外文翻译-一种关于小波变换的图像去噪的改进方法.doc

附件附件1 外文原文2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering (CMCE)An Improved Method of Image Denoising Based on Wavelet TransformXiwen Qin, Yang Yue, Xiaogang DongCollege of Basic SciencesChangchun University of Technology Changchun, P.R.China e-mail:Xinmin Wang , ZhanSheng Tao Institute of applied mathematics Changchun University of Technology Changchun, P.R.Chinae-mail:AbstractA new thresholding function is proposed for imagedenoising in the wavelet domain. The new threshold functionhas the same characteristic of continuity with the soft-thresholding function, and also overcomes the defects ofconstant bias between the estimate of the wavelet coefficientsand wavelet decomposition in the soft-thresholding function,the thresholding function effects are considered simultaneously. The simulation results show that the proposed thresholding function has superior features compared to conventional methods when used with an improved thresholding function. It also shows that using the new threshold function for image denoisinginherits the advantages of the hard and soft threshold function. This makes it an efficient method in image denoising applications,it can also remove the noise and retain the image details better.Keywords: image denoising; wavelet transform; threshold functionI.INTRODUCTIONImages may be affected by noise in capturing and transmission stages. Noise sources cover a wide range of unwanted distortions from additive Gaussian noise in natural images. Image denoising is a necessary step in image processing applications1.The wavelet transform has primary properties such as compression or sparsity which means that wavelet transforms of real-world signals tend to be sparse2. Therefore,they have a few large coefficients that contain the main energy of the signal and other small coefficients which can be ignored 3. Moreover, the energy of the noise is spread among all the coefficients in the wavelet domain. Due to the fact that the wavelet transform of a noisy signal is a linear combination of the wavelet transform of the noise and the original signal, the noise power can be suppressed significantly with a suitable threshold while the main signal features can be preserved. In the noise reduction method in the wavelet domain that is called wavelet shrinkage, the wavelet coefficients of a noisy image are divided into important and non-important coefficients and each of these groups are modified by certain rules.The functionality of the shrinkage process is due to the threshold value and the thresholding rule. Hard and soft thresholding functions that are the basic ones introduced by Donoho and Johnstone 4,5 together with garrote, and semisoft thresholding functions that are more powerful,are used in noise suppression applications 6,7. In these methods, the non-important coefficients are set to zero. In hard thresholding,the important coefficients remain unchanged. In softthresholding, the important coefficients are reduced by theabsolute threshold value.So the image denoising pre-processing is very essential. Because the wavelet transform has characteristics of low entropy, multi-resolution nature,decorrelation and flexibility in base selection etc., it plays an important role in the field. This method has been widely used as soon as it was put forward. However, this method has a shortage. There is pseudo-Gibbs phenomena in thediscontinuous neighborhood. Thus, Coifman and Donoho proposed shift-invariant denoising method8, through the cycle displacement of the signal to remove man-made artifacts. The experiment results show that translation invariance wavelet denoisingwas significantly better thanthe non-translation invariance wavelet de-noising. Bui and Chen had improved the method from shift-invariant to multi-wavelet 9. They found that through the translation invariant multi-wavelet denoising we could get better results than the translation invariant single wavelet denoising. Later, Cai and Silverman proposed a neighborhood threshold program based on coefficient neighborhood. Comparing with the traditional one by one wavelet denoising method, their experimental results show a