's method and Z method to process noised image, we can find that our method is also better than other four methods.The results above reveal that our method not only maintain more texture details and smooth transitions in the face but also suppress more noise than other methods after processing.Additionally, our method can increase more PSNR and decrease RESE than other methods.
At last, we give some other types of images.And we only present results recovered by our m thod.A MR w aintain all important information and filter out m wavelet transform filtering method method e image (see Figure 7 A fingerprint image is used in the last test.Figures 9(a)-(c) are the original image, the noised image with σ = 15, and the recovered result with our scheme.From the visual quality, it is obvious that restored image is as good as the original one.

Conclusions
We have introduced the dyadic  technique for denoising in image processing.Our filtering algorithm is superior to soft-thresholding method, hardthresholding method, Xu's method and Zheng's method because important edge features in the wavelet transform domain are preserved while much noise is suppressed.The other filtering methods perform very poorly in image denoising because they tends to remove the high-frequency component exclusively, which yields smooth images and blurs the image edge features.
13) At small scales, noise in the noised image is dominating except some sharp image edges.According to Xu and Zheng's ideas, if compare the value of at the point .We use a new matrix name  , n to represent the retained value,  , .n  Making comparison at th  1 ,

Figure 1 .
Figure 1.The effect of the new wavelet filtering at the scale m = 3 and  3 27.


according to the user's request.For th umber of iterations, when it is too small, we can not obtain a oth estimate.If the number of iterations is too large, most of the edge information of reconstructed image will e n smo eld the final filtered image.The be eliminated.Thus we should choose a tradeoff between the number of iterations and the estimation of filtered image.It is well known that the Lipschitz exponents of image and noise are different.At the finer scales such as 1 2 and 2 2 , the modulus maxima mainly produced by noise, while at coarser scale, most modulus maxima duced image.So if we set the different value of   k m pro by  at the different scale m, noise will be eliminated more effectively.In general, let parameter  

Figure 3 (
a) is the original image, Figure 3(b) is the noised image.Figures 4(a)-(e) show the results heng's processed by the five methods.
(a)) has been corrupted with white Gaussian noise (σ = 20) and become a noised image, see Figure7(b).After the noised image has been processedith our method, we can see that our restoration scheme is able to m uch noise, see Figure7(c).Figures8(a), (b) and (c) are an original building image, the noised image, and the processed result by our method.We can see that the recovered image can preserve more image edge details.

Table 2
is the comparison of the values of PSNR and RMSE for restored images.

Table 3
is the quantitative comparison among the five methods.