外文翻譯--最小方波在小波領(lǐng)域的展開

1、<p><b>　　中文3300字</b></p><p><b>　　附錄A：英文原文</b></p><p>　　Least squares phase unwrapping in wavelet domain</p><p>　　Abstract: </p><p>　　Least

2、squares phase unwrapping is one of the robust techniques used to solve two-dimensional phase unwrapping problems. However, owing to its sparse structure, the convergence rate is very slow, and some practical methods have

3、 been applied to improve this condition. In this paper, a new method for solving the least squares two-dimensional phase unwrapping problem is presented. This technique is based on the multiresolution representation of a

4、 linear system using the discrete wavelet transform. B</p><p>　　1 introduction</p><p>　　Two-dimensional phase unwrapping is an important processing step in some coherent imaging applications, s

5、uch as synthetic aperture radar interferometry(InSAR) and magnetic resonance imaging(MRI).In these processes, three-dimensional information of the measured objects can be extracted from the phase of the sensed signals ,H

6、owever, the obseryed phase data are wrapped principal values, which are restricted in a 2 modulus ,and they must be unwrapped to their true absolute phase values .This is the </p><p>　　The basic assumption o

7、f the general phase unwrapping methods is that the discrete derivatives of the unwrapped phase at all grid points are less than in absolute value .With this assumption satisfied ,the absolute phase can be reconstructed

8、perfectly by integrating the partial derivatives of the wrapped phase data. In the general case, however, it is not possible to recover unambiguously the absolute phase from the measured wrapped phase which is usually co

9、rrupted by noise or aliasing effects s</p><p>　　After Goldstein-et al introduced the concept of ‘residues’ in the two-dimensional phase unwrapping problem of InSAR, many phase unwrapping approaches to cope w

10、ith this problem have been investigated. Path-following (or integration-based) methods and least squares methods are the most representative two basic classes in this field. There have also been some other approaches suc

11、h as Green methods, Bayesian regularization methods ,image processing-based methods, and model-based methods.</p><p>　　Least squares phase unwrapping ,established by Ghiglia and Romero, is one of the most ro

12、bust techniques to solve the two-dimensional phase unwrapping problem. This method obtains an unwrapped solution by minimizing the differences between the partial derivatives of the wrapped phase data and the unwrapped s

13、olution .Least squares method is divided into unweighted and weighted least squares phase unwrapping. To isolate the phase inconsistencies, a weighted least squares method should be used, whic</p><p>　　The l

14、east squares method is well-defined mathematically and equivalent to the solution of Poisson’s partial differential equation, which can be expressed as a sparse linear equation. anterior method is usually used to solve t

15、his large linear equation. However, a large computation time is required and therefore improving the convergence rate is a very important task when using this method. Some numerical algorithms have been applied to this p

16、roblem to improve convergence conditions.</p><p>　　An approach for fast convergence of a sparse linear equation is to transfer the original equation system into a new system with larger supports .Multiresolu

17、tion or hierarchical representation concepts have often been used for this purpose. Recently, wavelet transform has been investigated deeply in science and engineering fields as a sophisticated tool for the multiresoluti

18、on analysis of signals and systems. It decomposes a signal space into its low-resolution subspace and the complementary deta</p><p>　　Weighted least squares phase unwrapping: a review</p><p>　　L

19、east squares phase obtains an unwrapped solution by minimizing the -norm between the discrete partial derivatives of the wrapped phase data and those of the unwrapped solution function. Given the wrapped phase on an M&#

20、215;N rectangular grid(,),the partial derivatives of the wrapped phase are defined as</p><p>　　, (1) </p><p>　　Where W is the wrapping operator that wraps the phase into the interval

21、.The differences between the partial derivatives of the solution and those in (1) can be minimized in the weighted least squares sense, by differentiating the sum</p><p><b>　　(2)</b></p>

22、<p>　　With respect to and setting the result to zero.</p><p>　　In (2),the gradient weights , and ,are used to prevent some phase values corrupted by noise or aliasing from degrading the unwrapping , an

23、d are defined by </p><p><b>　　,, (3)</b></p><p>　　The weighted least squares phase unwrapping problem is to find the solution that minimizes the sum of (2).The initial weight arra

24、y is user-defined and some methods for defining these weights are presented in [1,11]. When all the weights , the above equation is the unweighted phase unwrapping problem. Since weight array is related to the exactitud

