大連民族學院畢業設計外文資料翻譯所 在 學 院：機電信息工程學院專 業 (班級)： 自動化063 學 生 姓 名： 徐睿 指 導 教 師： 王培昌 2010年6月4日 復雜脊波圖像去噪作者：G. Y. Chen and B. Kegl刊名：Pattern Recognition;出版日期：20071.介紹小波變換已成功地應用于many scientific fields such as image compression, i許多科學領域，如圖像壓縮，圖像age denoising, signal processing, com去噪，信號處理，計算機圖形，ics, and pattern recognition, to name only a feIC和模式識別，僅舉幾例。Donoho和他的同事們提出了小波閾值去噪通過軟閾值和閾值.這種方法的出現對于大量的應用程序是一個好的選擇。這是因為一個小波變換能結合的能量,在一小部分的大型系數和大多數的小波系數中非常小,這樣他們可以設置為零。這個閾值的小波系數是可以做到的只有細節的小波分解子帶。我們有一些低頻波子帶不能碰觸，讓他們不閾值。眾所周知，Donoho提出的方法的優勢是光滑和自適應。然而，Coifman和Donoho指出，這種算法展示出一個視覺產出：吉布斯現象在鄰近的間斷。因此，他們提出對這些產出去噪通過平均抑制所有循環信號。實驗結果證實單目標識別小波消噪優于沒有目標識別的情況。Bui和Chen擴展了這個目標識別計劃，他們發現多小波的目標識別去噪的結果比單小波去噪的結果要好。蔡和西爾弗曼提出了一種閾值方案通過采取相鄰的系數。他們結果表現出的優勢超于了傳統的一對一小波消燥。Chen和Bui擴展這個相鄰小波閾值為多小波方法。他們聲稱對于某些標準測試信號和真實圖像相鄰的多小波降噪優于相鄰的單一小波去噪。陳等人提出一種圖像去噪是考慮方形相鄰的小波域。陳等人也嘗試對圖像去噪自定義小波域和閾值。實驗結果表明：這兩種方法產生更好的去噪效果。 研究脊波變換的數多年來打破了小波變換的局限性。將小波變換產生的二維圖像在每個規模大的小波系數的分解。有這么多的大系數，對于圖像去噪有很多困難。我們知道脊波變換已經成功用于分析數字圖像。不像小波變換,脊波變換過程首先計算積分在不同的方向和位置的數據。沿著“x1cos_ + x2sin_ = 常數” 一條線的脊波是不變的。在這些脊的方向正交小波變換是一。最近脊波已成功應用于圖像去噪。在本文中，我們結合dual-tree complex wavelet in the ridgelet transfo二元樹復小波的脊波變換中并將其應用到圖像降噪。這種近似二元樹性能的復雜變性小波和良好性能的脊波使我們有更好的方法去圖像去噪。實驗結果表明，采用二元樹復雜脊波在所有去噪圖像和許多不同噪音中我們的算法獲得較高的峰值信噪比（PSNR）。這篇文章大體是這樣的。在第二部分，我們將解釋如何將二元樹復雜的波變換成脊波去圖像去噪。實驗結果在第3節。第4節是最后得出的結論和未來需要做的工作。2.用復雜脊波圖像去噪離散脊波變換提供接近理想的稀松代表光滑的物體邊緣。高斯去噪是一個接近最優的方法。脊波變換能夠壓縮圖像能量成為少量的脊波系數。在另一方面,利用小波變換產生的多大的小波系數對每個尺度邊緣二維小波分解。這句話的意思是說許多小波系數進行重構在圖像的邊緣。我們知道近似氡轉化為數字數據可以基于離散傅立葉變換。普通的脊波變換即可達到如下：1. 計算出二維FFT的圖像。2. 替補的采樣傅里葉廣域上變換得到晶格和極性格的采樣值。3. 計算一維逆FFT每一個角的線。4. 執行一維標量小波對角線結果，獲取脊波系數。眾所周知，普通的離散小波變換在變換期間是不移位和不轉變的。輸入信號的一個小小的改變能夠引起輸出小波系數很大的變化。為了克服這個問題，Kingsbury發明了一種新型的小波變換，叫做二元樹復雜小波變換，它能夠轉移性能和提高近似角分辨率不變。由于標量波不是轉移不變的，在脊波變換中就更好的應用二元樹復雜小波變換這樣我們就可以叫我們的復雜脊波。這樣可以通過取代一維標量小波的一維二元樹復雜小波在最后一步進行脊波變換。用這種方法我們可以優秀品質的脊波變換用來替換二元樹發雜脊波。這個復雜的脊波變換可以應用到整體圖像，或者我們可以應用到分割圖像大量重疊的平方或者在每一平方上運用脊波變換。我們分解一組n*n的影像重疊順利進入邊長R的象素是重疊的是兩個相鄰長方形的數組大小為R/2*R兩者之間重疊的相鄰區域就是一個長方形的大小R*R/2。對于一個n*n的圖像，我們能夠計數2n=R對于不同方向的模塊，這個分區就引入了4倍的冗余。為了得到降噪的復雜脊波系數我們通常在當前象素地位對降噪的復雜脊波系數進行平均4份。復雜的脊波變換閾值類似于曲波閾值。當我們求閾值時一個不同是我們采取的是復雜的脊波系數。當y是帶噪的脊波系數。我們使用下列硬閾值規則估算未知的脊波系數。當y k, 我們令= .否則, y_ = 0.在這里，是通過用蒙特卡羅模擬接近。采用的系數k是依賴于噪聲系數。當這個小于30時，我們用k=5首先分解尺度和k=4分解其他尺度。當這個噪音系數大于30時，我們用k=6首次分解尺度和k=5分解其他尺度。這個復雜的脊波去噪算法能夠被描述如下：1. 圖像分割成R*R塊，兩個垂直相鄰的R /2*R重疊，兩個水平象素塊R*R/2重疊。2. 對于每一塊，應用所提出的復雜脊波，復雜脊波系數的閾值，復雜脊波的逆換算。3. 在同一位置以平均象素對圖像去噪。我們稱這種算法叫，同時我們使用普通的脊波。這個計算復雜度的ComRidgeletShrink是和小波RidgeletShrink的標量相似。唯一的區別是我們取代了一維小波變換與一維二元樹發雜小波變換。這個數量的計算是一維二元樹復數小波的變換是一維小波變換的兩倍。該算法的其他計算步驟保持相同。我們的實驗結果顯示ComRidgeletShrink優于V isuShrink, RidgeletShink, and 過濾器wiener2等所有測試案例。在某些情況下，我們在RidgeletShink中能夠提高0.8db的信噪比。通過V isuShrink,能夠改善更大的去噪圖像。這表明ComRidgeletSrink對于自然圖像去噪是一個很好的選擇。 3.實驗結果 我們通過對眾所周知的蕾娜進行處理，通過Donoho等人我們得到了這種圖片的自由軟體包WaveLab。帶有不同噪音的噪音圖像時通過對原無噪音圖像添加高斯白噪音得到的。與之相比，我們實行VisuShrink, RidgeletShrink, ComRidgeletShrink and wiener2。VisuShrink是通用軟閾值去噪技術。這個wiener2函數是可以從MatLab圖像工具箱得到，我們用一個5*5的相鄰圖像在每個象素中。該wiener2適用于一個維納濾波器（一種線性的濾波器）圖形自適應。剪裁自己的圖像局部方差。峰值信噪比的實驗結果顯示的表1.我們發現對于分區塊的大小32*32或者64*64是最好的選擇。表1是對蕾娜圖像進行去噪，根據不同的噪聲水平固定分區和一素塊為32*32。表格中的第一欄是原來帶噪圖片的信噪比，其他列是通過不同去噪方法得到的去噪圖像信噪比。這個信噪比被定義PSNR = 10 log10Pi;j (B(i; j) A(j)2n22552 :; 其中B是去噪圖像A是無噪音圖像。