clear advantage10. Chen and Bui had extended the wavelet neighborhood threshold to multiwavelets neighborhood threshold11. Through the denoising experiment of the standard test image, they found that neighborhood wavelet denoising is better than a single wavelet denoising. Taking into account the same sub-band wavelet coefficients based on the correlation, Sendui and Selesnick went further propose a dual-threshold program based on the correlation of parent coefficient .II.WAVELET THRESHOLD DENOISING PRINCIPLEThe soft and hard threshold function method proposed by Donoho has been widely used in practice. Using that wavelet transform, especially orthogonal wavelet transform has a strong nature of data decorrelation. In the wavelet domain, it can make the signal energy concentrate in a few large wavelet coefficients, while the noise energy is distributed throughout the wavelet domain. Therefore,by wavelet decomposition, the signal amplitude of the wavelet coefficients of magnitude greater than the noise factor, we can also say that the relatively large amplitude of the wavelet coefficients is mainly signal, while the relatively small amplitude coefficient is largely noise.Thus, using the threshold approach can keep the signal coefficient, reducing most of the noise figure coefficient to zero. If its threshold is bigger than the specified threshold,it can be seen that that this factor contains a signal component and is the result of both signal and noise, which shall be maintained, if its threshold is less than the specified threshold, it can be shown that this factor does not contain the signal component, but only the result of noise which should be filtered out. This method has its shortcomings. In the hard threshold method, the wavelet coefficient processed by the threshold value have discontinuous point on the threshold and , which may cause Gibbs shock to the useful reconstructed signal.In the soft-thresholding method, its continuity is good, but when the wavelet coefficients are greater than the threshold value, there will be a constant bias between the wavelet coefficients that have been processed and the original wavelet coefficients, making it impossible to maintain the original features of the images effectively.A. Select the appropriate waveletSelect the appropriate wavelet function for image wavelet decomposition to obtain the wavelet coefficients. The Fourier transform of is。When the function satisfy the conditions ，the function is called a basic wavelet. By scaling and translation the basic function ，you can get a wavelet sequence: （1）Where a,bR，a0，and a is defined as the stretch factor,，b is defined as the shift factor. The image signal is a two-dimensional function, so its continuous wavelet transform as follows: （2）and reconstruction formula which corresponding with the formula (2) is: （3）The decomposition and reconstruction of twodimensional discrete wavelet transform uses the Mallat fast algorithm suppose I is located two-dimensional image,the wavelet decomposition algorithm as follows: （4）Where, I1 is the low-frequency operator map obtainedc after the first level wavelet decomposition. is respectively horizontal, vertical, diagonal direction in the high-frequency operator map (fig.1.). Low-frequency subgraph reflects the general characteristics of the image,high-frequency sub-graph reflects the characteristics of image details. L and G is a filtering operator which determined by the chosing wavelet function. and is respectively stands for the functions on the rows andcolumns of two-dimensional image. Similarly, and respectively stands for the function G on the rows and columns. Having a continuous higher lever decomposition on the low frequency sub-images , we can get the various frequency domain sub-graph under the different scales.B. Threshold processThen threshold processing the coefficients and removing the noise.C. ReconstructionReconstructing the images according to the processed wavelet coefficients. Wavelet reconstruction algorithm as follows:Where is respectively the conjugate transpose operator of L,G .III. THRESHOLDING FUNCTION The selection of thresholding function is the key issue of wavelet