MATLAB毕业设计外文资料翻译.docx

1、MATLAB毕业设计外文资料翻译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

2、a wavelet denoising 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 co

3、eficients are very 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 th

4、e advantages of 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 ov

5、er the denoised 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 wa

6、velet denoising. 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 multiw

7、avelet case. They 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 cus

8、tomize the wavelet _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 la

9、rge wavelet coeficients 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 d

10、ata by first computing 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 th

11、e dual-tree complex 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-

12、tree complex ridgelets, 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

13、denoising. Experimental 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.

14、It is a near-optimal 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 decompo

15、sition. This means 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 th

16、e image.2. Substitute 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_cie

17、nts.It is well known 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 o

18、f wavelet transform, 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

19、what we call complex 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

20、dual-tree complex 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 p

21、ixels so that the 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 re

22、dundancy of 4 times. 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 ta

23、ke the magnitude 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 Mon

24、te-Carlo simulations. 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 decomp

25、osition scales.The 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 r

26、idgelets, threshold 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 computation

27、al complexity of 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 1

28、D scalar wavelet 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

29、Noise Ratio (PSNR) 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

30、image from the free 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 wiene

31、r2. VisuShrink is 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 adaptive

32、ly, tailoring itself 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 blo

33、ck size of 32 *32 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 and A is the noise-free image. From Table 1 we