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1、Complex Ridgelets for Image Denoising1 IntroductionWavelet transforms have been successfully used in many scientific fields such as image compression, image denoising, signal processing, computer graphics,and pattern recognition, to name only a few.Donoho and his coworkers pioneered a wavelet denois

2、ing scheme by using soft thresholding and hard thresholding. This approach appears to be a good choice for a number of applications. This is because a wavelet transform can compact the energy of the image to only a small number of large coefficients and the majority of the wavelet coeficients are ve

3、ry small so that they can be set to zero. The thresholding of the wavelet coeficients can be done at only the detail wavelet decomposition subbands. We keep a few low frequency wavelet subbands untouched so that they are not thresholded. It is well known that Donohos method offers the advantages of

4、smoothness and adaptation. However, as Coifmanand Donoho pointed out, this algorithm exhibits visual artifacts: Gibbs phenomena in the neighbourhood of discontinuities. Therefore, they propose in a translation invariant (TI) denoising scheme to suppress such artifacts by averaging over the denoised

5、signals of all circular shifts. The experimental results in confirm that single TI wavelet denoising performs better than the non-TI case. Bui and Chen extended this TI scheme to the multiwavelet case and they found that TI multiwavelet denoising gave better results than TI single wavelet denoising.

6、 Cai and Silverman proposed a thresholding scheme by taking the neighbour coeficients into account Their experimental results showed apparent advantages over the traditional term-by-term wavelet denoising.Chen and Bui extended this neighbouring wavelet thresholding idea to the multiwavelet case. The

7、y claimed that neighbour multiwavelet denoising outperforms neighbour single wavelet denoising for some standard test signals and real-life images.Chen et al. proposed an image denoising scheme by considering a square neighbourhood in the wavelet domain. Chen et al. also tried to customize the wavel

8、et _lter and the threshold for image denoising. Experimental results show that these two methods produce better denoising results. The ridgelet transform was developed over several years to break the limitations of the wavelet transform. The 2D wavelet transform of images produces large wavelet coef

9、icients at every scale of the decomposition.With so many large coe_cients, the denoising of noisy images faces a lot of diffculties. We know that the ridgelet transform has been successfully used to analyze digital images. Unlike wavelet transforms, the ridgelet transform processes data by first com

10、puting integrals over different orientations and locations. A ridgelet is constantalong the lines x1cos_ + x2sin_ = constant. In the direction orthogonal to these ridges it is a wavelet.Ridgelets have been successfully applied in image denoising recently. In this paper, we combine the dual-tree comp

11、lex wavelet in the ridgelet transform and apply it to image denoising. The approximate shift invariance property of the dual-tree complex wavelet and the good property of the ridgelet make our method a very good method for image denoising.Experimental results show that by using dual-tree complex rid

12、gelets, our algorithms obtain higher Peak Signal to Noise Ratio (PSNR) for all the denoised images with di_erent noise levels.The organization of this paper is as follows. In Section 2, we explain how to incorporate the dual-treecomplex wavelets into the ridgelet transform for image denoising. Exper

13、imental results are conducted in Section 3. Finally we give the conclusion and future work to be done in section 4.2 Image Denoising by using ComplexRidgelets Discrete ridgelet transform provides near-ideal sparsity of representation of both smooth objects and of objects with edges. It is a near-opt

14、imal method of denoising for Gaussian noise. The ridgelet transform can compress the energy of the image into a smaller number of ridgelet coe_cients. On the other hand, the wavelet transform produces many large wavelet coe_cients on the edges on every scale of the 2D wavelet decomposition. This mea

15、ns that many wavelet coe_cients are needed in order to reconstruct the edges in the image. We know that approximate Radon transforms for digital data can be based on discrete fast Fouriertransform. The ordinary ridgelet transform can be achieved as follows:1. Compute the 2D FFT of the image.2. Subst

16、itute the sampled values of the Fourier transform obtained on the square lattice with sampled values on a polar lattice.3. Compute the 1D inverse FFT on each angular line.4. Perform the 1D scalar wavelet transform on the resulting angular lines in order to obtain the ridgelet coe_cients.It is well k

17、nown that the ordinary discrete wavelet transform is not shift invariant because of the decimation operation during the transform. A small shift in the input signal can cause very di_erent output wavelet coe_cients. In order to overcome this problem, Kingsbury introduced a new kind of wavelet transf

