独立分量分析算法及其在信号处理中的应用研究
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摘要
独立分量分析(ICA)是二十世纪九十年代发展起来的一种多元统计和计算技术,目的是用来分离或提取随机变量、观测数据或信号混合物中具有独立特性的隐藏分量。ICA可以看作是主分量分析(PCA)和因子分析(FA)的扩展。与PCA和FA相比,ICA是一种更强有力的技术。当PCA和FA等经典方法失效时,ICA仍然能从具有统计独立特性的观测信号中挖掘出支撑数据的内在分量或因子。对于通常是以大型样本数据库形式给出的多元观测数据,ICA定义了一个生成模型,该模型假设所观测到的数据变量是未知源信号的线性或非线性混合。事实上,ICA模型中原始的源信号和实现混合的系统都是未知的。ICA还假设那些潜在变量是非高斯的且相互独立,并称它们为观测数据的独立分量。这些独立分量也可以称作为源信号或因子,它们可以通过ICA相关方法分离或提取出来。
近年来,由于在语音处理、生物医学信号处理、图像特征提取和无线通信等领域潜在的影响力,基于ICA的盲源分离(BSS)和盲源提取(BSE)已经引起了社会各界高度的关注。许多科研机构都在致力于盲源分离/盲源提取方法的开发和应用,并已在ICA相关理论和应用中取得了很多有价值的研究成果。然而,ICA的研究目前尚处于发展阶段,ICA算法和应用中仍然存在若干尚未解决的问题,这就限制了ICA技术的发展和应用。总的来说,ICA技术仍然需要进一步加强和完善。
本文介绍了国内外ICA的发展历史、研究现状以及应用情况,阐述了ICA的理论基础,包括ICA的数学定义、基本假设以及相关的数学理论基础和实现途径等,并针对扩展ICA现存的几个问题。例如：对具有时间结构特性感兴趣信号的盲源提取、噪声环境下基于高斯矩和参考信号的盲源提取和基于感兴趣信号归一化峭度值范围的盲源提取等进行了比较深入的研究,提出了几个较为有效的算法。
本文的核心内容概括如下：
提出了一种针对源信号具有时间结构特性的基于极大似然估计技术的盲源提取算法。该算法可以有效地从线性混合的源信号混合物中提取出具有特定时间结构特性的感兴趣信号。基于时间结构特性的盲源提取(TBSE)可以看作是标准ICA的扩展。在生物医学信号测量中,很多感兴趣信号具有不同程度的周期特性。因此,TBSE将有非常广阔的应用空间。为了弥补现有的基于时间结构特性盲源提取算法的计算需求量大和提取精度低等缺陷,本文提出一种改良的基于源信号时间结构特性的盲源提取算法。
在实际应用中,传统的基于信号时间结构特性的盲源提取算法会遇到若干与观测数据有关的问题。例如：时间相关关系不能得到完全满足；尽管感兴趣信号在特定的时间滞延处有强烈的时间相关性,有时其它信号也会在该时间滞延处有较弱的相关性,其它信号甚至也会在该时间滞延处时间相关。因此,传统的基于信号时间结构特性的盲源提取算法所提取的信号经常混杂有其它不感兴趣的信号或者噪声。极大似然估计是统计估计领域中的一种流行的高阶统计(HOS)技术。如果源信号是非高斯的且具有时间相关特性,极大似然估计可以开发有效地盲源提取方法。该类算法可以从信号混合物中提取出潜在的信号,但由于局部最大化或算法随机初始化等因素的影响,基于极大似然估计的盲源提取算法常常收敛到某一个局部极大值,所提取的信号不能保证是感兴趣信号。
为了从测量到的源信号混合物中排他性地提取出感兴趣信号,本文提出一种基于源信号时间结构特性和极大似然估计技术的综合性盲源提取算法。整个提取过程分为两个阶段。第一阶段利用感兴趣信号的周期性信息从其线性混合物中提取出具有特定时间结构特性的信号。所提取的信号虽然逼近了感兴趣信号,但常混杂有若干其它信号甚至噪声。因此,该阶段只能看作是对感兴趣信号的粗略提取。第二阶段,基于源信号的统计独立特性,我们把第一阶段所提取的信号在极大似然估计框架下通过引进一个参数密度模型进行优化处理。所设计的指数密度函数束能与源信号的边际概率密度相匹配,因而可以对第一阶段所提取的信号在未知源信号概率密度分布情况下实施优化处理,从而提取出稳定有效的感兴趣信号。基于生物医学信号的计算机仿真实验验证了本文提出算法的有效性,与其它盲源提取算法的对比进一步说明了算法的可靠性和鲁棒性。
与传统的盲源分离方法相比,盲源提取具有许多优良特性,如计算负载少和处理速度快。因此,盲源提取广泛应用于解决源信号众多而感兴趣信号很少情况下的盲信号分离问题。在实际应用中,感兴趣信号总是被其它信号甚至噪声所干扰。例如：在现实世界中,许多测得的生物医学信号不但包含众多源信号而且感兴趣信号还常常被其它信号甚至噪声所污染。噪声经常会造成错误的临床诊断,有时甚至会造成死亡事件的发生。
