盲源分离的若干算法及应用研究
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摘要
盲源分离(BSS)在数字图像处理、语音信号处理、医学信号处理、地球信号处理、通信信号处理、遥感图像处理等等邻域取得了广泛的应用。在混合过程和源信号未知的前提下,盲源分离技术试图对传感器观测信号进行分解而得到了解混的源信号,这个看似不可思议的问题,在一定的假设下取得了巨大的成功,各种新颖而有效的解决方法层出不穷。本文围绕这个研究热点,做了以下研究工作：
1.第三章对文献中的盲源提取(BSE)算法进行分析。着重讨论了具有时间结构源信号的盲源提取问题,针对文献中的两种代表性算法进行了详细的分析,得出两点结论：①其中一个算法声称对加性高斯白噪声更鲁棒的说法是不对的,理论分析表明两种算法的抗噪性能是一致的；②两种算法是殊途同归的,即代价函数都可转化为观测信号的自相关函数。在此基础上我们得到几个更稳健的算法,仿真实验表明它们对时间结构信息的估计值更鲁棒,提取效果更好。
2.参考独立分量分析或约束独立分量分析是一类重要的盲源提取算法。但这类算法存在明显的不足：①算法耗时;②算法中一个关键的阈值参数对算法的性能影响很大,一旦设置不当的话算法就失效。文献中的多个改进版本同样存在这些不足。第四章针对参考独立分量分析方法提出了三个改进算法,这些算法较好地弥补了以上不足,仿真实验表明它们收敛都较快,耗时降低两个数量级,同时分离效果显著提高。
3.第五章研究基于广义特征值分解的盲源分离算法。着重讨论了基于线性预测的算法、基于可预测性的算法和"On blind source separation using generalized eigenvalues with a new metric"一文提出的算法。这三类算法都是通过优化广义Rayleigh商形式的代价函数来获得解混矩阵,在实际计算时把广义Rayleigh商的优化转化为相应的广义特征值分解,由特征向量构成解混矩阵。第三类算法的原文证明了最大化广义Rayleigh商能分离出一个源信号,但广义Rayleigh商其它的极值点和临界点是否一一对应呢?这是把用于解决盲源分离的广义Rayleigh商的优化转化为相应的广义特征值分解的基本前提,但我们的分析表明该文给出的证明不够严谨。本章给出了基于线性预测的三个类似的代价函数,其物理意义更明确。针对第三类算法也给出了一个算法实例,进而结合原文的工作给出了一个统一框架并加以证明。提出的框架包罗了通过优化广义Rayleigh商来解决盲源分离的所有算法,为最终用广义特征值分解来获得解混矩阵奠定了理论基础,也为构造Rayleigh商形式的代价函数提供了范本。
4.第六章研究了基于矩阵联合对角化的盲源分离算法。早期的算法基于正交联合对角化,但正交的约束对分离效果有一定的影响,所以近年来都研究非正交的联合对角化算法。记各个对角化处理后的矩阵为B,,衡量对角化程度的一个很自然的准则是：J(W)=∑off(Bi),式中off(B)表示矩阵B所有非对角元素的平方和。为了避免分离矩阵陷入奇异解或退化解,文献中的大量算法在以上简单的准则加入了一个约束项。在本章中我们也通过加入约束项得到了两个简单的非正交联合对角化算法。此外我们还提出一个新颖的对角化准则：J(W)=∑off(Bi)‖Bi‖,式中‖·‖为Euclidean范数。该准则同时考虑了对角元素和非对角元素,仿真实验表明算法能获得更好的对角化效果和盲源分离效果。
Blind source separation (BSS) has found many applications in digital image processing, speech signal processing, medical signal processing, geophysical signal processing, communication signal processing, remote sensing image processing, etc. When the mixing process and the original signals are unknown, BSS tries decomposing the observed sensor signals in order to obtain the unmixed source signals, as seems mysterious. However, given some assumptions, BSS has had great success and many novel and effective methods have been emerging. The thesis concerning this hot research area does some research work as follows:
(1) The blind source extraction (BSE) methods are analyzed in the third chapter. BSE for signal with temporal structure is focused on. After analysis in depth of two representative algorithms in the literature, two conclusions are reached:firstly, one of them claims that its method is more robust to the additive Gaussian white noise, as is incorrect. Our theoretical analysis shows that anti-noise performance of these two method are the same. Secondly, these two algorithms reach the same goal by different routes, and their cost functions can both be transformed to the autocorrelation function of the observed signals. Based on these points, a few more robust algorithms are obtained, which are more robust to the estimation of the temporal structure information and have better extraction results.