18、orm, called the dual-tree complex wavelet transform, that exhibits approximate shift invariant property and improved angular resolution. Since the scalar wavelet is not shift invariant, it is better to apply the dual-tree complex wavelet in the ridgelet transform so that we can have what we call com

19、plex ridgelets. This can be done by replacing the 1D scalar wavelet with the 1D dualtree complex wavelet transform in the last step of the ridgelet transform. In this way, we can combine the good property of the ridgelet transform with the approximate shift invariant property of the dual-tree comple

20、x wavelets.The complex ridgelet transform can be applied to the entire image or we can partition the image into a number of overlapping squares and we apply the ridgelet transform to each square. We decompose the original n _ n image into smoothly overlapping blocks of sidelength R pixels so that th

21、e overlap between two vertically adjacent blocks is a rectangular array of size R=2 _ R and the overlap between two horizontally adjacent blocks is a rectangular array of size R _ R=2 . For an n _ n image, we count 2n=R such blocks in each direction. This partitioning introduces a redundancy of 4 ti

22、mes. In order to get the denoised complex ridgelet coe_cient, we use the average of the four denoised complex ridgelet coe_cients in the current pixel location.The thresholding for the complex ridgelet transform is similar to the curvelet thresholding 10. One difference is that we take the magnitude

23、 of the complex ridgelet coe_cients when we do the thresholding. Let y_ be the noisy ridgelet coe_cients. We use the following hard thresholding rule for estimating the unknown ridgelet coe_cients. When jy_j k_, we let y_ = y_. Otherwise, y_ = 0. Here, It is approximated by using Monte-Carlo simulat

24、ions. The constant k used is dependent on the noise . When the noise is less than 30, we use k = 5 for the first decomposition scale and k = 4 for other decomposition scales. When the noise _ is greater than 30, we use k = 6 for the _rst decomposition scale and k = 5 for other decomposition scales.T

25、he complex ridgelet image denoising algorithm can be described as follows:1. Partition the image into R*R blocks with two vertically adjacent blocks overlapping R=2*R pixels and two horizontally adjacent blocks overlapping R _ R=2 pixels2. For each block, Apply the proposed complex ridgelets, thresh

26、old the complex ridgelet coefficients, and perform inverse complex ridgelet transform.3. Take the average of the denoising image pixel values at the same location.We call this algorithm ComRidgeletShrink,while the algorithm using the ordinary ridgelets RidgeletShrink. The computational complexity of

27、 ComRidgeletShrink is similar to that of RidgeletShrink by using the scalar wavelets. The only di_erence is that we replaced the 1D wavelet transform with the 1D dual-tree complex wavelet transform. The amount of computation for the 1D dual-tree complex wavelet is twice that of the 1D scalar wavelet

28、 transform. However, other steps of the algorithm keep the same amount of computation. Our experimental results show that ComRidgeletShrink outperforms V isuShrink, RidgeletShink, and wiener2 _lter for all testing cases. Under some case, we obtain 0.8dB improvement in Peak Signal to Noise Ratio (PSN

29、R) over RidgeletShrink. The improvement over V isuShink is even bigger for denoising all images. This indicates that ComRidgeletShrink is an excellent choice for denoising natural noisy images.3 Experimental ResultsWe perform our experiments on the well-known image Lena. We get this image from the f

30、ree software package WaveLab developed by Donoho et al. at Stanford University. Noisy images with di_erent noise levels are generated by adding Gaussian white noise to the original noise-free images. For comparison, we implement VisuShrink, RidgeletShrink, ComRidgeletShrink and wiener2. VisuShrink i

31、s the universal soft-thresholding denoising technique. The wiener2 function is available in the MATLAB Image Processing Toolbox, and we use a 5*5 neighborhood of each pixel in the image for it. The wiener2 function applies a Wiener _lter (a type of linear filter) to an image adaptively, tailoring it

32、self to the local image variance. The experimental results in Peak Signal to Noise Ratio (PSNR) are shown in Table 1. We find that the partition block size of 32 * 32 or 64 *64 is our best choice. Table 1 is for denoising image Lena, for di_erent noise levels and afixed partition block size of 32 *3

