The application of regularization technique based on partial optimization in the nearfield acoustic holography

The regularization technique for stabilizing the reconstruction based on the nearfield acoustic holography(NAH) was investigated on the basis of the equivalent source method.In order to obtain higher regularization effect,a regularization method based on the idea of partial optimization was proposed,which inherits the advantages of the Tikhonov and another regularization method—truncated singular value decomposition(TSVD).Through the numerical simulation,it is proved that the proposed method is stabler than the Tikhonov,and more precise than the TSVD.Finally the validity and the feasibility of the proposed method are demonstrated by an experiment carried out in a semi-anechoic room with two speakers.     

半自由声场的全息重建和预测实验研究

半自由声场环境下的声源重建和声场预测研究对声全息技术走向实际应用具有非常重要的意义.在提出基于分布源边界点法的半自由声场全息重建和预测方法的基础上，对此展开了实验研究.并将重建和预测的结果与常规方法重建和预测的结果进行了比较和讨论，说明了重建预测过程中反射声压的影响和考虑反射声压的必要性，证明了所提出方法在解决半自由声场环境下存在地面反射时的声源重建和声场预测时的有效性和准确性.还提出了采用奇异值截断滤波和Tikhonov正则化方法来削弱测量误差的影响，从而进一步优化了重建结果，提高了全息成像的可信度. 关键词： 声全息 半自由场 边界点 声辐射 反射声     

Information operator approach and iterative regularization methods for atmospheric remote sensing

In this study, we present the main features of the information operator approach for solving linear inverse problems arising in atmospheric remote sensing. This method is superior to the stochastic version of the Tikhonov regularization (or the optimal estimation method) due to its capability to filter out the noise-dominated components of the solution generated by an inappropriate choice of the regularization parameter. We extend this approach to iterative methods for nonlinear ill-posed problems and derive the truncated versions of the Gauss-Newton and Levenberg-Marquardt methods. Although the paper mostly focuses on discussing the mathematical details of the inverse method, retrieval results have been provided, which exemplify the performances of the methods. These results correspond to the NO₂ retrieval from SCIAMACHY limb scatter measurements and have been obtained by using the retrieval processors developed at the German Aerospace Center Oberpfaffenhofen and Institute of Environmental Physics of the University of Bremen.     

A robust algorithm for high-resolution dynamic MRI based on the partially separable functions model

A recently developed partially separable functions (PSF) model can be used to generate high-resolution dynamic magnetic resonance imaging (MRI). However, this method could not robustly reconstruct high-quality MR images because the estimation of the PSF parameters is often interfered by the noise of the sampled MR data. To improve the robustness of MRI reconstruction using the PSF model, we proposed a new algorithm to estimate the PSF parameters by jointly using robust principal component analysis and modified truncated singular value decomposition regularization methods, instead of using the least square fitting method in the original PSF model. The experiment results of in vivo cardiac MRI demonstrated that the proposed algorithm can robustly reconstruct dynamic MR images with higher signal-to-noise ratio and clearer anatomical structures in comparison with the previous PSF model.     

A state space force identification method based on Markov parameters precise computation and regularization technique

A time domain force identification approach for linear system is proposed. This approach can found a highly precise force identification model within the scope of general computer precision while it does not cost much computing time. Although the force identification model is accurate, the force identification process, like other inverse methods, is still ill-posed due to the inversion process and the white noise in measured structural responses. The singular value decomposition is used to reveal the intrinsically matter of the ill-posedness of force identification problem and a regularization technique is utilized to deal with this issue. Finally, the proposed method with the aid of regularization technique is successfully applied to identify the input forces in two numerical simulations.     