作为一种重要的非高斯性量度,归一化峭度广泛用于设计解决盲源分离／盲源提取问题的目标函数。尽管在理论和应用上已经证明了该类目标函数的有效性,目前的基于归一化峭度的盲源提取方法大多是在无噪声环境下推导出来的,这在实际应用中是不现实的。近年来,学者们提出了几个从噪声环境下的信号混合物中根据归一化峭度提取感兴趣信号的方法,然而这些算法大都需要事先知道感兴趣信号的归一化峭度值。我们在现实世界中经常会碰到这样的情况：不能事先确定感兴趣信号准确的归一化峭度值,但可以事先获取到感兴趣信号归一化峭度所在的区间范围,且其它信号的归一化峭度值不在该区间范围内。到目前为止,尚没有相应的盲源提取算法能在噪声环境下使用该类区间范围作为前验信息提取出感兴趣信号。
本文首先设计出一个基于信号归一化峭度的目标函数,然后使用拉格郎日乘子法最大化该目标函数,进而构建出一个基于感兴趣信号归一化峭度值区间范围的盲源提取算法。只要事先获取到感兴趣信号归一化峭度值所在的区间范围,且其它信号的归一化峭度值不在该区间范围内,即使当多个信号的归一化峭度值非常接近,该算法也可以从噪声环境下具有统计独立特性的源信号混合物中提取出感兴趣信号。
在许多BSS/BSE应用中,人们经常可以事先获取到感兴趣信号的某些前验信息。例如：感兴趣信号的形态、相位、踪迹或发生时间等。这些前验信息是与感兴趣信号紧密相关的,如果它们携带的信息能够把感兴趣信号从观测到的信号混合物中有效区分出来,就称其为参考信号。总的来说,参考信号被认为是根据某一距离量度离感兴趣信号最近的信号。
近年来,学者们提出了若干基于参考信号的盲源提取算法。例如：Lu等人提出一种称作为ICA with reference(ICA-R)或constrained ICA(cICA)的盲源提取方法。ICA-R是通过最小化一个欠完备的目标函数和最大化利用参考信号中的前验信息而构建的。通过把部分前验信息以参考信号形式嵌入到著名的FastlCA算法中,ICA-R可以从大量的源信号混合物中提取出距离参考信号最近的感兴趣信号。作为一种经典地利用参考信号的盲源提取算法,ICA-R已经成功地应用到了功能磁共振成像(fMRI)处理领域中。然而,ICA-R在设计时并未考虑到噪声的存在。在很多情况下由于噪声污染的影响,算法的性能并不是很好。
参考信号携带着足够的前验信息能够从源信号混合物中排他性地区分出感兴趣信号。在实际应用中,感兴趣信号通常总是被各种噪声所污染。本文提出一种改进的基于参考信号的盲源提取算法。我们首先把参考信号作为限制性条件系统化地嵌入到一个适用于噪声数据的目标函数中,从而构建出一个限制性最优化问题,然后使用拉格郎日乘子法和梯度最优化技术求解该最优化问题,进而导出一个噪声环境下基于参考信号的盲源提取算法。计算机仿真实验验证了算法的有效性和可靠性。
Independent component analysis (ICA) developed in1990s is a multivariate statistical and computational technique. Its basic task is to separate or extract independently hidden components that underlie sets of random variables, measurements or signal mixtures. ICA can be considered an extension to principal component analysis (PCA) and factor analysis (FA). In contrast to PCA and FA, ICA is a more powerful technique, which can reveal the underlying components or factors of the observed data when these classical methods fail completely. ICA defines a generative model for the observed multivariate mixtures, which are often given as a large database of samples. In the ICA model, the observed data variables are assumed to be linear or nonlinear source signal mixtures. In fact, neither the original sources nor the mixing system is known in advance. The latent variables are called the independent components of the observed data, which are assumed non-Gaussian and mutually independent. These independent components, which are also called source signals or factors, can be separated or extracted by ICA methods.