從表1.我們可以看出VisuShrink ,ComRidgeletShrink是優于不同RidgeletShrink和wiener2在所有案例中。當噪音低時VisuShrink沒有去噪能力。在這樣的情況下，VisuShrink將產生比原來的去噪圖像更糟的結果。然而，ComRidgeletShrink在這種情況下取得較好的效果。在某些情況下，ComRidgeletShrink能夠比普通RidgeletShrink多提供給我們0.8db。這表明，我們把二元樹結合復數的小波變換成脊波變換能夠明顯的改善我們圖像去噪的效果。ComRidgeletShrink超越VisuShrink的表現更重要的是所有噪音水平和圖像測試。圖一顯示的是在無噪音圖像，添加噪音的圖像，用VisuShrink去噪的圖像，用RidgeletShrink去噪的圖像，用ComRidgeletShrink去噪的圖像，用wiener2去噪的圖像，在一個分區大小為32*32的象素塊中。ComRidgeletShrink在視覺上產生的效果比VisuShrink ，wiener2 RidgeletShrink更清晰具有高線性和恢復曲線的特點。4結論和未來工作在這篇文章中我們研究了用復雜脊波對圖像去噪。復雜脊波變換是通過執行一維二元樹復雜小波在空氣中氡的變換系數獲得的。氡變換是通過投影片定理得到的。對于圖像去噪近似轉換不變性質的二元樹復數小波變換對于復雜小波變換是一個很好的選擇。復雜脊波變換提供了近乎完美的對于光滑的物體和表現對象與邊緣。這使噪音閾值的脊波系數更接近高斯白噪音的消噪。我們測試了我們新的去噪方法和幾個標準圖像和加入高斯白噪音的圖像。用一個非常簡單的硬閾值復雜脊波系數。在這些脊波實驗中，實驗結果表明復雜的脊波有更好的去噪能力比起VisuShrink和普通的wiener2。我們建議ComRidgeletShrink用于實際的圖像去噪中。未來工作主要是考慮在復雜圖像應用曲波復雜脊波。同樣，復雜脊波還可以應用的不變特征提取模式識別方法。Complex Ridgelets for Image DenoisingG. Y. Chen and B. Kegl1 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 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 coeficients 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 the 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 over 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 wavelet 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 multiwavelet 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 customize 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 large 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 data 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 the 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-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 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. 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 decomposition. 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 the 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_cients.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 of 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 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 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 pixels 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 redundancy 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 take 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 Monte-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 decomposition 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 ridgelets, 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 computational 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 1D 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 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 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 wiener2. 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 adaptively, 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 block 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 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 the 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 improvement 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 denoised image with ComRi