联合波叠加法的全息理论与实验研究

当空间声场中同时存在多个相干声源时，运用常规近场声全息方法无法重建每个相干声源表面的声学信息，当然也无法预测每个声源单独产生的空间声场，相干声场的全息重建与预测已成为全息技术推广应用过程中亟待解决的问题.在提出联合波叠加法并将其应用于空间声场变换的基础上，对其进行了实验研究.通过对实际相干声场的全息重建与预测，验证了常规波叠加法在相干声场重建中的局限性、联合波叠加法在相干声场全息重建与预测过程的可行性和准确性，还研究了Tikhonov正则化方法在抑制声学逆问题的非适定性中的有效性和滤波系数的选择原则的可行性，以提高全息重建与预测的精度. 关键词： 近场声全息 联合波叠加 相干声场 Tikhonov正则化     

Compensation for discarded singular values in vibro-acoustic inverse methods

The paper addresses the inverse problem where source strengths are back-calculated from a sound pressure field sampled at several points. Regularization techniques, such as singular value discarding or Tikhonov regularization, are commonly used to improve estimates of source strength in such situations. However, over-regularization can result in even worse errors. A simple procedure is proposed here to compensate for errors of over-regularization. The basis is to constrain the solution such that the spatial mean of the measured and reconstructed sound pressure are equal. In other words, to set the overall sound power of the equivalent (calculated) sources equal to that of the real source. It is argued that the overall sound power is the most stable and reliable quantity on which to base source strength estimates. Examples of both singular value discarding and Tikhonov regularization are given.     

炉膛三维温度场重建中Tikhonov正则化和截断奇异值分解算法比较

在煤粉锅炉诊断中火焰辐射能图像扮演着越来越重要的角色, 通过电荷耦合器件(CCD)获得的辐射能图像可以重建出炉内火焰三维温度场, CCD 用于获取视场角内的辐射能图像. 温度场重建的矩阵方程是一个严重病态的方程, 本文使用两种算法(Tikhonov正则化算法和截断奇异值分解(TSVD)算法)来重建温度场. 应用广义交叉检验算法来选取正确的正则化参数. 数值模拟的环境为一个10 m×10 m×10 m的三维炉膛, 系统被划分为10×10×10的1000个网格, 每个网格单元都是边长为1 m的立方体. 在正问题求解所得到的CCD接受信号基础上加上不同随机误差以模拟测量时的CCD接受信号. 研究两种算法重建后的温度重建误差、两者的重建时间, 以及最高温度的重建效果. 初步的研究结果显示, 一般情况下基于Tikhonov算法重建的温度场比基于TSVD算法重建的温度场误差要小, 计算所需时间短, 最高温度重建更准确.     

Reconstruction of vibroacoustic fields in half-space by using hybrid near-field acoustical holography

In this paper we examine the accuracy and efficiency of reconstructing the vibroacoustic quantities generated by a vibrating structure in half-space by using hybrid near-field acoustic holography (NAH) and modified Helmholtz equation least squares (HELS) formulations. In hybrid NAH, we combine modified HELS with an inverse boundary element method (IBEM) to reconstruct a vibroacoustic field. The main advantage of this approach is that the majority of the input data can be regenerated but not measured, thus the efficiency is greatly enhanced. In modified HELS, we expand the field acoustic pressure in terms of outgoing and incoming spherical waves and specify the corresponding expansion coefficients by solving a system of equations obtained by matching the assumed-form solution to the measured acoustic pressure. Here the system of equations is ill conditioned and Tikhonov regularization is implemented through singular value decomposition (SVD) and the generalized cross-validation (GCV) method. Numerical examples of a dilating and oscillating spheres and finite cylinder are demonstrated. Test results show that hybrid NAH can yield a more accurate reconstruction than does a modified HELS, but a modified HELS is more efficient than is hybrid NAH [Work supported by NSF].     

字典奇异值分解加权压缩感知多径信号参数估计

为提高水声信道多径参数估计的分辨率,提出了一种基于字典奇异值分解的加权压缩感知算法。对于有源声呐,根据发射信号构造字典,对字典进行奇异值分解,利用大特征值对应的特征向量构造信号子空间,然后使用信号子空间对接收信号进行滤波。对滤波结果进行加权压缩感知参数估计,得出最终时延估计结果。仿真实验表明,所提方法能够对水声多径参数进行超分辨估计,适用于任何脉冲信号。湖试处理结果显示,混响背景下该方法也有较好的多径参数估计性能,能够降低接收数据的噪声成分,提高对水声信道的多径时延、个数和幅度的估计精度。      

A comparison of filter design structures for multi-channel acoustic communication systems