Recently, blind source separation (BSS) and blind source extraction (BSE) on the basis of ICA have received much research attention due to their potential applications in the fields of speech processing, biomedical signal processing, image feature extraction and wireless telecommunication system, etc. Much effort has been devoted to the development and application of BSS/BSE methods. As a result, there are some progress in ICA theories and applications. However, ICA is still in an initial stage of development and many unsolved problems about its theory and application still exist, which restrict its development and application. In general, ICA technique needs to be further enhanced and improved.
In this dissertation, we first briefly introduce the development history and the current research status and applications of ICA both at home and abroad. Then some mathematical preliminaries in ICA technique are provided, including the mathematical definition of ICA, the assumptions made about ICA problems and the mathematical theories and methods currently used in ICA, etc. At last, some problems of extended ICA (for example, the blind extraction of desired temporally correlated signals, noisy component extraction on the basis of Gaussian moments and reference signals, and blind source extraction on the basis of the normalized kurtosis value range of the desired signal) are investigated deeply and several efficient algorithms are introduced.
The main works in this dissertation can be introduced as follows:
Based on the maximum likelihood estimation (MLE) technique and the assumption that the desired source signal is temporally correlated, a reliable method is proposed for blind extraction of temporally correlated signals from linear mixtures. The problem of blind source extraction for temporally correlated signals (TBSE) is an extension of standard ICA. Since majority of measurements obtained from biomedical applications exhibit some degree of periodicity, TBSE technique will have widely potential applications. The existing BSE methods for temporally correlated signal mixtures have many limitations such as computation-demanding and imprecise estimation. To help mitigate these limitations, we propose an improved TBSE method.
In practical applications, the conventional TBSE methods may have many problems associated with the measured recordings. For example, temporally correlated relations are not strictly satisfied. Although the desired signal is strongly temporally correlated at a specific time delay, sometimes it also weakly temporally correlates with other source signals or noise. Furthermore, some of other source signals may be also temporally correlated at the same time delay. Therefore, the extracted signal is often contaminated by some undesired signals or noise. Maximum likelihood estimation (MLE) is a fundamental technique of higher order statistics (HOS) estimation. If the source signals are not Gaussian and time dependent, MLE can be efficiently utilized to develop BSE methods. The MLE based BSE algorithm can extract the underlying signal from source signal mixtures. Due to local maximization and random initialization, the signal extracted by MLE based BSE algorithm often converges to a local maximum, which is not necessarily the desired one.
To extract the desired signal from the observed signal mixtures exclusively, we propose a hybrid algorithm by combining the TBSE technique and the MLE technique. The whole extraction procedure is divided into two stages. In the first stage, period property of the original source is employed to extract the desired signal from its linear mixtures. However, the extracted signal is often mixed with some undesired signals or noise, so this stage can only be considered to be a rough extraction process. In the second stage, further extraction is accomplished under a maximum likelihood framework by introducing a parametric density model and exploiting the statistical independence of the source signals. This model is constructed with an exponential power family of density functions. As these functions can adaptively match the source marginal probability densities, the signal obtained in the first stage can be further processed without any precise knowledge of the source probability distribution. As a result, the extracted signal is stable and efficient. Computer simulations on biomedical signals confirm the effectiveness of the proposed algorithm. Further comparison with other algorithms in existence verifies its reliability and robustness.
In contrast, BSE has many advantages over conventional BSS method such as the low computational load and fast processing speed. Therefore, BSE has been widely used to solve blind signal separation problem where there are a lot of source signals while only one or a few are desired. In practice, the desired signal is always contaminated by other signals or noise. For example, measured biomedical signals are seldom recorded in isolation and are almost certainly contaminated by other signals or noise. Noise often results in wrong clinical diagnosis. Sometimes incorrect interpretation of noise may lead to death.
