【正文】 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 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 puting integrals over different orientations and locations. A ridgelet is constant along the lines x1cos_ + x2sin_ = constant. In the direction orthogonal to these ridges it is a have been successfully applied in image denoising recently. In this paper, we bine the dualtree plex wavelet in the ridgelet transform and apply it to image denoising. The approximate shift invariance property of the dualtree plex wavelet and the good property of the ridgelet make our method a very good method for image results show that by using dualtree plex ridgelets, our algorithms obtain higher Peak Signal to Noise Ratio (PSNR) for all the denoised images with di_erent noise anization of this paper is as follows. In Section 2, we explain how to incorporate the dualtree plex 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. IMAGE DENOISING BY USING COMPLEX Ridgelets Discrete ridgelet transform provides nearideal sparsity of representation of both smooth objects and of objects with edges. It is a nearoptimal method of denoising for Gaussian noise. The ridgelet transform can press 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 deposition. 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 overe this problem, Kingsbury introduced a new kind of wavelet transform, called the dualtree plex wavelet transform, that exhibits approximate shift invariant 外文文獻原文 6 property and improved angular resolution. Since the scalar wavelet is not shift invariant, it is better to apply the dualtree plex wavelet in the ridgelet transform so that we can have what we call plex ridgelets. This can be done by replacing the 1D scalar wavelet with the 1D dualtree plex wavelet transform in the last step of the ridgelet transform. In this way, we can bine the good property of the ridgelet transform with the approximate shift invariant property of the dualtree plex wavelets. The plex 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 depose 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 plex ridgelet coe_cient, we use the average of the four denoised plex ridgelet coe_cients in the current pixel location. The thresholding for the plex ridgelet transform is similar to the curvelet thresholding [10]. One difference is that we take the magnitude of the plex ridgelet coe_cients when we do t。 ComRidgeletShrink 超越VisuShrink的表現(xiàn)更重要的是所有噪音水平和圖像測試。j (B(i。 VisuShrink 是通用軟閾值去噪技術。該算法的其他計算步驟保持相同。當這個噪音系數(shù)大于 30 時，我們用k=6首次分解尺度和 k=5分解其他尺度。為了得到降噪的復雜脊波系數(shù)我們通常在當前象素地位 對降噪的復雜脊波系數(shù)進行平均 4 份。輸入信號的一個小小的改變能夠引起輸出小波系數(shù)很大的變化。在另一方面 ,利用小波變換產(chǎn)生的多大的小波系數(shù)對每個尺度邊緣二維小波分解。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。將小波變換產(chǎn)生的二維圖像在每個規(guī)模大的小波系數(shù)的分解。蔡和西爾弗曼提出了一種閾值方案通過采取相 鄰的系數(shù)。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(shù)和大多數(shù)的小波系數(shù)中非常小 ,這樣他們可以設置為零。由于實驗研究者對給定小波選擇問題有一些經(jīng)驗，所以他們對于開發(fā)選擇可靠適合的特定信號表示基的方法可參考曾經(jīng)的試驗經(jīng)驗。 計算小波包分解。 外文文獻譯文 3 在特定的信號分析中任何性能的變化都是高度基于基礎功能的變換。 雖然時間和頻率分辨率的問題是一種物理現(xiàn)象（海森堡測不準原理）無論是否使用變換，它都存在，但是它可以使用替代方法分析，稱為信號多分辨率分析（ MRA）。這個詞意味著母波在支持不同類型波的轉型過程中起主要作用，或者叫母小波。小波分析和 STFT的分析方法類似，在這個意義上說，就是信號和一個函數(shù)相乘， {它的小波 }，類似的 STFT的窗口功能，并轉換為不同分段的時域信號。因此，這些信息將提交本節(jié)。外文文獻譯文 1 附錄三： 外文文獻譯文 一種新型基于小波圖像去噪法 （一） 基礎知識介紹 近年來 ,小波理論得到了非常迅速的發(fā)展 ,而且由于其具備良好的時頻特性 ,實際應用也非常廣泛。,crit,thr,keepapp)。 %基于小波包的消噪處理 thr=10。 X1=X+10*randn(size(X))。 title(39。 imshow(y3)。添加高斯噪聲后的圖像 39。 pepper39。)。 a=imread(39。 X2=wprcoef(NT,1)。 axis square。,thr)。)。seed39。 image(X)。 image(X2)。db1039。 subplot(3,3,8)。 程序源代碼 4 T=wpdec2(X1,1,39。 X2=wprcoef(NT,1)。 axis square。,thr)。)。s39。sym2小波去噪圖像 39。 程序源代碼 3 NT=wpthcoef(T,0,39。 title(39。 init=2055615866。 2. 研究不同小波基的程序： load flujet。 X2=wprcoef(NT,1)。 axis square。,thr)。)。s39。小波分解 1層 39。 NT=wpthcoef(T,0,39。 title(39。 init=2055615866。 五 .參考文獻 [1]李世雄 .小波變換及應用 [M].北京：高等教育出版社， 1997． [2]彭玉華 .小波變換與工程應用 [M].北京：科學出版社， 1999． [3]趙瑞珍 .小波理論及其在圖像信號處理中的算法研究 [M].西安：西安電子科技大學， 2020. [4]章毓晉 .圖像處理和分析基礎 [M].北京 :高等教育出版社， 2020. [5]李弼程，羅建書 .小波分析及其應用 [