threshold de-noising. In addition, you must also select the appropriate wavelet basis function to decompose, and to determine the number of decomposition levels. Only when these factors complement each ther ,we can then achieve the desired effect of denoising.A. Traditional threshold functionOne is the hard threshold function: (5)One is the soft-thresholding function: (6)Where, stands for the wavelet coefficients , stands for the threshold value, stands for the wavelet coefficients after treatment.B. New threshold function By a simple soft thresholding function is constructed wavelet coefficient is smaller than its absolute value.Therefore, in order to improve the denoising results, we are trying to reduce the fixed-bias; but they can not be reduced to zero (At this point is the hard threshold function). Therefore, we have made the estimated wavelet coefficients betweenand.Based on the above analysis, we use the weighted method to construct the threshold function, its expression is as follows: （7）Where m is an positive constant. The improved thresholding function has the same characteristic of continuity with soft-threshold function, and overcomes the soft threshold function with a constant deviation between the shortcomings. When m, the formula (7) stands for the soft threshold function; when m 0 , formula (7) stands for the hard-thresholding function, you can get practical and effective threshold function by changing the values of m . If using a constant to process all the wavelet coefficients , it will result in too much kill wavelet coefficients, which will seriously affect the quality of reconstructed image. Therefore, the changeable threshold is used in algorithm in this paper. （8）where，N stands for the image size, stands for the noise variance,J stands for the current decomposition level, decreases with the increase of J . Standard deviation cannot be the first to obtain in the actual applications, so it must be estimated from the noisy images. The classical estimate is，wherestands for high-frequency wavelet coefficients of the median figure. With the increase of in scale decomposition J decreases, which is corresponding with the theory that the noise energy in the wavelet domain decreases rapidly with the increase of the decomposition scale. In summary, the adaptive threshold function in this paper is as follows: （9）IV. EXPERIMENTAL RESULTSIn order to illustrate the effectiveness of the new threshold function in the image threshold denoising algorithm, This article is mainly based on MATLABR2009a implementation of image denoising used several thresholding function. In the experiment, a typical test image Lena was adopted. The image size is 512512,grayscale image is 8 bit, adding mean is zero, and the variance is 0.02 Gaussian white noise. Suppose the original image isreconstruction denoising images is.The denoising performance indicators were described by peak signal to noise ratio and mean square error to describe. The following is the definition expression given to them(Eq. (10),(11). （10） （11） Figure 1. Comparison of different thresholding functions:(a). Noisy image,(b)Proposed thresholding function.(c)Hard thresholding function (d)Soft thresholding function. By using the DB1 orthogonal wavelet function to decompose wavelet, it can make the wavelet coefficients decomposition of all levels and between levels achieve a very small degree of correlation, and too much wavelet decomposition levels will increase the amount of unnecessary calculation, and will also affect the reconstruction accuracy, so it usually uses the three layers of wavelet decomposition. The simulation results shown in Figure1,the decomposition levers of the threshold shown in Table 1. Table 1 Comparison of three denoising methodsindexSoft threshold Hard thresholdProposed thresholdPSNR30.9930.8732.47MSE51.7353.1936.85 From the simulation results and experimental data we can find that comparing with the soft and hard threshold function method, the improved version of the threshold function method proposed in this paper not only removes a certain degree of Gaussian noise effectively, but also better preserves the detail information of the images