(2) Independent component analysis with reference (ICA-R) is an important method for BSE. However, our analysis indicates that it has two deficiencies. Firstly, it is time-consuming; secondly, its threshold parameter has an important effect on its performance, once its value is improperly set, it will fail. Many improved versions of ICA-R still have these drawbacks. Three improved algorithms for ICA-R are proposed in the fourth chapter, which completely avoid inherent drawbacks on ICA-R. Simulation experiments demonstrate their better performance.
(3) The fifth chapter is about BSS methods based on generalized eigen-value decomposition (GED). Three algorithms, which is based on linear prediction, temporal predictability, and is proposed in " On blind source separation using generalized eigenvalues with a new metric", respectively, are investigated. They all acquire the demixing matrix through optimizing some cost function with the form of generalized Rayleigh quotient (GRQ). The optimization is transformed to the corresponding GED, whose eigen-vectors constitutes the demixing matrix. The original article of the third algorithm proved that one source signal can be obtained by maximizing of the cost function with the form of GRQ. But the problem is: whether any other extreme point of GEQ corresponds to its critical point, respectively. It is the fundamental premise for the transformation from the optimization of GRQ for BSS to GED. Our analysis shows that the proof in the original article of the third algorithm is not rigorous. We give three cost functions based on linear prediction, whose meanings are much more clear. We also give an algorithm based on the third method above, then we give a unifying framework and prove it. The framework covers all BSS methods based on the optimization of some cost function with the form of GRQ, which provides a solid theoretical basis for methods with the help of GED, also an sample for building a cost function with the form of GRQ.
(4) Methods for BSS based on joint diagonalization (JD) of matrices are studied in the sixth chapter. Early algorithms are all based on orthogonal JD, but the constraint of orthogonality influences separation result, so in recent years all research is on the methods based on non-orthogonal JD. Denoting the diagonlized matrix as Bi whose its kl-th element is bkl, a very natural criterion for measurement of diagonalization extent is J(W)=∑off(Bi), where off(B)=∑k≠l(bkl)2 In order to avoid singular or degenerate solution, some constrained term is added to the aforementioned simple criterion. By this means, two simple non-orthogonal JD algorithms are proposed. Moreover, we give a novel JD criterion as follows:J(W)=∑off(Bi)/‖B‖, where‖.‖denote Euclidean norm. It simultaneously considers the diagonal elements and off- diagonal elements and its better diagonalization and BSS performance is validated by simulation experiments.
1.第三章对文献中的盲源提取(BSE)算法进行分析。着重讨论了具有时间结构源信号的盲源提取问题,针对文献中的两种代表性算法进行了详细的分析,得出两点结论：①其中一个算法声称对加性高斯白噪声更鲁棒的说法是不对的,理论分析表明两种算法的抗噪性能是一致的；②两种算法是殊途同归的,即代价函数都可转化为观测信号的自相关函数。在此基础上我们得到几个更稳健的算法,仿真实验表明它们对时间结构信息的估计值更鲁棒,提取效果更好。
2.参考独立分量分析或约束独立分量分析是一类重要的盲源提取算法。但这类算法存在明显的不足：①算法耗时;②算法中一个关键的阈值参数对算法的性能影响很大,一旦设置不当的话算法就失效。文献中的多个改进版本同样存在这些不足。第四章针对参考独立分量分析方法提出了三个改进算法,这些算法较好地弥补了以上不足,仿真实验表明它们收敛都较快,耗时降低两个数量级,同时分离效果显著提高。
3.第五章研究基于广义特征值分解的盲源分离算法。着重讨论了基于线性预测的算法、基于可预测性的算法和"On blind source separation using generalized eigenvalues with a new metric"一文提出的算法。这三类算法都是通过优化广义Rayleigh商形式的代价函数来获得解混矩阵,在实际计算时把广义Rayleigh商的优化转化为相应的广义特征值分解,由特征向量构成解混矩阵。第三类算法的原文证明了最大化广义Rayleigh商能分离出一个源信号,但广义Rayleigh商其它的极值点和临界点是否一一对应呢?这是把用于解决盲源分离的广义Rayleigh商的优化转化为相应的广义特征值分解的基本前提,但我们的分析表明该文给出的证明不够严谨。本章给出了基于线性预测的三个类似的代价函数,其物理意义更明确。针对第三类算法也给出了一个算法实例,进而结合原文的工作给出了一个统一框架并加以证明。提出的框架包罗了通过优化广义Rayleigh商来解决盲源分离的所有算法,为最终用广义特征值分解来获得解混矩阵奠定了理论基础,也为构造Rayleigh商形式的代价函数提供了范本。
4.第六章研究了基于矩阵联合对角化的盲源分离算法。早期的算法基于正交联合对角化,但正交的约束对分离效果有一定的影响,所以近年来都研究非正交的联合对角化算法。记各个对角化处理后的矩阵为B,,衡量对角化程度的一个很自然的准则是：J(W)=∑off(Bi),式中off(B)表示矩阵B所有非对角元素的平方和。为了避免分离矩阵陷入奇异解或退化解,文献中的大量算法在以上简单的准则加入了一个约束项。在本章中我们也通过加入约束项得到了两个简单的非正交联合对角化算法。此外我们还提出一个新颖的对角化准则：J(W)=∑off(Bi)‖Bi‖,式中‖·‖为Euclidean范数。该准则同时考虑了对角元素和非对角元素,仿真实验表明算法能获得更好的对角化效果和盲源分离效果。
