参数化的超声回波模型及其参数估计
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摘要
在无损检测中，通过换能器发射超声脉冲照射目标，并接收目标的反向散射回波来完成对目标的探测。通过对获得的目标回波进行分析，确定反射目标的物理特性如几何形状、尺寸以及传输路径上的各种信息，因此精确估计回波信号的参数至关重要。然而，在实际中，由于材料结构的复杂性和噪声对目标回波信号的影响，所接收的回波信号总会发生一定的变化，而且，随着超声射入目标的深度增加，回波衰减加重。另外，超声回波的多层材料检测技术，主要是应用层与层之间的时延来估计层的厚度，但随着层厚度减小，不同层之间的反射回波会发生重叠，造成时间测量困难和误差。现存的很多算法在低信噪比时估计精度较差，计算量较大，并且在低信噪比时性能与克拉美罗（CRLB）界仍存在一定的距离，尤其是对重叠的多回波信号的参数估计，误差较大。因而在中低信噪比时，性能逼近CRLB界且计算量小的超声回波信号参数估计算法，尤其是重叠的多回波信号的参数估计是研究的重点和难点。
本文研究的算法旨在提高算法的抗噪声特性和估计精度，尤其是对重叠的多回波信号的估计精度，使之在低信噪比情况下性能贴近CRLB界，并取得以下一些创新性成果：
1）提出了一种基于包络的超声回波信号的参数估计算法，并详细推导了加性高斯白噪声对算法性能的影响，得到了关于高斯信号的CRLB界的理论表达式。由于高斯信号的包络包含了无损检测中重要的因素——传播时间（TOF），因此可以只是通过包络来估计TOF，这不仅提高精度，还可以降低计算量。为此，文中首先通过希尔伯特变换来抽取回波信号的包络，研究发现噪声对包络的提取影响很大，因此引入小波去噪的方法，提高提取包络的精度，最后用改进的高斯牛顿法求解关于信号包络参数的最小二乘估计问题以达到估计TOF的目的。在对多回波信号的参数估计中，采用了空间交替广义期望最大(SAGE)算法，以提高运算速度。文中深入讨论了算法对噪声的鲁棒性。与经典算法相比，本文所提的算法估计精度高，并且运算量低，也适用于信噪比低的情况。算法的弊端是当两个回波重叠很严重时，无法有效估计两个回波的TOF。
2）根据Gabor变换时频分析的特点，提出了Gabor变换域上超声回波信号参数估计方法。通过建立回波信号与Gabor变换分析窗函数相似度（即距离）模型，将模型相似度最小化问题转化为求解回波信号Gabor变换系数模的最大值来估计回波信号的传播时间（TOF）和中心频率(CF)。在估计超声回波信号的参数时，首先通过网格搜索方法粗略估计TOF和CF，然后用拟牛顿法迭代搜索得到精确的TOF和CF估计值，最后通过Gabor变换函数表达式推导出与TOF和CF相关的其他估计量，进而估计剩余参数。在估计多回波信号的参数时，最强的信号分量的参数先估计出来，接着估计次强的信号分量的参数，最后估计最弱的信号分量的参数。通过理论分析和仿真实验验证了本文所提算法的性能逼近CRLB界，且优于互相关算法和MUSIC算法。
3）基于分数阶傅立叶变换表达信号具有良好的时频聚集性，提出了一种基于分数阶傅立叶变换的超声回波信号参数估计算法。将重叠的多回波高斯信号的各个分量在分数域平面(α u)上的能量谱投影在分数域轴u后，通过最小化输出信号与参考信号之间的均方误差来确定最优变换阶α用于有效分解重叠的多回波高斯信号，最后利用互相关法估计各个分量的TOF。另外，详细讨论了分数阶域上的加窗问题。通过仿真实验验证了所提出的算法在分解多回波超声信号的有效性，以及在估计各个分量的TOF的性能优于经典的互相关法。
In nondestructive evaluation (NDE) testing, target can be detected by the transducertransmitting ultrasonic pulse and target backscattering echo. By inspecting the backscatteredechoes, the patterns of ultrasonic backscattered echoes represent valuable informationpertaining to the geometric shape, size, and orientation of the reflectors as well as themicrostructure of the propagation path. Accurate estimation of the ultrasonic echo pattern isessential. In practice, however, because of the complexity of the material structure and noiseon the influence of the echo signal, the received echo signal will be corrupted. Furthermore,the echo attenuation becomes more serious as the depth of the target increases. Multilayermaterials detection technology by ultrasonic echo aims at estimating the thickness of the layerby time delay. However, when the layer thickness is reduced, echoes of different layers willoverlap, and it will result in difficulties and errors for time measurement. Furthermore, theexisting methods still have performance gap compared with the CRLB when SNR is low.Especially for superimposed echoes, the error is larger. Thus, method for estimatingparameters of ultrasonic echoes which approach the CRLB when SNR is low are the keypoint in the study of parameter estimation.
