【文章內(nèi)容簡介】 波變換產(chǎn)生的多大的小波系數(shù)對(duì)每個(gè)尺度邊緣二維小波分解。這句話的意思是說許多小波系數(shù)進(jìn)行重構(gòu)在圖像的邊緣。我們知道近似氡轉(zhuǎn)化為數(shù)字?jǐn)?shù)據(jù)可以基于離散傅立葉變換。普通的脊波變換即可達(dá)到如下： FFT的圖像。 。 FFT每一個(gè)角的線。 ，獲取脊波系數(shù)。 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(zhuǎn)變的。輸入信號(hào)的一個(gè)小小的改變能夠引起輸出小波系數(shù)很大的變化。為了克服這個(gè)問題， Kingsbury 發(fā)明了一種新型的小波變換，叫做二元樹復(fù)雜小波變換，它能夠轉(zhuǎn)移性能和提高近似角分辨率不變。由于標(biāo)量波不是轉(zhuǎn)移不變的，在脊波變換中就更好的應(yīng)用二元樹復(fù)雜小波變換這樣我們就可以叫我們的復(fù)雜脊波。這樣可以通過取代 一維標(biāo)量小波的一維二元樹復(fù)雜小波在最后一步進(jìn)行脊波變換。用這種方法我們可以優(yōu)秀品質(zhì)的脊波變換用來替換二元樹發(fā)雜脊波。 這個(gè)復(fù)雜的脊波變換可以應(yīng)用到整體圖像，或者我們可以應(yīng)用到分割圖像大量重疊的平方或者在每一平方上運(yùn)用脊波變換。我們分解一組 n*n 的影像重疊順利進(jìn)入邊長 R的象素是重疊的是兩個(gè)相鄰長方形的數(shù)組大小為 R/2*R 兩者之間重疊的相鄰區(qū)域就是一個(gè)長方形的大小 R*R/2。對(duì)于一個(gè)外文文獻(xiàn)譯文 5 n*n的圖像，我們能夠計(jì)數(shù) 2n=R對(duì)于不同方向的模塊，這個(gè)分區(qū)就引入了 4倍的冗余。為了得到降噪的復(fù)雜脊波系數(shù)我們通常在當(dāng)前象素地位 對(duì)降噪的復(fù)雜脊波系數(shù)進(jìn)行平均 4 份。復(fù)雜的脊波變換閾值類似于曲波閾值。當(dāng)我們求閾值時(shí)一個(gè)不同是我們采取的是復(fù)雜的脊波系數(shù)。當(dāng) yλ是帶噪的脊波系數(shù)。我們使用下列硬閾值規(guī)則估算未知的脊波系數(shù)。當(dāng)│ yλ┃ kσ ? , 我們令λ = ?λλ .否則 ， ^y_ = ， ? σ是通過用蒙特卡羅模擬接近。采用的系數(shù) k 是依賴于噪聲系數(shù)。當(dāng)這個(gè)小于 30時(shí)，我們用 k=5首先分解尺度和 k=4 分解其他尺度。當(dāng)這個(gè)噪音系數(shù)大于 30 時(shí)，我們用k=6首次分解尺度和 k=5分解其他尺度。這個(gè)復(fù)雜的脊波去噪算法能夠被描述如下： R*R塊，兩個(gè)垂直相鄰的 R/2*R重疊，兩個(gè)水平象素塊 R*R/2重疊。 ，應(yīng)用所提出的復(fù)雜脊波，復(fù)雜脊波系數(shù)的閾值，復(fù)雜脊波的逆換算。 。 我們稱這種算法叫，同時(shí)我們使用普通的脊波。這個(gè)計(jì)算復(fù)雜度的 ComRidgeletShrink 是和小波 RidgeletShrink 的標(biāo)量相似。唯一的區(qū)別是我們?nèi)〈艘痪S小波變換與一維二元樹發(fā)雜小波變換。這個(gè)數(shù)量的計(jì)算是一維二元樹復(fù)數(shù)小波的變換是一維小波變換的兩倍。該算法的其他計(jì)算步驟保持相同。我們的實(shí)驗(yàn)結(jié)果顯示 ComRidgeletShrink優(yōu)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 的信噪比。通過 V isuShrink,能夠改善更大的去噪圖像。這表明 ComRidgeletSrink 對(duì)于自然圖像去噪是一個(gè)很好的選擇。 我們通過對(duì)眾所周知的蕾娜進(jìn)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(shí)通過對(duì)原無噪音圖像添加高斯白噪音得到的。與之 相比，我們實(shí)行 VisuShrink, RidgeletShrink, ComRidgeletShrink and wiener2。 VisuShrink 是通用軟閾值去噪技術(shù)。這個(gè) wiener2函數(shù)是可以從 MatLab圖像工具箱得到，我們用一個(gè) 5*5的相鄰圖像在每個(gè)象素中。該 wiener2 適用于一個(gè)維納濾波器（一種線性的濾波器）圖形自適應(yīng)。剪裁自己的圖像局部方差。峰值信噪比的實(shí)驗(yàn)結(jié)果顯示的表 32*32或者 64*64是最好的選擇。表 1 是對(duì)蕾娜圖像進(jìn)行去噪，根據(jù)不同的噪聲水平固定分區(qū)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(gè)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(yōu)于不同 RidgeletShrink和 wiener2在所有案例中。當(dāng)噪音低時(shí) VisuShrink 沒有去噪能力。在這樣的情況下， VisuShrink 將產(chǎn)生比原來的去噪圖像更糟的結(jié)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 。這表明，我們把二元樹結(jié)合復(fù)數(shù)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(xiàn)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(gè)分區(qū)大小為 32*32 的象素塊中。 ComRidgeletShrink 在視覺上產(chǎn)生的效果比 VisuShrink ， wiener2 RidgeletShrink更清晰具有高線性和恢復(fù)曲線的特點(diǎn)。 外文文獻(xiàn)譯文 6 （三） 總結(jié) 本文基于小波變換對(duì)信號(hào)去噪進(jìn)行了深入地分析和研究 ,結(jié)合去噪原理討論和比較了實(shí)際應(yīng)用中對(duì)小波基及閾值規(guī)則的合理選取問題。實(shí)驗(yàn)結(jié)果表明，利用小波去噪能實(shí)現(xiàn)對(duì)各種信號(hào)的去噪，且效果比較明顯。 