The application of inverse filter designs as a means of providing improved communication performance in acoustic environments is investigated. Tikhonov regularized inverse filters of channel transfer functions calculated in the frequency domain are used as a means of obtaining multi-channel filters. Three classifications of inverse filter structures have been considered using time-domain simulations. The performance of Tikhonov regularized inverse filters designed according to each of these classifications is compared with each other and against a filter design developed by Stojanovic [Stojanovic, M. (2005). "Retrofocusing techniques for high rate acoustic communications," J. Acoust. Soc. Am. 117, 1173-1185]. It is shown that the filter design developed by Stojanovic requires less regularization and outperforms the Tikhonov regularized inverse filter designs when communicating over a single channel. While the filter developed by Stojanovic is designed to use multiple transmitters to transmit to a single receiver, the filter was implemented in a multi-channel system and proposed to have a focusing similar to that obtained using time-reversal. It was found that for the scenario used in the simulation, the Tikhonov regularized inverse design for full multi-channel inversion achieved better focusing than the design by Stojanovic, where simulation results show 20 dB less cross-talk at the expense of around 2 dB loss in signal strength.     

心磁信号广义S变换域奇异值分解滤波方法

针对心磁信号工频及背景噪声干扰问题, 提出了广义S变换奇异值分解(singular value decomposition, SVD)滤波方法.在离散S变换基础上, 导出了广义矩阵S变换和逆变换公式. 通过对采样信号进行广义S变换, 调节时频分辨率, 利用SVD分解方法确定有效心磁信 号区域, 实现自适应时频滤波. 实验结果表明, 该方法能有效滤除工频及背景噪声干 扰, 且在较少奇异值个数情况下可获得更好的滤波性能. 关键词： 心磁信号 S变换 奇异值分解 时频滤波     

Numerical evaluation of linearized image reconstruction based on finite element method for biomedical photoacoustic imaging

An image reconstruction algorithm for biomedical photoacoustic imaging is discussed. The algorithm solves the inverse problem of the photoacoustic phenomenon in biological media and images the distribution of large optical absorption coefficients, which can indicate diseased tissues such as cancers with angiogenesis and the tissues labeled by exogenous photon absorbers. The linearized forward problem, which relates the absorption coefficients to the detected photoacoustic signals, is formulated by using photon diffusion and photoacoustic wave equations. Both partial differential equations are solved by a finite element method. The inverse problem is solved by truncated singular value decomposition, which reduces the effects of the measurement noise and the errors between forward modeling and actual measurement systems. The spatial resolution and the robustness to various factors affecting the image reconstruction are evaluated by numerical experiments with 2D geometry.     

The quantification of structure-borne transmission paths by inverse methods. Part 2: Use of regularization techniques

The inversion of an ill-conditioned matrix of measured data lies at the heart of procedures for the quantification of structure-borne sources and transmission paths. In an earlier paper the use of over-determination, singular value decomposition and the rejection of small singular values was discussed. In the present paper alternative techniques for regularizing the matrix inversion are considered. Such techniques have been used in the field of digital image processing and more recently in relation to nearfield acoustic holography. The application to structure-borne sound transmission involves matrices, which vary much more with frequency and from one element to another. In this study Tikhonov regularization is used with the ordinary cross-validation method for selecting the regularization parameter. An iterative inversion technique is also studied. Here a form of cross-validation is developed allowing an optimum value of the iteration parameter to be selected. Simulations are carried out using a rectangular plate structure to assess the relative merits of these techniques. Experiments are also performed to validate the results. Both techniques are found to give considerably improved results compared to singular value rejection.     

Sound source reconstruction using inverse boundary element calculations

Whereas standard boundary element calculations focus on the forward problem of computing the radiated acoustic field from a vibrating structure, the aim in this work is to reverse the process, i.e., to determine vibration from acoustic field data. This inverse problem is brought on a form suited for solution by means of an inverse boundary element method. Since the numerical treatment of the inverse source reconstruction results in a discrete ill-posed problem, regularization is imposed to avoid unstable solutions dominated by errors. In the present work the emphasis is on Tikhonov regularization and parameter-choice methods not requiring an error-norm estimate for choosing the right amount of regularization. Several parameter-choice strategies have been presented lately, but it still remains to be seen how well these can handle industrial applications with real measurement data. In the present work it is demonstrated that the L-curve criterion is robust with respect to the errors in a real measurement situation. In particular, it is shown that the L-curve criterion is superior to the more conventional generalized cross-validation (GCV) approach for the present tire noise studies.     

Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions

In this article, a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed, which allows the definition of a Tikhonov functional on a space of level-set functions. We provide convergence analysis for the Tikhonov approach, including stability and convergence results. Moreover, a numerical investigation of the proposed Tikhonov regularization method is presented. Newton-type methods are used for the solution of the optimality systems, which can be interpreted as stabilized versions of algorithms in a previous work and yield a substantial improvement in performance. The whole approach is focused on three dimensional models, better suited for real life applications.     

Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of the MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of the MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.     

Spatial resolution limits for the reconstruction of acoustic source strength by inverse methods

One method for deducing the strength of an acoustic source distribution from measurement of the radiated field involves the inversion of the matrix of frequency response functions relating the field measurement points to the strengths of a number of point sources used to represent the source distribution. In practice, the frequency response function matrix to be inverted may very often be ill-conditioned. This ill-conditioning will also often result in an ill-posed problem and thus regularization algorithms are used to produce reasonable solutions. For this purpose, Tikhonov regularization has been applied, and generalized cross-validation (GCV) has been introduced as an effective method for determining the proper amount of regularization without prior knowledge of either the source distribution or the contaminating errors. In the present work, the emphasis is placed on the relationship between the spatial resolution of the reconstructed source distribution and the small singular values of the frequency response function matrix to be inverted. However, the use of Tikhonov regularization often suppresses the effect of small singular values and these are in turn often associated with high spatial frequencies of the source distribution. Thus, the process of regularization produces a useful estimate of the acoustic source strength distribution but with a limited spatial resolution. Furthermore, in the field of Fourier acoustics, the spatial resolution of the reconstructed source distribution is usually limited by the wavelength of the radiation. This paper expresses the relationship between estimation accuracy, spatial resolution, noise-level and source/sensor geometry, when a range of inverse sound radiation problems are regularised using Tikhonov regularization with GCV. The results presented form the basis of guidelines that enable the reconstruction of acoustic source strength with a resolution that is finer than the intrinsic half-wavelength limit.     

基于奇异值分解和中位数绝对偏差的拉曼成像数据去噪方法

拉曼成像是一种无损伤、无需标记的光谱成像技术,它可以提供样品的不同组分的分子指纹信息以及空间分布特征,相比其他成像技术有着更重要的应用。但是拉曼散射的截面积小,灵敏度低,加上在很多实验中为了观察某些组分的动态分布而缩短扫描时间,导致最终得到的成像数据被噪声干扰,因此往往需要对信号进行去噪处理。常规的算法一般都是基于一个给定的数学模型对光谱进行处理,容易造成过滤波,使得信号失真;另外,在处理拉曼成像数据时,常规算法往往是对数据进行逐条光谱去噪,从而忽略了多条光谱之间的相互关系,导致最终的拉曼图像仍然受许多噪点干扰。因此,提出了一种基于奇异值分解和中位数绝对偏差的拉曼成像的信号处理方法,用于拉曼成像数据的去噪处理。该方法首先对拉曼成像数据进行奇异值分解,获得一个奇异值矩阵与两个正交矩阵;然后通过中位数绝对偏差法对奇异值矩阵中的各奇异值进行离群值检测,选取前k个被连续标记的离群值作为要保留的奇异值,并将其余的奇异值赋值为零,得到新的奇异值矩阵;最后用新的奇异值矩阵与两个正交矩阵重新求解得到去噪后的拉曼成像数据。实验中,首先验证了中位数绝对偏差法确定前k个奇异值的正确性,其次分别从处理后的图像质量和信号波形两方面对比了该算法与常规算法的去噪效果。结果证明,中位数绝对偏差法可以快速地确定出合理的k值大小,而且,依据该k值处理后的成像数据不仅在成像质量上消除了大量的噪点,使得组分的空间分布特征清晰可见,也在信号波形上较完美地保留了微小谱峰,并恢复光谱信号。该算法不同于常规算法,能同时对整个拉曼成像数据进行处理,并保留光谱之间的统计特征,是一种更加有效的拉曼成像数据的去噪方法。