In this study, algorithm of research on parameter estimation aims to improve the anti noisecharacteristic and the estimation precision, especially for superimposed echoes. The followinginnovative results are obtained.
1) A novel method for estimating parameters of ultrasonic echoes based on envelope isproposed. We also deduce the effects of White Gaussian Noise (WGN) on the performanceand CRLB of Gaussian signal. By considering only the envelope of the ultrasonic signalswhich contains Time-Of-Flight (TOF), the number of parameters can be reduced, thecomputational burden can be reduced and the estimation accuracy can be improved. Firstly,the Hilbert transform (HT) is used in ultrasonic signal processing in order to extract theenvelope of the echo and to reduce the computational burden. Then, the wavelet denoisingtechnique is applied to the extracted noisy envelope to improve the estimation accuracy.Finally, the echo parameters are estimated by using a Modified Gauss Newton (MGN) basednonlinear Least Squares (LS) estimation method. And, the Space Alternating GeneralizedExpectation-Maximization (SAGE) algorithm is adopted to estimate the parameters ofM-superimposed echoes. Compared with the other exsiting methods, the proposed method ismore efficient and is fit for the cases of low SNRs. The drawback of the proposed method isthat TOFs of two overlaping echoes can not be effectively estimated.
2) a novel method for parameter estimation based on the Gabor transform (GT) isproposed according to the characteristics of time-frequency analysis in Gabor transform. Thesimilarity (i.e. distance) for echo signal and the Gabor transform window function is modeled.Time of Flight (TOF) and Center Frequency (CF) of echo signal are estimated by translatingmodel for solving the minimum into solving the maximum of Gabor transform coefficientmodulus. Firstly, TOF and CF are estimated by grid search method. Then, Quasi-Newtonmethod is employed to improve the estimation accuracy and reduce computational complexity.Gabor transform function of the signal has an analytic expression which contributes to deriveother estimators relating to TOF and CF such as bandwidth factor, phase and amplitude. Theparameters of multi-component echo signal are estimated by estimating the parameters ofeach component successively. The Monte Carlo simulation and experimental results show thatthe proposed method outperforms the existing cross-correlation algorithm and MUSICalgorithm and approaches the CRLB.
3) The fractional Fourier transform (FRFT) is an effective technique to display thetime-frequency information of signals. A method for estimating parameters of ultrasonicechoes based on FRFT is proposed. Fractional energy spectrum of each componentoverlapping ultrasonic Gaussian echo signals on parameter plane (α u)is projected ontothe fractional domain u. In order to separate overlapping ultrasonic echo signals, the optimaltransform order is obtained by minimizing the mean square error (MSE) between the outputand the reference signal. TOF of each component is estimated by cross-correlation algorithm.Furthermore, windowing in FRFT domain is discussed. Numerical simulation results showthat the proposed method in separating signals overlapping in time and estimating TOF ofeach component outperforms the cross-correlation algorithm.
本文研究的算法旨在提高算法的抗噪声特性和估计精度，尤其是对重叠的多回波信号的估计精度，使之在低信噪比情况下性能贴近CRLB界，并取得以下一些创新性成果：
1）提出了一种基于包络的超声回波信号的参数估计算法，并详细推导了加性高斯白噪声对算法性能的影响，得到了关于高斯信号的CRLB界的理论表达式。由于高斯信号的包络包含了无损检测中重要的因素——传播时间（TOF），因此可以只是通过包络来估计TOF，这不仅提高精度，还可以降低计算量。为此，文中首先通过希尔伯特变换来抽取回波信号的包络，研究发现噪声对包络的提取影响很大，因此引入小波去噪的方法，提高提取包络的精度，最后用改进的高斯牛顿法求解关于信号包络参数的最小二乘估计问题以达到估计TOF的目的。在对多回波信号的参数估计中，采用了空间交替广义期望最大(SAGE)算法，以提高运算速度。文中深入讨论了算法对噪声的鲁棒性。与经典算法相比，本文所提的算法估计精度高，并且运算量低，也适用于信噪比低的情况。算法的弊端是当两个回波重叠很严重时，无法有效估计两个回波的TOF。
2）根据Gabor变换时频分析的特点，提出了Gabor变换域上超声回波信号参数估计方法。通过建立回波信号与Gabor变换分析窗函数相似度（即距离）模型，将模型相似度最小化问题转化为求解回波信号Gabor变换系数模的最大值来估计回波信号的传播时间（TOF）和中心频率(CF)。在估计超声回波信号的参数时，首先通过网格搜索方法粗略估计TOF和CF，然后用拟牛顿法迭代搜索得到精确的TOF和CF估计值，最后通过Gabor变换函数表达式推导出与TOF和CF相关的其他估计量，进而估计剩余参数。在估计多回波信号的参数时，最强的信号分量的参数先估计出来，接着估计次强的信号分量的参数，最后估计最弱的信号分量的参数。通过理论分析和仿真实验验证了本文所提算法的性能逼近CRLB界，且优于互相关算法和MUSIC算法。
3）基于分数阶傅立叶变换表达信号具有良好的时频聚集性，提出了一种基于分数阶傅立叶变换的超声回波信号参数估计算法。将重叠的多回波高斯信号的各个分量在分数域平面(α u)上的能量谱投影在分数域轴u后，通过最小化输出信号与参考信号之间的均方误差来确定最优变换阶α用于有效分解重叠的多回波高斯信号，最后利用互相关法估计各个分量的TOF。另外，详细讨论了分数阶域上的加窗问题。通过仿真实验验证了所提出的算法在分解多回波超声信号的有效性，以及在估计各个分量的TOF的性能优于经典的互相关法。