As an important non-Gaussian measuring index, the normalized kurtosis has been wide utilized as the objective function for BSS/BSE problem. Despite of being theoretically well justified, vast majority of the existing normalized kurtosis based methods are conducted in noise-free environment, which is not realistic in practice. Recently, a few source extraction algorithms have been proposed on the basis of normalized kurtosis to extract a desired source signal from noisy mixtures. However, most of these algorithms need to obtain a specific normalized kurtosis value of the desired signal in advance. In real world, we often meet with cases that although the precise normalized kurtosis value of the desired signal cannot be obtained in advance, we are blessed with some foresight that the normalized kurtosis of the desired signal generally lies in a specific value range while other unwanted source signals do not belong to this range. By now, it seems that no existing BSE algorithms can be utilized for such cases in noisy environment.
A new objective function on the basis of normalized kurtosis is introduced in this dissertation. Maximizing this objective function and adopting Lagrange multiplier method, a robust BSE algorithm is developed which can extract the desired signal as the first output with coarse estimation of its normalized kurtosis value range. If one knows that the normalized kurtosis of the desired signal lies in a specific value range, whereas other unwanted signals do not belong to this range, he can extract the desired signal with the proposed algorithm from its mutually independent mixtures in noisy environment even if sometimes the normalized kurtosis values of different signals are very approximate.
In many BSS/BSE applications, especially for biomedical signal processing problems, one may obtain some prior information (e.g. the morphology, the phase, the trace and the occurrence time) about the desired signal in advance. If such prior information, which is closely related to the desired signal, carries enough valuable information to efficiently distinguish the desired signal from the observed signal mixtures, it is called the reference signal. In general, the reference signal is always considered to be the closest one to the desired signal in terms of a proper closeness measure.
Nowadays, several BSE methods have been developed by using the reference signal. Lu et al. proposed a good candidate called ICA with reference (ICA-R) or constrained ICA (cICA) for extracting a source signal from a large number of signal mixtures. ICA-R is constructed by minimizing the less-complete objective function and making the best of traces of the desired signal. By incorporating traces of the desired signal into the famous FastICA algorithm, ICA-R may extract the desired signal, which is the closest one to the reference signal. As a classical BSE algorithm for exploiting the reference signal, ICA-R has been sucessfully used in the field of functional Magnetic Resonance Imaging (fMRI). However, ICA-R does not take into account the existence of noise and cannot work well in many cases due to the existence of noise.
The reference signal carries enough prior information to distinguish the desired signal from signal mixtures exclusively. In practice, the desired signal is often contaminated by noise. An improved BSE algorithm on the basis of the reference signal is proposed for extraction of the desired signal from noisy measurements. We incorporate the reference signal as an additional constraint into a noisy objective function so as to form a constrained optimization problem. According to the Lagrange multiplier method and the gradient optimization technique, we develop a new BSE algorithm on the basis of the reference signal which can work well even in noisy environment. Computer simulations confirm the effectiveness and reliability of the proposed algorithm.