Blind source separation (BSS) has found many applications in digital image processing, speech signal processing, medical signal processing, geophysical signal processing, communication signal processing, remote sensing image processing, etc. When the mixing process and the original signals are unknown, BSS tries decomposing the observed sensor signals in order to obtain the unmixed source signals, as seems mysterious. However, given some assumptions, BSS has had great success and many novel and effective methods have been emerging. The thesis concerning this hot research area does some research work as follows:
(1) The blind source extraction (BSE) methods are analyzed in the third chapter. BSE for signal with temporal structure is focused on. After analysis in depth of two representative algorithms in the literature, two conclusions are reached:firstly, one of them claims that its method is more robust to the additive Gaussian white noise, as is incorrect. Our theoretical analysis shows that anti-noise performance of these two method are the same. Secondly, these two algorithms reach the same goal by different routes, and their cost functions can both be transformed to the autocorrelation function of the observed signals. Based on these points, a few more robust algorithms are obtained, which are more robust to the estimation of the temporal structure information and have better extraction results.
(2) Independent component analysis with reference (ICA-R) is an important method for BSE. However, our analysis indicates that it has two deficiencies. Firstly, it is time-consuming; secondly, its threshold parameter has an important effect on its performance, once its value is improperly set, it will fail. Many improved versions of ICA-R still have these drawbacks. Three improved algorithms for ICA-R are proposed in the fourth chapter, which completely avoid inherent drawbacks on ICA-R. Simulation experiments demonstrate their better performance.
(3) The fifth chapter is about BSS methods based on generalized eigen-value decomposition (GED). Three algorithms, which is based on linear prediction, temporal predictability, and is proposed in " On blind source separation using generalized eigenvalues with a new metric", respectively, are investigated. They all acquire the demixing matrix through optimizing some cost function with the form of generalized Rayleigh quotient (GRQ). The optimization is transformed to the corresponding GED, whose eigen-vectors constitutes the demixing matrix. The original article of the third algorithm proved that one source signal can be obtained by maximizing of the cost function with the form of GRQ. But the problem is: whether any other extreme point of GEQ corresponds to its critical point, respectively. It is the fundamental premise for the transformation from the optimization of GRQ for BSS to GED. Our analysis shows that the proof in the original article of the third algorithm is not rigorous. We give three cost functions based on linear prediction, whose meanings are much more clear. We also give an algorithm based on the third method above, then we give a unifying framework and prove it. The framework covers all BSS methods based on the optimization of some cost function with the form of GRQ, which provides a solid theoretical basis for methods with the help of GED, also an sample for building a cost function with the form of GRQ.
(4) Methods for BSS based on joint diagonalization (JD) of matrices are studied in the sixth chapter. Early algorithms are all based on orthogonal JD, but the constraint of orthogonality influences separation result, so in recent years all research is on the methods based on non-orthogonal JD. Denoting the diagonlized matrix as Bi whose its kl-th element is bkl, a very natural criterion for measurement of diagonalization extent is J(W)=∑off(Bi), where off(B)=∑k≠l(bkl)2 In order to avoid singular or degenerate solution, some constrained term is added to the aforementioned simple criterion. By this means, two simple non-orthogonal JD algorithms are proposed. Moreover, we give a novel JD criterion as follows:J(W)=∑off(Bi)/‖B‖, where‖.‖denote Euclidean norm. It simultaneously considers the diagonal elements and off- diagonal elements and its better diagonalization and BSS performance is validated by simulation experiments.
引文
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