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  本文選題：超聲層析成像 + 不適定��； 參考：《中北大學》2017年碩士論文

【摘要】：超聲層析成像技術是利用介質(zhì)外部接收到的散射波數(shù)據(jù),依照一定的物理和數(shù)學關系對介質(zhì)內(nèi)部結構進行反演的一種技術。本文由連續(xù)介質(zhì)中超聲波傳播的波動理論,推導出超聲波穿過被測介質(zhì)時的前向散射方程以及逆散射方程,以此來反演物體內(nèi)部結構。運用迭代算法解決逆散射方程的非線性問題,對于迭代過程中逆散射方程的不適定問題,則引入正則化方法進行處理。第一類方法為直接正則化方法,適用于解中、小型線性離散不適定系統(tǒng)。此類方法通常借助于矩陣分解,其中最常用的為奇異值分解法(SVD),因為該分解處理簡潔且分解數(shù)值穩(wěn)定。TSVD和TTLS是基于SVD分解的較為流行的正則化方法,在求解過程中舍棄系數(shù)矩陣中較小的奇異值,保留問題的可靠部分。這兩種方法與經(jīng)典的Tikhonov正則化方法相比具有不需要先驗信息、正則化參數(shù)選取方便等優(yōu)點。文中將TSVD正則化方法和Tikhonov-Gaussian正則化方法結合,對TSVD正則化方法進行改進。改進的TSVD方法的主要思想是:引入截斷參數(shù)將系數(shù)矩陣分為較大奇異值和較小奇異值,即可靠部分和不可靠部分,再利用Tikhonov-Gaussian方法只對問題的不可靠部分進行修正。這樣既抑制了小奇異值對數(shù)據(jù)端噪聲的放大作用,又避免了模型的可靠部分受到修正的影響。第二類方法為迭代正則化方法,此類方法在處理不適定問題時可減少計算量,加快運算的速度。對大規(guī)模的線性離散不適定系統(tǒng),這類方法是一個不錯的選擇。CGLS和LSQR是兩種常用的Krylov子空間方法。CGLS方法實質(zhì)是應用共軛梯度法來求解原問題的法方程。LSQR方法則是用Lanczos雙對角化方法求解原問題的法方程�？紤]到CGLS方法的半收斂的特性,文中對CGLS方法進行了改進。通過適當?shù)男拚蜃幼饔糜跉埐钕蛄?在CGLS迭代中通過平衡殘差達到抑制殘差中噪聲擴散的目的,進而克服原CGLS方法半收斂現(xiàn)象,得到更好的重建效果。通過實驗仿真以及結果分析得到:(1)總體而言迭代正則化方法收斂速度快于直接正則化方法,且對模型的擬合程度好于直接正則化方法。(2)TSVD、改進的TSVD和TTLS方法都能實現(xiàn)逆散射問題的正則化處理。其中改進的TSVD方法最逼近原問題的真實解,TSVD方法次之,TTLS方法最差。(3)CGLS方法和LSQR方法具有相似的數(shù)值結果,LSQR方法的存儲量小于CGLS方法,計算量大于CGLS方法,且具有更好的數(shù)值穩(wěn)定性。(4)改進的CGLS方法在沒有明顯增加計算量和存儲量的前提下克服半收斂現(xiàn)象,數(shù)值穩(wěn)定性和數(shù)據(jù)擬合程度好于CGLS和LSQR方法。綜上所述,上述幾種正則化方法都可以在散射比較強的情況下,實現(xiàn)對比度為20%的被測物體的內(nèi)部結構的反演重建,且獲得了良好的仿真結果。
[Abstract]:Ultrasonic tomography is a technique for retrieving the internal structure of the medium according to the physical and mathematical relations by using the scattering wave data received from the external media. Based on the wave theory of ultrasonic wave propagation in continuous medium, the forward scattering equation and inverse scattering equation of ultrasonic wave passing through the measured medium are derived in this paper, so as to invert the internal structure of the object. The nonlinear problem of inverse scattering equation is solved by iterative algorithm. The regularization method is introduced to deal with the ill-posed problem of inverse scattering equation in iterative process. The first method is a direct regularization method, which is suitable for small linear discrete-time ill-posed systems. This kind of method usually relies on matrix decomposition, the most commonly used method is the singular value decomposition (SVD), because the decomposition is concise and the decomposition is numerically stable. TSVD and TTLS are popular regularization methods based on SVD decomposition. The smaller singular value in the coefficient matrix is abandoned and the reliable part of the problem is preserved. Compared with the classical Tikhonov regularization method, these two methods have the advantages of no prior information and convenient selection of regularization parameters. In this paper, the TSVD regularization method is combined with Tikhonov-Gaussian regularization method to improve the TSVD regularization method. The main idea of the improved TSVD method is to introduce truncation parameters to divide the coefficient matrix into larger singular values and smaller singular values, which can be divided into partial and unreliable parts, and then the Tikhonov-Gaussian method is used to modify only the unreliable part of the problem. This not only restrains the amplification effect of the small singular value on the data end noise, but also avoids the correction of the reliable part of the model. The second kind of method is iterative regularization method, which can reduce the computational cost and speed up the operation when dealing with ill-posed problems. For large scale linear discrete ill-posed systems, This kind of method is a good choice. CGLS and LSQR are two commonly used Krylov subspace methods. CGLS method is essentially a normal equation using conjugate gradient method to solve the original problem. The LSQR method is a normal equation using Lanczos double diagonalization method to solve the original problem. Considering the semi-convergence of CGLS method, the CGLS method is improved in this paper. By applying the appropriate correction factor to the residual vector and balancing the residual error in the CGLS iteration, the noise diffusion in the residual is restrained, and the semi-convergence phenomenon of the original CGLS method is overcome, and a better reconstruction result is obtained. By experimental simulation and result analysis, it is found that the convergence speed of the iterative regularization method is faster than that of the direct regularization method. The fitting degree of the model is better than that of the direct regularization method. The improved TSVD and TTLS methods can be used to regularize the inverse scattering problem. The improved TSVD method approximates the real solution of the original problem. The TSVD method is the worst. The TTLS method is the worst, and the LSQR method has similar numerical results. The LSQR method has less memory than the CGLS method, and the computational complexity is greater than that of the CGLS method. And the improved CGLS method has better numerical stability and better fitting degree than the CGLS and LSQR methods without increasing the computation and storage capacity obviously, and the numerical stability is better than that of the CGLS and LSQR method, and the numerical stability and the data fitting degree are better than that of the CGLS and LSQR methods. In conclusion, the above regularization methods can be used to reconstruct the internal structure of the measured object with a contrast of 20% under strong scattering, and good simulation results are obtained.
【學位授予單位】：中北大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：TP391.41;TB559