25、e of the resultant unwrapped solution , it must be defined properly. In this paper, however, it is assumed that the weight array is defined already fo</p><p>　　The least squares solution to this problem yiel

26、ds the following equation:</p><p><b>　　(4)</b></p><p>　　Where is the weighted phase Laplacian’ defined by</p><p><b>　　(5)</b></p><p>　　The unwr

27、apped solution is obtained by iteratively solving the following equation </p><p><b>　　(6)</b></p><p>　　Equation (4) is the weighted and discrete version of the Poisson’s partial dif

28、ferential equation (PDE),.By concatenating all the nodal variables into MN×1 one column vector , the above equation is expressed as a linear system</p><p><b>　　(7)</b></p><p>　

29、　Where the system matrix A is of size K×K(K=MN) and is a column vector of , That is ,the solution of the least squares phase unwrapping problem can be obtained by solving this linear system, and for given A and ,w

30、hich are defined from the weight array and the measured wrapped phase the unwrapped phase has the unique solution ,But since A is a very large matrix, the direct inverse operation is practically impossible. The struct

31、ure of the system matrix A is very sparse and most of the off-diag</p><p>　　Direct methods based on the fast Fourier transform(FFT) or the discrete cosine transform (DCT) can be applied to solve the unweight

32、ed phase unwrapping problem. However, in the weighted case, iterative methods should be adopted. The classical iterative method for solving the linear system is the Gauss-Seidel relaxation, which solves (6) by simple ite

33、ration until it converges. However, this method is not practical owing to its extremely slow convergence, which is caused by the sparse characteristi</p><p>　　There are other approaches to solve a sparse lin

34、ear system problem efficiently, In these approaches, a system is converted into another equivalent system with better convergence condition .The convergence speed of the system is characterised by the system matrix A.

35、The structure of the system matrix of the least squares phase unwrapping problem is very sparse. In the iterative solving methods, the local connections between the nodal variables slow down the progress of the solution

36、in iteration</p><p>　　Wavelet transform is the most sophisticated method to represent a system in multiresolution concept. In this paper, an efficient method to solve the least squares phase unwrapping probl

37、em is proposed, by using the discrete wavelet transform (DWT).This is an extension of the work presented in literature. Some literature on work in the domain of wavelet approaches to the solution of partial differential

38、equations can be found. Those studies deal with the PDE structure itself in wavelet domain to s</p><p>　　conclusions</p><p>　　An efficient method to solve the weighted least squares two-dimensio

39、nal phase unwrapping has been presented. Biorthogonal wavelet transform is applied to transfer the original system into the new equivalent system in wavelet domain with low-frequency and high-frequency portions decompose

40、d. Separately solving the low-frequency portion of the new system speeds up the overall system convergence rate. The convergence improvement has been shown by experiments with some synthetic phase images.</p><

41、p>　　The proposed method provides better results than those obtained by using the Gauss-Seidel relaxation and the multigrid method. Another advantage of this method is that the new system is mathematically equivalent t

42、o the original matrix. so that its solution is exact to the original equation both for the weighted and unweighted least squares phase unwrapping problems.</p><p><b>　　附錄B：漢語(yǔ)翻譯</b></p><

43、;p>　　最小方波在小波領(lǐng)域的展開</p><p><b>　　摘要: </b></p><p>　　最小方波的展開是過去一直解決二維小波展開問題的關(guān)鍵技術(shù)之一。 然而，它的稀疏結(jié)構(gòu)，集中率非常低，因此就需要一些更實(shí)際的方法來改善這情況。 在本文中，提出了一個(gè)解決最小二維方波展開問題的新方法。該技術(shù)是以不連續(xù)的小波變換 的線系統(tǒng)的多途徑為基礎(chǔ)，通過小波變換，原始

44、系統(tǒng)被分解成模糊和精確兩部分。 在展開的模糊部分的快速集中作全部的系統(tǒng)集中是非?？斓?。</p><p><b>　　1 介紹： </b></p><p>　　二維小波的展開在一些數(shù)字圖像處理, 例如綜合性的孔雷達(dá)干涉測(cè)量法 (InSAR) 和磁性共嗚圖像處理 (磁共振成像) 中是很重要的部分 。在這些處理步驟中，被測(cè)量物體的三維信息能從被感覺的信號(hào)的相位中被提取 ,然

