超声层析成像中正则化方法的研究

发布时间：2018-06-14 11:33

本文选题：超声层析成像 + 不适定　；参考：《中北大学》2017年硕士论文

【摘要】：超声层析成像技术是利用介质外部接收到的散射波数据,依照一定的物理和数学关系对介质内部结构进行反演的一种技术。本文由连续介质中超声波传播的波动理论,推导出超声波穿过被测介质时的前向散射方程以及逆散射方程,以此来反演物体内部结构。运用迭代算法解决逆散射方程的非线性问题,对于迭代过程中逆散射方程的不适定问题,则引入正则化方法进行处理。第一类方法为直接正则化方法,适用于解中、小型线性离散不适定系统。此类方法通常借助于矩阵分解,其中最常用的为奇异值分解法(SVD),因为该分解处理简洁且分解数值稳定。TSVD和TTLS是基于SVD分解的较为流行的正则化方法,在求解过程中舍弃系数矩阵中较小的奇异值,保留问题的可靠部分。这两种方法与经典的Tikhonov正则化方法相比具有不需要先验信息、正则化参数选取方便等优点。文中将TSVD正则化方法和Tikhonov-Gaussian正则化方法结合,对TSVD正则化方法进行改进。改进的TSVD方法的主要思想是:引入截断参数将系数矩阵分为较大奇异值和较小奇异值,即可靠部分和不可靠部分,再利用Tikhonov-Gaussian方法只对问题的不可靠部分进行修正。这样既抑制了小奇异值对数据端噪声的放大作用,又避免了模型的可靠部分受到修正的影响。第二类方法为迭代正则化方法,此类方法在处理不适定问题时可减少计算量,加快运算的速度。对大规模的线性离散不适定系统,这类方法是一个不错的选择。CGLS和LSQR是两种常用的Krylov子空间方法。CGLS方法实质是应用共轭梯度法来求解原问题的法方程。LSQR方法则是用Lanczos双对角化方法求解原问题的法方程。考虑到CGLS方法的半收敛的特性,文中对CGLS方法进行了改进。通过适当的修正因子作用于残差向量,在CGLS迭代中通过平衡残差达到抑制残差中噪声扩散的目的,进而克服原CGLS方法半收敛现象,得到更好的重建效果。通过实验仿真以及结果分析得到:(1)总体而言迭代正则化方法收敛速度快于直接正则化方法,且对模型的拟合程度好于直接正则化方法。(2)TSVD、改进的TSVD和TTLS方法都能实现逆散射问题的正则化处理。其中改进的TSVD方法最逼近原问题的真实解,TSVD方法次之,TTLS方法最差。(3)CGLS方法和LSQR方法具有相似的数值结果,LSQR方法的存储量小于CGLS方法,计算量大于CGLS方法,且具有更好的数值稳定性。(4)改进的CGLS方法在没有明显增加计算量和存储量的前提下克服半收敛现象,数值稳定性和数据拟合程度好于CGLS和LSQR方法。综上所述,上述几种正则化方法都可以在散射比较强的情况下,实现对比度为20%的被测物体的内部结构的反演重建,且获得了良好的仿真结果。
[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.
【学位授予单位】：中北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP391.41;TB559

【参考文献】