近年来,由于在语音处理、生物医学信号处理、图像特征提取和无线通信等领域潜在的影响力,基于ICA的盲源分离(BSS)和盲源提取(BSE)已经引起了社会各界高度的关注。许多科研机构都在致力于盲源分离/盲源提取方法的开发和应用,并已在ICA相关理论和应用中取得了很多有价值的研究成果。然而,ICA的研究目前尚处于发展阶段,ICA算法和应用中仍然存在若干尚未解决的问题,这就限制了ICA技术的发展和应用。总的来说,ICA技术仍然需要进一步加强和完善。
本文介绍了国内外ICA的发展历史、研究现状以及应用情况,阐述了ICA的理论基础,包括ICA的数学定义、基本假设以及相关的数学理论基础和实现途径等,并针对扩展ICA现存的几个问题。例如：对具有时间结构特性感兴趣信号的盲源提取、噪声环境下基于高斯矩和参考信号的盲源提取和基于感兴趣信号归一化峭度值范围的盲源提取等进行了比较深入的研究,提出了几个较为有效的算法。
本文的核心内容概括如下：
提出了一种针对源信号具有时间结构特性的基于极大似然估计技术的盲源提取算法。该算法可以有效地从线性混合的源信号混合物中提取出具有特定时间结构特性的感兴趣信号。基于时间结构特性的盲源提取(TBSE)可以看作是标准ICA的扩展。在生物医学信号测量中,很多感兴趣信号具有不同程度的周期特性。因此,TBSE将有非常广阔的应用空间。为了弥补现有的基于时间结构特性盲源提取算法的计算需求量大和提取精度低等缺陷,本文提出一种改良的基于源信号时间结构特性的盲源提取算法。
在实际应用中,传统的基于信号时间结构特性的盲源提取算法会遇到若干与观测数据有关的问题。例如：时间相关关系不能得到完全满足；尽管感兴趣信号在特定的时间滞延处有强烈的时间相关性,有时其它信号也会在该时间滞延处有较弱的相关性,其它信号甚至也会在该时间滞延处时间相关。因此,传统的基于信号时间结构特性的盲源提取算法所提取的信号经常混杂有其它不感兴趣的信号或者噪声。极大似然估计是统计估计领域中的一种流行的高阶统计(HOS)技术。如果源信号是非高斯的且具有时间相关特性,极大似然估计可以开发有效地盲源提取方法。该类算法可以从信号混合物中提取出潜在的信号,但由于局部最大化或算法随机初始化等因素的影响,基于极大似然估计的盲源提取算法常常收敛到某一个局部极大值,所提取的信号不能保证是感兴趣信号。
为了从测量到的源信号混合物中排他性地提取出感兴趣信号,本文提出一种基于源信号时间结构特性和极大似然估计技术的综合性盲源提取算法。整个提取过程分为两个阶段。第一阶段利用感兴趣信号的周期性信息从其线性混合物中提取出具有特定时间结构特性的信号。所提取的信号虽然逼近了感兴趣信号,但常混杂有若干其它信号甚至噪声。因此,该阶段只能看作是对感兴趣信号的粗略提取。第二阶段,基于源信号的统计独立特性,我们把第一阶段所提取的信号在极大似然估计框架下通过引进一个参数密度模型进行优化处理。所设计的指数密度函数束能与源信号的边际概率密度相匹配,因而可以对第一阶段所提取的信号在未知源信号概率密度分布情况下实施优化处理,从而提取出稳定有效的感兴趣信号。基于生物医学信号的计算机仿真实验验证了本文提出算法的有效性,与其它盲源提取算法的对比进一步说明了算法的可靠性和鲁棒性。
与传统的盲源分离方法相比,盲源提取具有许多优良特性,如计算负载少和处理速度快。因此,盲源提取广泛应用于解决源信号众多而感兴趣信号很少情况下的盲信号分离问题。在实际应用中,感兴趣信号总是被其它信号甚至噪声所干扰。例如：在现实世界中,许多测得的生物医学信号不但包含众多源信号而且感兴趣信号还常常被其它信号甚至噪声所污染。噪声经常会造成错误的临床诊断,有时甚至会造成死亡事件的发生。
作为一种重要的非高斯性量度,归一化峭度广泛用于设计解决盲源分离／盲源提取问题的目标函数。尽管在理论和应用上已经证明了该类目标函数的有效性,目前的基于归一化峭度的盲源提取方法大多是在无噪声环境下推导出来的,这在实际应用中是不现实的。近年来,学者们提出了几个从噪声环境下的信号混合物中根据归一化峭度提取感兴趣信号的方法,然而这些算法大都需要事先知道感兴趣信号的归一化峭度值。我们在现实世界中经常会碰到这样的情况：不能事先确定感兴趣信号准确的归一化峭度值,但可以事先获取到感兴趣信号归一化峭度所在的区间范围,且其它信号的归一化峭度值不在该区间范围内。到目前为止,尚没有相应的盲源提取算法能在噪声环境下使用该类区间范围作为前验信息提取出感兴趣信号。
本文首先设计出一个基于信号归一化峭度的目标函数,然后使用拉格郎日乘子法最大化该目标函数,进而构建出一个基于感兴趣信号归一化峭度值区间范围的盲源提取算法。只要事先获取到感兴趣信号归一化峭度值所在的区间范围,且其它信号的归一化峭度值不在该区间范围内,即使当多个信号的归一化峭度值非常接近,该算法也可以从噪声环境下具有统计独立特性的源信号混合物中提取出感兴趣信号。
在许多BSS/BSE应用中,人们经常可以事先获取到感兴趣信号的某些前验信息。例如：感兴趣信号的形态、相位、踪迹或发生时间等。这些前验信息是与感兴趣信号紧密相关的,如果它们携带的信息能够把感兴趣信号从观测到的信号混合物中有效区分出来,就称其为参考信号。总的来说,参考信号被认为是根据某一距离量度离感兴趣信号最近的信号。
近年来,学者们提出了若干基于参考信号的盲源提取算法。例如：Lu等人提出一种称作为ICA with reference(ICA-R)或constrained ICA(cICA)的盲源提取方法。ICA-R是通过最小化一个欠完备的目标函数和最大化利用参考信号中的前验信息而构建的。通过把部分前验信息以参考信号形式嵌入到著名的FastlCA算法中,ICA-R可以从大量的源信号混合物中提取出距离参考信号最近的感兴趣信号。作为一种经典地利用参考信号的盲源提取算法,ICA-R已经成功地应用到了功能磁共振成像(fMRI)处理领域中。然而,ICA-R在设计时并未考虑到噪声的存在。在很多情况下由于噪声污染的影响,算法的性能并不是很好。
参考信号携带着足够的前验信息能够从源信号混合物中排他性地区分出感兴趣信号。在实际应用中,感兴趣信号通常总是被各种噪声所污染。本文提出一种改进的基于参考信号的盲源提取算法。我们首先把参考信号作为限制性条件系统化地嵌入到一个适用于噪声数据的目标函数中,从而构建出一个限制性最优化问题,然后使用拉格郎日乘子法和梯度最优化技术求解该最优化问题,进而导出一个噪声环境下基于参考信号的盲源提取算法。计算机仿真实验验证了算法的有效性和可靠性。
Independent component analysis (ICA) developed in1990s is a multivariate statistical and computational technique. Its basic task is to separate or extract independently hidden components that underlie sets of random variables, measurements or signal mixtures. ICA can be considered an extension to principal component analysis (PCA) and factor analysis (FA). In contrast to PCA and FA, ICA is a more powerful technique, which can reveal the underlying components or factors of the observed data when these classical methods fail completely. ICA defines a generative model for the observed multivariate mixtures, which are often given as a large database of samples. In the ICA model, the observed data variables are assumed to be linear or nonlinear source signal mixtures. In fact, neither the original sources nor the mixing system is known in advance. The latent variables are called the independent components of the observed data, which are assumed non-Gaussian and mutually independent. These independent components, which are also called source signals or factors, can be separated or extracted by ICA methods.