45、而, 信號(hào)的被包裝主要的價(jià)值被限制在2相位數(shù)據(jù)中,因此它們的真實(shí)絕對(duì)相位價(jià)值一定要展開。這就是相位展開的問題, 特別是二維情況。</p><p>　　一般的相位展開方法的基本假定是在所有的被展開的不連續(xù)相位格子點(diǎn)的引出之物要少于在中展開的絕對(duì)值。為了滿足這一項(xiàng)假定，絕對(duì)相位能通過被包裝的相位數(shù)據(jù)部分的引出之物的整合來完全地重建。然而，在一般的情形下，從被噪音腐爛或被別的處理如圖像、短暫中斷等等影響過的被包裝的標(biāo)準(zhǔn)

46、相位復(fù)原是不可能的。在如此的情況，基本的假定被違犯，同時(shí)由于污染所引起的相位不一致，簡(jiǎn)單的整合過程也不能夠被運(yùn)用。</p><p>　　在高思頓以及其他人在用孔雷達(dá)干涉測(cè)量法展開二維相位的問題中介紹了 ‘殘留物' 的觀念之后, 許多處理這個(gè)問題的相位展開方法已經(jīng)被調(diào)查。以整合為基礎(chǔ)的途徑跟蹤法和最小的方波法，在這一個(gè)領(lǐng)域中是最代表性的二個(gè)基本的類型。 不過另外也有一些其他的方法，比如格林方法,貝斯定理的規(guī)

47、則化方法 ,圖像以處理為基礎(chǔ)的方法、和以型號(hào)為基礎(chǔ)的方法。</p><p>　　Ghiglia 和 Romero 提出最小方波逐步展開法,是解決二維相位展開問題最強(qiáng)健的技術(shù)之一。 這個(gè)方法包含了將被包裝的和被展開的相位部分引出之物數(shù)據(jù)之間不同減到最少獲得展開的方法。最小方波法被劃分為非傾斜和傾斜的最小方波逐步展開。為了要隔離狀態(tài)不一致 , 應(yīng)該用一個(gè)傾斜的最小方波方法, 它通過使用權(quán)衡排列能削弱污染的影響。 格林

48、法和貝斯定理的方法也是以最小方波方案為基礎(chǔ)的。但是這些方法不同于那些在相位不一致處理方法。 因此，這篇文章只與Ghiglia提出的最小方波逐步展開之類的問題有關(guān)。</p><p>　　最小的方波法是定義明確的和對(duì) Poisson 的部分微分方程式的算術(shù)地解決, 能被表示成一個(gè)稀疏的一次方程序的同等物。通常用來解決這個(gè)大的一次方程序有較多的方法。 然而,這需要很長(zhǎng)的計(jì)算時(shí)間，因此在使用這一個(gè)方法時(shí)，提高集中率是一件

49、非常重要的工作。一些數(shù)字的運(yùn)算法則已經(jīng)被應(yīng)用于改善集中情況這一個(gè)問題。</p><p>　　為一個(gè)稀疏的一次方程快速集中的方法是盡量將最初的相等系統(tǒng)轉(zhuǎn)變?yōu)橐粋€(gè)新的系統(tǒng)之內(nèi)。多分辨率或階層的表現(xiàn)觀念已經(jīng)經(jīng)常作為這一個(gè)目的。 最近，小波變換已經(jīng)為信號(hào)和系統(tǒng)的多分辨率分析在作為一個(gè)復(fù)雜的工具科學(xué)和工程領(lǐng)域中被深深地調(diào)查。 它把信號(hào)空間分解為低分辨率次空間和補(bǔ)充的細(xì)節(jié)次空間兩部分。 在我們所說的方法中，不連續(xù)的小波變換被

50、適用于表現(xiàn)獨(dú)立的多分辨率空間的最初的系統(tǒng)的最小方波相位展開問題的線系統(tǒng)。在這個(gè)新的轉(zhuǎn)移系統(tǒng)，能達(dá)到一種較好的集中情況。 在本文中簡(jiǎn)短的介紹了這一個(gè)方法,被提議的方法只適用于非傾斜相位的展開問題，在本文中，這個(gè)新的方法被延伸到傾斜的最小方波展開問題。 同時(shí)在這里也全面描述了被提議的這種方法。</p><p>　　傾斜最小方波相位的展開: 歷史回顧</p><p>　　最小方波相位的展開方法是