【參考文獻】

相關期刊論文 前10條

1 曾小牛;劉代志;李夕海;蘇娟;陳鼎新;齊瑋;;一種改進的病態(tài)問題奇異值修正法[J];武漢大學學報(信息科學版);2015年10期

2 閔濤;胡剛;晏立剛;;第一類弱奇異核Fredholm積分方程的數(shù)值求解[J];應用泛函分析學報;2014年02期

3 王倩;戴華;;求解離散不適定問題的正則化GMERR方法[J];計算數(shù)學;2013年02期

4 謝親;李自強;閆士舉;;基于C型臂超短掃描路徑錐束投影的圖像合成[J];中國組織工程研究;2013年04期

5 張培;王浩全;李媛;;超聲CT層析成像方法[J];信息通信;2011年03期

6 向洪平;葛建芳;張藍月;;超聲波輔助萃取功能性天然色素的研究與應用進展[J];江蘇農(nóng)業(yè)科學;2010年03期

7 王在華;劉杰;周龍泉;王海濤;;天山中部地殼及上地幔三維速度層析成像[J];內(nèi)陸地震;2008年03期

8 史曉倫;;波動理論的實驗設計[J];實驗室科學;2008年03期

9 溫君芳;馮玉田;仇清海;;超聲層析成像的迭代重建[J];聲學技術;2008年03期

10 李萬萬;裴正林;;井間地震彈性波傳播特征數(shù)值模擬[J];物探與化探;2008年02期

，

本文編號：2017236