33、2 pixels.The first column in these tables is the PSNR of the original noisy images, while other columns are the PSNR of the denoised images by using di_erent denoising methods. The PSNR is de_ned as PSNR = 􀀀10 log10 Pi;j (B(i; j) 􀀀 A(i; j)2 n22552 : where B is the denoised image an

34、d A is the noise-free image. From Table 1 we can see that ComRidgeletShrink outperforms VisuShrink, the ordinary RidgeletShrink, and wiener2 for all cases. VisuShrink does not have any denoising power when the noise level is low. Under such a condition, VisuShrink produces even worse results than th

35、e original noisy images. However, ComRidgeletShrink performs very well in this case. For some case, ComRidgeletShrink gives us about 0.8 dB improvement over the ordinary RidgeletShink. This indicates that by combining the dual-tree complex wavelet into the ridgelet transform we obtain signi_cant imp

36、rovement in image denoising. The improvement of ComRidgeletShrink over V isuShrink is even more signi_cant for all noisy levels and testing images. Figure 1 shows the noise free image, the image with noise added, the denoised image with VisuShrink, the denoised image with RidgeletShrink, the denoise

37、d image with ComRidgeletShrink, and the denoised image with wiener2 for images Lena, at a partition block size of 32*32 pixels. It can be seen that ComRidgeletShrink produces visually sharper denoised images than V isuShrink, the ordinary RidgeletShrink, and wiener2 filter, in terms of higher qualit

38、y recovery of edges and linear and curvilinear features.4 Conclusions and Future WorkIn this paper, we study image denoising by using complex ridgelets. Our complex ridgelet transform is obtained by performing 1D dual-tree complex wavelet onto the Radon transform coe_cients. The Radon transform is d

39、one by means of the projection-slice theorem. The approximate shift invariant property of the dual-tree complex wavelet transform makes the complex ridgelet transform an excellent choice for image denoising. The complex ridgelet transform provides near-ideal sparsity of representation for both smoot

40、h objects and objects with edges. This makes the thresholding of noisy ridgelet coe_cients a near-optimal method of denoising for Gaussian white noise. We test our new denoising method with several standard images with Gaussian white noise added to the images. A very simple hard thresholding of the

41、complex ridgelet coe_cients is used. Experimental results show that complex ridgelets give better denoising results than VisuShrink, wiener2, and the ordinary ridgelets under all experiments. We suggest that ComRidgeletShrink be used for practical image denoising applications. Future work will be do

42、ne by considering complex ridgelets in curvelet image denoising. Also, complex ridgelets could be applied to extract invariant features for pattern recognition.复杂脊波图像去噪1.介绍小波变换已成功地应用于many scientific fields such as image compression, i许多科学领域，如图像压缩，图像age denoising, signal processing, com去噪，信号处理，计算机图形，

43、ics, and pattern recognition, to name only a feIC和模式识别，仅举几例。Donoho和他的同事们提出了小波阈值去噪通过软阈值和阈值.这种方法的出现对于大量的应用程序是一个好的选择。这是因为一个小波变换能结合的能量,在一小部分的大型系数和大多数的小波系数中非常小,这样他们可以设置为零。这个阈值的小波系数是可以做到的只有细节的小波分解子带。我们有一些低频波子带不能碰触，让他们不阈值。众所周知，Donoho提出的方法的优势是光滑和自适应。然而，Coifman和Donoho指出，这种算法展示出一个视觉产出：吉布斯现象在邻近的间断。因此，他们提出对这些产出

44、去噪通过平均抑制所有循环信号。实验结果证实单目标识别小波消噪优于没有目标识别的情况。Bui和Chen扩展了这个目标识别计划，他们发现多小波的目标识别去噪的结果比单小波去噪的结果要好。蔡和西尔弗曼提出了一种阈值方案通过采取相邻的系数。他们结果表现出的优势超于了传统的一对一小波消燥。Chen和Bui扩展这个相邻小波阈值为多小波方法。他们声称对于某些标准测试信号和真实图像相邻的多小波降噪优于相邻的单一小波去噪。陈等人提出一种图像去噪是考虑方形相邻的小波域。陈等人也尝试对图像去噪自定义小波域和阈值。实验结果表明：这两种方法产生更好的去噪效果。 研究脊波变换的数多年来打破了小波变换的局限性。将小波变换产