and improves the peak signal to noise ratio of the images.V. CONCLUSION In this paper, some efficient wavelet-based image denoising methods are proposed and utilized in image denoising applications. A remarkable point is that Gaussian noise reduction subjects are considered together in this paper. The proposed methods are efficient in suppressing both cases. Another important consideration is that the proposed methods are utilized in universalthreshold and subband-adaptive cases and compared with similar methods. Implementing the method in spatial adaptive case can be the subject of future work.Furthermore, these methods only consider the primary properties of wavelet transform. Utilizing secondary wavelet properties such as parent-child relation in network learning can be another topic for future work and may improve the efficiency of the method.ACKNOWLEDGMENTThis research was supported by College of Basic Sciences, Changchun University of Technology, We thank Changchun University of Technology that funded this work. The authors are grateful for the helpful suggestions and comments provided by the anonymous reviewers of this work.REFERENCES1. Stephane Mallat. A wavelet tour of signal processing.Elsevier,2009.2. R.C. Gonzalez, R.E. Woods, Digital Image Processing,second ed., Prentice-Hall, Inc., 2002.3. M.S. Crouse, R.D. Nowak, R.G. Baraniuk, Wavelet-based signal processing using hidden Markov models, IEEE Trans. Signal Process. 1998,vol.46 ,pp. 886-902.4. D.L. Donoho, I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika,1994, vol.81 (3) ,pp. 425-455.5. D.L. Donoho, I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, J. Am. Stat.Assoc.1995,vol.90 (432) ,pp. 1200-1224.6. H. Gao, Wavelet shrinkage denoising using the nonnegative garrote, J. Comput. Graph. Stat. 1998,vol.7,pp. 469-488.7. D.L. Donoho. Denoising by Soft thresholding, IEEE Trans.Inform Theory, 1995,vol.41(3),pp. 613-627.8. R.R.Coifman,D.L.Donoho,Translation invariant denoising,in:Wavelets and Statistics, Springer Lecture Notesin Statistics,vol.103, Springer, New York, 1995,pp.125-150.9. T.D.Bui,G.Y.Chen, Translation invariant denoising using multiwavelets, IEEE.Trans.Signal Process. 1998,vol.46(12),pp.3414-3420.10. T. T. Cai and B. W. Silverman, Incorporating information on neighboring coefficients into wavelet estimation,Sankhya A 63 ,2001, 127148.11. G. Y. Chen and T. D. Bui, Multiwavelet denoising using neighboring coefficients,IEEE Signal Process. Lett.10 ,2003,211214.附件2 外文翻译 一种关于小波变换的图像去噪的改进方法摘要 在小波领域一个新的用于图像去噪阈值函数被提了出来。这个新的阈值函数和软阈值函数有同样的连续特性，同时也克服了在软阈值函数中的小波系数和小波分解的预估之间的恒定偏差的缺陷，我们认为阈值函数的影响是同时的。 模拟结果显示，这个改进后的阈值函数有优于传统方法的突出特点。同时也表明用这种新的阈值函数来进行图像去噪继承了硬阈值函数和软阈值函数的优点。这使得它变成了图像去噪应用的一个有效方法，也使得去噪和保存图像细节变得更好。关键词：图像去噪，小波变换，阈值函数.介绍 图像在获取和传输过程中可能会被噪声影响。噪声来源涵盖了来自自然图像中的附加高斯噪声的失真这样一个广阔的范围，这个失真是不被需要的。在图像处理应用中图像去噪是一个必须的步骤。小波变换有这样的基本特性，例如：压缩性或稀疏性，这意味真正信号的小波变换是稀疏的。因此，小波变换有一个很大的系数，这个系数包含了信号的主要能量，所以其他的小系数可以被忽略。此外，在小波领域中噪声的能量存在于所有的系数中。噪声信号的小波变换是噪声和原始信号的线性组合，由于这个事实，噪声能量可以用一个合适的阈值有效地抑制同时主要的信号特征被保留。在小波领域噪声减少方法被称为小波收缩，一个有噪图像的小波系数被分为重要的和不重要的系数，同时这些组的每一个都被特定的规则所更改。收缩过程的函数性是由于阈值的值和阈值规则。硬和软阈值函数是被多诺霍、约翰斯通和加罗特介绍的最基本的，半软阈值函数更有效，被用于噪声压缩应用中。在这些方法中，不重要的系数被设为零。在硬阈值中，重要的系数保持不变。在软阈值中，重要的系数被绝对阈值减小。因此，图像预处理是必不可少的。因为小波变换具有低熵、多分辨率特性、在基本选择方面有去相关和适应性等等，所以它在图像去噪方面有重要作用。这个方法一被提出就被广泛运用。然而，这个方法有一个缺点。有一个伪吉布斯现象在不连续的附近。因此，夸夫曼和多诺霍提出移不变去噪方法，通过信号的循环替换来去除人为噪声。实验结果表明平移不变性小波去噪比非平移不变性小波去噪要显著好得多。布伊和陈把移不变方法改进成了多小波法。他们发现通过平移不变多小波去噪，我们能够得到比平衡不变单小波去噪效果好的结果。后来，蔡和西尔弗曼提出了依据邻域系数的邻阈计划。与单小波去噪的传统去噪方法相比，他们的实验结果展示出了一个显著的优势。陈和布伊将单小波邻域阈值扩展noise figure coefficient为了多小波领域阈值。通过标准测试图像的去噪实验，他们发现邻域小波去噪比单个小波去噪要好。考虑到根据据相关性的子带小波系数是相同的，森迪尼和斯里尼克进一步提出了一个根据父系数相关性的双阈值计划。.小波阈值去噪规则被多诺霍提出的软阈值函数和硬阈值函数已经被广泛应用到现实中。用小波变换，尤其是正交小波变换有很强的数据去相关特性。在小波领域，小波变换可以使信号的能量集中在非常大的小波系数里，同时使噪声能量分布贯穿整个小波领域。然而，通过小波分解，在小波系数中信号幅值比噪声大，也可以说小波系数的振幅相对较大的主要是信号，同时振幅系数相对较小的主要是噪声。因此，用阈值方法可以保持信号系数，使大部分的噪声条数减少到零。如果它的阈值比规定的的阈值大，可以认为这个因素包含信号成分，这个因素是信号和噪声的结果，将会被保持；如果阈值比规定的阈值小的话，这个因素则不包含信号成分，但是仅仅噪声会被滤除。但是这种方法有它的缺点。在硬阈值方法中，被阈值的值处理的小波系数在阈值和-，这可能使吉布斯对有用的重建信号来说成为震惊。在软阈值方法中，它的连续性是好的，但是当小波系数比阈值的值大的话，在被处理的小波系数和原始小波系数之间将会有一个恒定的偏差，这使得更有效地保持图像的原始特征变得可能。 A 选择合适的小波 选择合适的小波函数用于图像小波分解来获得小波系数。的傅里叶变换是。当函数满足条件，函数被叫做基本小波。通过缩放和变换基本函数，你可以得到一个小波公式： （1）条件是a,bR，a0，而且a被定义为伸展因子，b被定义为平衡因子。图像信号是一个