Recently, blind source separation (BSS) and blind source extraction (BSE) on the basis of ICA have received much research attention due to their potential applications in the fields of speech processing, biomedical signal processing, image feature extraction and wireless telecommunication system, etc. Much effort has been devoted to the development and application of BSS/BSE methods. As a result, there are some progress in ICA theories and applications. However, ICA is still in an initial stage of development and many unsolved problems about its theory and application still exist, which restrict its development and application. In general, ICA technique needs to be further enhanced and improved.
In this dissertation, we first briefly introduce the development history and the current research status and applications of ICA both at home and abroad. Then some mathematical preliminaries in ICA technique are provided, including the mathematical definition of ICA, the assumptions made about ICA problems and the mathematical theories and methods currently used in ICA, etc. At last, some problems of extended ICA (for example, the blind extraction of desired temporally correlated signals, noisy component extraction on the basis of Gaussian moments and reference signals, and blind source extraction on the basis of the normalized kurtosis value range of the desired signal) are investigated deeply and several efficient algorithms are introduced.
The main works in this dissertation can be introduced as follows:
Based on the maximum likelihood estimation (MLE) technique and the assumption that the desired source signal is temporally correlated, a reliable method is proposed for blind extraction of temporally correlated signals from linear mixtures. The problem of blind source extraction for temporally correlated signals (TBSE) is an extension of standard ICA. Since majority of measurements obtained from biomedical applications exhibit some degree of periodicity, TBSE technique will have widely potential applications. The existing BSE methods for temporally correlated signal mixtures have many limitations such as computation-demanding and imprecise estimation. To help mitigate these limitations, we propose an improved TBSE method.
In practical applications, the conventional TBSE methods may have many problems associated with the measured recordings. For example, temporally correlated relations are not strictly satisfied. Although the desired signal is strongly temporally correlated at a specific time delay, sometimes it also weakly temporally correlates with other source signals or noise. Furthermore, some of other source signals may be also temporally correlated at the same time delay. Therefore, the extracted signal is often contaminated by some undesired signals or noise. Maximum likelihood estimation (MLE) is a fundamental technique of higher order statistics (HOS) estimation. If the source signals are not Gaussian and time dependent, MLE can be efficiently utilized to develop BSE methods. The MLE based BSE algorithm can extract the underlying signal from source signal mixtures. Due to local maximization and random initialization, the signal extracted by MLE based BSE algorithm often converges to a local maximum, which is not necessarily the desired one.