45、生的二维图像在每个规模大的小波系数的分解。有这么多的大系数，对于图像去噪有很多困难。我们知道脊波变换已经成功用于分析数字图像。不像小波变换,脊波变换过程首先计算积分在不同的方向和位置的数据。沿着“x1cos_ + x2sin_ = 常数” 一条线的脊波是不变的。在这些脊的方向正交小波变换是一。最近脊波已成功应用于图像去噪。在本文中，我们结合dual-tree complex wavelet in the ridgelet transfo二元树复小波的脊波变换中并将其应用到图像降噪。这种近似二元树性能的复杂变性小波和良好性能的脊波使我们有更好的方法去图像去噪。实验结果表明，采用二元树复杂脊波在所

46、有去噪图像和许多不同噪音中我们的算法获得较高的峰值信噪比（PSNR）。这篇文章大体是这样的。在第二部分，我们将解释如何将二元树复杂的波变换成脊波去图像去噪。实验结果在第3节。第4节是最后得出的结论和未来需要做的工作。2.用复杂脊波图像去噪离散脊波变换提供接近理想的稀松代表光滑的物体边缘。高斯去噪是一个接近最优的方法。脊波变换能够压缩图像能量成为少量的脊波系数。在另一方面,利用小波变换产生的多大的小波系数对每个尺度边缘二维小波分解。这句话的意思是说许多小波系数进行重构在图像的边缘。我们知道近似氡转化为数字数据可以基于离散傅立叶变换。普通的脊波变换即可达到如下：1. 计算出二维FFT的图像。2.

47、替补的采样傅里叶广域上变换得到晶格和极性格的采样值。3. 计算一维逆FFT每一个角的线。4. 执行一维标量小波对角线结果，获取脊波系数。众所周知，普通的离散小波变换在变换期间是不移位和不转变的。输入信号的一个小小的改变能够引起输出小波系数很大的变化。为了克服这个问题，Kingsbury发明了一种新型的小波变换，叫做二元树复杂小波变换，它能够转移性能和提高近似角分辨率不变。由于标量波不是转移不变的，在脊波变换中就更好的应用二元树复杂小波变换这样我们就可以叫我们的复杂脊波。这样可以通过取代一维标量小波的一维二元树复杂小波在最后一步进行脊波变换。用这种方法我们可以优秀品质的脊波变换用来替换二元树发杂

48、脊波。这个复杂的脊波变换可以应用到整体图像，或者我们可以应用到分割图像大量重叠的平方或者在每一平方上运用脊波变换。我们分解一组n*n的影像重叠顺利进入边长R的象素是重叠的是两个相邻长方形的数组大小为R/2*R两者之间重叠的相邻区域就是一个长方形的大小R*R/2。对于一个n*n的图像，我们能够计数2n=R对于不同方向的模块，这个分区就引入了4倍的冗余。为了得到降噪的复杂脊波系数我们通常在当前象素地位对降噪的复杂脊波系数进行平均4份。复杂的脊波变换阈值类似于曲波阈值。当我们求阈值时一个不同是我们采取的是复杂的脊波系数。当y是带噪的脊波系数。我们使用下列硬阈值规则估算未知的脊波系数。当y k, 我们

49、令= .否则, y_ = 0.在这里，是通过用蒙特卡罗模拟接近。采用的系数k是依赖于噪声系数。当这个小于30时，我们用k=5首先分解尺度和k=4分解其他尺度。当这个噪音系数大于30时，我们用k=6首次分解尺度和k=5分解其他尺度。这个复杂的脊波去噪算法能够被描述如下：1. 图像分割成R*R块，两个垂直相邻的R /2*R重叠，两个水平象素块R*R/2重叠。2. 对于每一块，应用所提出的复杂脊波，复杂脊波系数的阈值，复杂脊波的逆换算。3. 在同一位置以平均象素对图像去噪。我们称这种算法叫，同时我们使用普通的脊波。这个计算复杂度的ComRidgeletShrink是和小波RidgeletShrink的标量相似。唯一的区别是我们取代了一维小波变换与一维二元树发杂小波变换。这个数量的计算是一维二元树复数小波的变换是一维小波变换的两倍。该算法的其他计算步骤保持相同。我们的实验结果显示ComRidgeletShrink优于V isuShrink, RidgeletShink, and 过滤器wiener2等所有测试案例。在某些情况下，我们在RidgeletShink中能够提高0.8db的信噪比。通过V isuS