51、通過減小包裝的不連續(xù)部分派生物的相位數(shù)據(jù)和那些展開的解決功能之間的基準(zhǔn)。 在M × N 矩形格子 (,) 上給出包裝的相位,包裝的相位部分派生物被定義為：</p><p>　　, (1)</p><p>　　關(guān)于和結(jié)果設(shè)定為零。</p><p>　　在公式(2)中,傾斜的和,用來避免許多被噪音腐蝕的相位價(jià)值或者從降低那展開等級(jí)別名,

52、而且被定義為</p><p><b>　　,, (3)</b></p><p>　　傾斜的最小方波的展開問題的目的是找到求的方法去減小公式（2）中的總數(shù)。原始的傾斜數(shù)組已經(jīng)被使用者定義過并且定義這些傾斜數(shù)組的方法已經(jīng)在文獻(xiàn)中被陳述過了.當(dāng)所有的傾斜數(shù)組,上述的等式就是解決非傾斜相位展開問題。因?yàn)閮A斜數(shù)組是和最后結(jié)果的展開方法的提取相關(guān),因此，它一定被恰當(dāng)?shù)囟x。然

53、而，在本文中，假定的傾斜數(shù)組已經(jīng)為給定的相位數(shù)據(jù)被定義而且該如何定義它在這里沒被闡述。在這里只闡述了傾斜最小方波的展開問題相關(guān)問題的集中率。</p><p>　　對(duì)這一個(gè)問題的最小方波的解決辦法產(chǎn)生下列的等式:</p><p><b>　　(4)</b></p><p>　　傾斜相位被拉普拉斯定義為下式：</p><p>

54、;<b>　　(5)</b></p><p>　　展開的方法已經(jīng)在下面的公式包括：</p><p><b>　?。?）</b></p><p>　　公式(4) 是Poisson 的部分微分方程式 (PDE) 的傾斜和不連續(xù)譯本, .上述一連串不同的在M×N×1的空間中轉(zhuǎn)變?yōu)椋陨系牡仁娇梢员磉_(dá)為一個(gè)線性

55、等式：</p><p><b>　　(7)</b></p><p>　　其中系統(tǒng)點(diǎn)陣式A為 K × K(K=MN)矩陣而且是總稱矢量 , 也就是說 ,最小方波展開問題的相位能通過解決線性系統(tǒng)被獲得, 給定的A和,是從傾斜數(shù)組和標(biāo)準(zhǔn)相位定義被展開包裝的相位得到的, 。但是A是一個(gè)非常大的點(diǎn)陣式, 直接的倒轉(zhuǎn)操作實(shí)際不可能。系統(tǒng)點(diǎn)陣式A的結(jié)構(gòu)是非常稀疏而大部份的

56、對(duì)角線的元素是零,公式(4)就能證實(shí)。</p><p>　　直接方法是以快速的傅立葉變換（FFT）或者不連續(xù)的余弦變換 (DCT) 為基礎(chǔ)，被應(yīng)用以解決非傾斜相位的展開問題。然而，在傾斜的情形下，反復(fù)的方法應(yīng)該被采用。解決線性系統(tǒng)的傳統(tǒng)的反復(fù)方法是高斯西頓的釋放,根據(jù)公式(6)簡(jiǎn)單重復(fù)計(jì)算，直到它聚合。然而，這一個(gè)方法由于它的集中速度極端地慢而不實(shí)際的, 而這些由系統(tǒng)點(diǎn)陣式 A的稀疏特性引起的。一些數(shù)字的運(yùn)算法則

57、比如事先具備條件結(jié)合傾斜度 (PCG),或多格子方法被應(yīng)用于實(shí)現(xiàn)傾斜的最小方波逐步運(yùn)行展開。 PCG 方法在非傾斜的相位展開問題快速地聚合打開問題或者傾斜問題不有大的相位斷絕的問題。 然而，在數(shù)據(jù)上，大的相位斷絕問題需要許多重復(fù)聚合。在解決線性系統(tǒng)的問題方面，多格子方法是一個(gè)有效率的運(yùn)算法則，而且在解決最小方波方面逐步運(yùn)行展開問題方面也比高斯西頓的方法和 PCG 方法好的多。然而，在傾斜的情形下，方法需要另外的傾斜限制,這操作非常復(fù)雜而