外文文獻(xiàn)原文 1 附錄四： 外文文獻(xiàn)原文 A New Ways Of Signals Denoised By Wavelet (Ⅰ ) BASIC THEORY In recent years,wavelet theory has been very rapid development,but also because of its good timefrequency character istics of awide range of practical applications. Here wish to take advantage of the selfwavelet features,in the reduction of noise at the same time,to keep the details of the image itself and the edge of useful information,thus ensuring the best of image wavelet thresholding denoising method can be said that many image denoising methods are the best. THE WAVELET THEORY: A MATHEMATICAL APPROACH This section describes the main idea of wavelet analysis theory, which can also be considered to be the underlying concept of most of the signal analysis techniques. The FT defined by Fourier use basis functions to analyze and reconstruct a function. Every vector in a vector space can be written as a linear bination of the basis vectors in that vector space , ., by multiplying the vectors by some constant numbers, and then by taking the summation of the products. The analysis of the signal involves the estimation of these constant numbers (transform coefficients, or Fourier coefficients, wavelet coefficients, etc). The synthesis, or the reconstruction, corresponds to puting the linear bination equation. All the definitions and theorems related to this subject can be found in Keiser39。s book, A Friendly Guide to Wavelets but an introductory level knowledge of how basis functions work is necessary to understand the underlying principles of the wavelet theory. Therefore, this information will be presented in this section. THE WAVELET SYNTHESIS The continuous wavelet transform is a reversible transform, provided that Equation 2 is satisfied. Fortunately, this is a very nonrestrictive requirement. The continuous wavelet transform is reversible if Equation 2 is satisfied, even though the basis functions are in general may not be orthonormal. The reconstruction is possible by using the following reconstruction formula: Equation 1 Inverse Wavelet Transform where C_psi is a constant that depends on the wavelet used. The success of the reconstruction depends on this constant called, the admissibility constant , to satisfy the following admissibility condition : 外文文獻(xiàn)原文 2 Equation 2 Admissibility Condition where psi^hat(xi) is the FT of psi(t). Equation 2 implies that psi^hat(0) = 0, which is: Equation 3 As stated above, Equation 3 is not a very restrictive requirement since many wavelet functions can be found whose integral is zero. For Equation 3 to be satisfied, the wavelet must be oscillatory. THE CONTINUOUS WAVELET TRANSFORM The continuous wavelet transform was developed as an alternative approach to the short time Fourier transform to overe the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, {it the wavelet}, similar to the window function in the STFT, and the transform is puted separately for different segments of the timedomain signal. However, there are two main differences between the STFT and the CWT: 1. The Fourier transforms of the windowed signals are not taken, and therefore single peak will be seen corresponding to a sinusoid, ., negative frequencies are not puted. 2. The width of