To extract the desired signal from the observed signal mixtures exclusively, we propose a hybrid algorithm by combining the TBSE technique and the MLE technique. The whole extraction procedure is divided into two stages. In the first stage, period property of the original source is employed to extract the desired signal from its linear mixtures. However, the extracted signal is often mixed with some undesired signals or noise, so this stage can only be considered to be a rough extraction process. In the second stage, further extraction is accomplished under a maximum likelihood framework by introducing a parametric density model and exploiting the statistical independence of the source signals. This model is constructed with an exponential power family of density functions. As these functions can adaptively match the source marginal probability densities, the signal obtained in the first stage can be further processed without any precise knowledge of the source probability distribution. As a result, the extracted signal is stable and efficient. Computer simulations on biomedical signals confirm the effectiveness of the proposed algorithm. Further comparison with other algorithms in existence verifies its reliability and robustness.
In contrast, BSE has many advantages over conventional BSS method such as the low computational load and fast processing speed. Therefore, BSE has been widely used to solve blind signal separation problem where there are a lot of source signals while only one or a few are desired. In practice, the desired signal is always contaminated by other signals or noise. For example, measured biomedical signals are seldom recorded in isolation and are almost certainly contaminated by other signals or noise. Noise often results in wrong clinical diagnosis. Sometimes incorrect interpretation of noise may lead to death.
As an important non-Gaussian measuring index, the normalized kurtosis has been wide utilized as the objective function for BSS/BSE problem. Despite of being theoretically well justified, vast majority of the existing normalized kurtosis based methods are conducted in noise-free environment, which is not realistic in practice. Recently, a few source extraction algorithms have been proposed on the basis of normalized kurtosis to extract a desired source signal from noisy mixtures. However, most of these algorithms need to obtain a specific normalized kurtosis value of the desired signal in advance. In real world, we often meet with cases that although the precise normalized kurtosis value of the desired signal cannot be obtained in advance, we are blessed with some foresight that the normalized kurtosis of the desired signal generally lies in a specific value range while other unwanted source signals do not belong to this range. By now, it seems that no existing BSE algorithms can be utilized for such cases in noisy environment.
A new objective function on the basis of normalized kurtosis is introduced in this dissertation. Maximizing this objective function and adopting Lagrange multiplier method, a robust BSE algorithm is developed which can extract the desired signal as the first output with coarse estimation of its normalized kurtosis value range. If one knows that the normalized kurtosis of the desired signal lies in a specific value range, whereas other unwanted signals do not belong to this range, he can extract the desired signal with the proposed algorithm from its mutually independent mixtures in noisy environment even if sometimes the normalized kurtosis values of different signals are very approximate.
In many BSS/BSE applications, especially for biomedical signal processing problems, one may obtain some prior information (e.g. the morphology, the phase, the trace and the occurrence time) about the desired signal in advance. If such prior information, which is closely related to the desired signal, carries enough valuable information to efficiently distinguish the desired signal from the observed signal mixtures, it is called the reference signal. In general, the reference signal is always considered to be the closest one to the desired signal in terms of a proper closeness measure.
Nowadays, several BSE methods have been developed by using the reference signal. Lu et al. proposed a good candidate called ICA with reference (ICA-R) or constrained ICA (cICA) for extracting a source signal from a large number of signal mixtures. ICA-R is constructed by minimizing the less-complete objective function and making the best of traces of the desired signal. By incorporating traces of the desired signal into the famous FastICA algorithm, ICA-R may extract the desired signal, which is the closest one to the reference signal. As a classical BSE algorithm for exploiting the reference signal, ICA-R has been sucessfully used in the field of functional Magnetic Resonance Imaging (fMRI). However, ICA-R does not take into account the existence of noise and cannot work well in many cases due to the existence of noise.
The reference signal carries enough prior information to distinguish the desired signal from signal mixtures exclusively. In practice, the desired signal is often contaminated by noise. An improved BSE algorithm on the basis of the reference signal is proposed for extraction of the desired signal from noisy measurements. We incorporate the reference signal as an additional constraint into a noisy objective function so as to form a constrained optimization problem. According to the Lagrange multiplier method and the gradient optimization technique, we develop a new BSE algorithm on the basis of the reference signal which can work well even in noisy environment. Computer simulations confirm the effectiveness and reliability of the proposed algorithm.
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