In nondestructive evaluation (NDE) testing, target can be detected by the transducertransmitting ultrasonic pulse and target backscattering echo. By inspecting the backscatteredechoes, the patterns of ultrasonic backscattered echoes represent valuable informationpertaining to the geometric shape, size, and orientation of the reflectors as well as themicrostructure of the propagation path. Accurate estimation of the ultrasonic echo pattern isessential. In practice, however, because of the complexity of the material structure and noiseon the influence of the echo signal, the received echo signal will be corrupted. Furthermore,the echo attenuation becomes more serious as the depth of the target increases. Multilayermaterials detection technology by ultrasonic echo aims at estimating the thickness of the layerby time delay. However, when the layer thickness is reduced, echoes of different layers willoverlap, and it will result in difficulties and errors for time measurement. Furthermore, theexisting methods still have performance gap compared with the CRLB when SNR is low.Especially for superimposed echoes, the error is larger. Thus, method for estimatingparameters of ultrasonic echoes which approach the CRLB when SNR is low are the keypoint in the study of parameter estimation.
In this study, algorithm of research on parameter estimation aims to improve the anti noisecharacteristic and the estimation precision, especially for superimposed echoes. The followinginnovative results are obtained.
1) A novel method for estimating parameters of ultrasonic echoes based on envelope isproposed. We also deduce the effects of White Gaussian Noise (WGN) on the performanceand CRLB of Gaussian signal. By considering only the envelope of the ultrasonic signalswhich contains Time-Of-Flight (TOF), the number of parameters can be reduced, thecomputational burden can be reduced and the estimation accuracy can be improved. Firstly,the Hilbert transform (HT) is used in ultrasonic signal processing in order to extract theenvelope of the echo and to reduce the computational burden. Then, the wavelet denoisingtechnique is applied to the extracted noisy envelope to improve the estimation accuracy.Finally, the echo parameters are estimated by using a Modified Gauss Newton (MGN) basednonlinear Least Squares (LS) estimation method. And, the Space Alternating GeneralizedExpectation-Maximization (SAGE) algorithm is adopted to estimate the parameters ofM-superimposed echoes. Compared with the other exsiting methods, the proposed method ismore efficient and is fit for the cases of low SNRs. The drawback of the proposed method isthat TOFs of two overlaping echoes can not be effectively estimated.
2) a novel method for parameter estimation based on the Gabor transform (GT) isproposed according to the characteristics of time-frequency analysis in Gabor transform. Thesimilarity (i.e. distance) for echo signal and the Gabor transform window function is modeled.Time of Flight (TOF) and Center Frequency (CF) of echo signal are estimated by translatingmodel for solving the minimum into solving the maximum of Gabor transform coefficientmodulus. Firstly, TOF and CF are estimated by grid search method. Then, Quasi-Newtonmethod is employed to improve the estimation accuracy and reduce computational complexity.Gabor transform function of the signal has an analytic expression which contributes to deriveother estimators relating to TOF and CF such as bandwidth factor, phase and amplitude. Theparameters of multi-component echo signal are estimated by estimating the parameters ofeach component successively. The Monte Carlo simulation and experimental results show thatthe proposed method outperforms the existing cross-correlation algorithm and MUSICalgorithm and approaches the CRLB.
3) The fractional Fourier transform (FRFT) is an effective technique to display thetime-frequency information of signals. A method for estimating parameters of ultrasonicechoes based on FRFT is proposed. Fractional energy spectrum of each componentoverlapping ultrasonic Gaussian echo signals on parameter plane (α u)is projected ontothe fractional domain u. In order to separate overlapping ultrasonic echo signals, the optimaltransform order is obtained by minimizing the mean square error (MSE) between the outputand the reference signal. TOF of each component is estimated by cross-correlation algorithm.Furthermore, windowing in FRFT domain is discussed. Numerical simulation results showthat the proposed method in separating signals overlapping in time and estimating TOF ofeach component outperforms the cross-correlation algorithm.
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