58、且它在相關(guān)的文獻(xiàn)中被適當(dāng)?shù)卦O(shè)計(jì)，然而在限制期間可能有一些錯(cuò)誤。</p><p>　　除了這些方法外,還有其他的方法更有效率地解決一個(gè)稀疏的線性 系統(tǒng)問題 , 一個(gè)系統(tǒng)以較好的集中情況轉(zhuǎn)換成另外的一個(gè)相等的系統(tǒng)。系統(tǒng)的集中速度是以系統(tǒng)點(diǎn)陣式 A為特點(diǎn)。最小方波的展開問題的非常稀疏的系統(tǒng)點(diǎn)陣式A的結(jié)構(gòu)是很有特色的。在解決反復(fù)方法中,節(jié)的變數(shù)之間的當(dāng)?shù)剡B結(jié)慢地下來在重復(fù)中的解決的進(jìn)步而且造成低的集中率。 換句話說，高斯

59、西頓的方法提取來自每個(gè)節(jié)的價(jià)值的只有四個(gè)鄰居的表面的當(dāng)?shù)馗咧懿〝?shù)據(jù)。 因此，整體的低頻率的表面數(shù)據(jù)非常慢慢地繁殖,它才是稀疏問題的低集中率的主要理由。低頻率部分支配問題在最小方波展開問題的計(jì)算速度問題是最主要的,而且獲得一個(gè)快速的集中率, 問題的低頻率部分應(yīng)該被提取。這一個(gè)觀念以多分辨率為基礎(chǔ)，在該觀念中一個(gè)信號(hào)就代表不同的頻率，例如粗糙而精細(xì)的頻帶。分開地解決低頻率的部分將會(huì)加速全部的系統(tǒng)集中率。</p><p&g

60、t;　　小波變換是在多頻率觀念中表現(xiàn)一個(gè)最復(fù)雜的系統(tǒng)的方法。在本文中 ,提出了一個(gè)解決的最小方波展開問題的有效方法,利用不連續(xù)的小浪轉(zhuǎn)換 (DWT).這是在文獻(xiàn)中呈現(xiàn)的工作的擴(kuò)展。在部分微分方程式的解決能在被發(fā)現(xiàn)的小波達(dá)成的方式的領(lǐng)域的工作上的一些文學(xué). 那些研究處理在小浪領(lǐng)域中的 PDE 結(jié)構(gòu)本身有效率地解決問題。 然而，本文應(yīng)用小波變換改革被從 PDE 吸取 , 而且不處理 PDE 問題本身的線系統(tǒng)的結(jié)構(gòu)。</p>&

61、lt;p>　　小波變換進(jìn)行二重的程序分解(分析)和重建(綜合)。在分解程序，一個(gè)信號(hào)被分為它的低頻部分(細(xì)節(jié)).小浪系數(shù)的合量組是最初信號(hào)的多頻率信號(hào)。最多接近的成份位于最低的決議水平，而且其他的水平有對(duì)應(yīng)細(xì)節(jié)成份 ,最初的信號(hào)被藉由綜合這些在重建程序恢復(fù)接近并且細(xì)說成份。</p><p><b>　　結(jié)論：</b></p><p>　　一個(gè)有效解決傾斜的二維最

62、小方波的展開問題的方法已經(jīng)被提出。雙正交的小波變換被應(yīng)用于在小波領(lǐng)域中被分解的低頻的和高頻的部分和原始系統(tǒng)轉(zhuǎn)變?yōu)樾碌牡韧到y(tǒng)之內(nèi)。分別地解決新系統(tǒng)的低頻部分加速系統(tǒng)全部的集中率。集中率的增強(qiáng)已經(jīng)被實(shí)驗(yàn)用一些綜合性的狀態(tài)圖像顯示。</p><p>　　被提議的方法得到了較好的結(jié)果勝于使用高斯-西頓的松弛和多格子方法被獲得的那些結(jié)果。這一個(gè)方法的另一個(gè)優(yōu)點(diǎn)是新的系統(tǒng)對(duì)最初的點(diǎn)陣式算術(shù)相等。因此這種解決方法和精確解決傾