源信号自适应的独立成分分析算法应用与研究
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摘要
独立成分分析是信号处理领域在20世纪90年代所发展起来的一个新兴方向,它是解决盲源分离问题最为有效的方法之一.独立成分分析不仅在语音与图像处理方面有着广泛的应用,而且在心电图(electrocardiographic, ECG)、肌电图(electromyography, EMG)、脑电图(electroencephalograms, EEG)、脑磁图(magnetoencephalograms, MEG)等方面得到广泛应用.本文对独立成分分析算法及其应用进行了一些研究,提出了几种自适应独立成分分析算法,并将其应用到语音与图像去噪中.主要研究结果如下.
1.在分析了现有ICA算法优缺点的基础上,提出了通过求解梯度方程来解决ICA问题的方法.为求出梯度方程的根,提出了一种牛顿迭代算法,该算法不需要设置学习速率,结构简单,仅需求解一个矩阵方程就可通过迭代方法来得到梯度方程的根.为使算法对源信号具有自适应特性,我们使用非参数方法来估计源信号的统计特征,包括概率密度函数以及其一阶二阶导数.
2.为克服标准核密度估计方法对于大样本估计问题运算量大的缺点,提出了改进的核密度方法对源信号的概率密度及其一二阶导数进行自适应估计.该方法将源信号的直方图作为桥梁,直接估计出核函数的各个参数,使得当样本量较大时的算法速度得到了较大提高.
3.一般情况下, ICA中的解混合矩阵所在的参数空间并非欧式空间,而是黎曼空间,因此,传统梯度所得到的方向并非最速方向.为得到黎曼空间中的最速方向,我们从自然梯度角度(或相对梯度角度)出发,使用李群不变性这一准则,得到两种形式的自然梯度(或相对梯度),并以此为基础,得到估计方程,通过求解估计方程,并结合不同的自适应概率密度估计方法,得到自适应ICA算法.由于该方法采用自然梯度来获得最速方向,因此,算法具有superefficiency特性,能达到Fisher有效性.
4.提出一种自适应定点ICA算法,该方法引入白化预处理,使解混合阵具有正交约束,在此约束下,每次学习后得到的解混合阵必须经过重正交化,使其满足正交约束.该算法计算复杂度小,并且收敛速度快.
5.将独立成分分析算法应用到语音与图像去噪中.通过ICA算法得到含噪信号的独立分量,并在独立分量域中使用收缩算法对其进行去噪处理,然后经反变换得到去噪的声音或图像信号.
Independent component analysis (ICA) is a new direction in the signal pro-cessing fields which was developed in the 1990’s, and it is one of the most ef-ficient method for solving blind source separation (BSS) problems. ICA is notonly widely applied to the fields of sound signal processing and image process-ing, it has also demonstrated to be successful in various fields such as electro-cardiographic (ECG) processing, electromyography (EMG) processing, magneto-encephalograms (MEG) processing. This dissertation is devoted to the study ofICA algorithm and its application. Several adaptive ICA algorithms are proposed.At the same time, ICA is applied to denoise the sound signal and image data. Themain contributions of the dissertation are as follows.
1. After the analysis of the advantage and disadvantage of some traditionalICA algorithms, we propose an ICA algorithm through solving the gradi-ent equation. To solve the gradient equation, an iterative method based onNewton’s method is proposed where no learning rate is needed. Meanwhile,this method is very simple because the iterative equation can be obtained bymeans of solving a linear equation, which makes the algorithm very ease touse. At the same time, nonparametric density method is used to estimate theprobability density functions of the sources as well as their first and secondderivatives, which makes the algorithm adaptive to the source distributions.
2. A modified kernel density method is proposed to overcome the high com-putation cost of the standard kernel density method especially when thedata size is very large. This modified kernel density method utilize the his-tograms of the sources to directly estimated the parameters of the kernel func-tion. Thus the computation speed is faster than the standard kernel densitymethod especially when the data size is very large.
3. In general cases, the demixing matrix in ICA is not in the standard Euclideanspace but in the Riemannian space. Thus the tradition gradient direction is not the speedest direction. To obtain the speedest direction in Riemannianspace, we start from the viewpoint of natural gradient (or relative gradient)to derive two forms of natural gradient by means of invariant rule of theLie groups. Two forms of estimating equation are proposed based on thetwo forms of natural gradient. The source adaptive ICA algorithm can beobtained by solving the estimating equation iteratively where the probabilitydensity function is estimated adaptively. The algorithm based on solving theestimating equation has the property of superefficiency which can reach theFisher efficiency since the steepest direction is obtained by natural gradient.
4. After introducing the whitening pre-propessing, the demixing matrix mustbe orthogonal. Under this constraint, the learned demixing matrix must bere-orthogonized after each step, which leads to a fixed-point ICA algorithm.The computation complexity of the fixed-point algorithm is low, and the con-vergence speed is fast.
5. The ICA algorithm is applied to denoise the sound signal and image data. Theindependent components of the noisy signal is obtained by ICA algorithmand the noise is removed in the independent component domain, and thedenoised signal is obtained after the inverse transform.
1.在分析了现有ICA算法优缺点的基础上,提出了通过求解梯度方程来解决ICA问题的方法.为求出梯度方程的根,提出了一种牛顿迭代算法,该算法不需要设置学习速率,结构简单,仅需求解一个矩阵方程就可通过迭代方法来得到梯度方程的根.为使算法对源信号具有自适应特性,我们使用非参数方法来估计源信号的统计特征,包括概率密度函数以及其一阶二阶导数.
2.为克服标准核密度估计方法对于大样本估计问题运算量大的缺点,提出了改进的核密度方法对源信号的概率密度及其一二阶导数进行自适应估计.该方法将源信号的直方图作为桥梁,直接估计出核函数的各个参数,使得当样本量较大时的算法速度得到了较大提高.
3.一般情况下, ICA中的解混合矩阵所在的参数空间并非欧式空间,而是黎曼空间,因此,传统梯度所得到的方向并非最速方向.为得到黎曼空间中的最速方向,我们从自然梯度角度(或相对梯度角度)出发,使用李群不变性这一准则,得到两种形式的自然梯度(或相对梯度),并以此为基础,得到估计方程,通过求解估计方程,并结合不同的自适应概率密度估计方法,得到自适应ICA算法.由于该方法采用自然梯度来获得最速方向,因此,算法具有superefficiency特性,能达到Fisher有效性.
4.提出一种自适应定点ICA算法,该方法引入白化预处理,使解混合阵具有正交约束,在此约束下,每次学习后得到的解混合阵必须经过重正交化,使其满足正交约束.该算法计算复杂度小,并且收敛速度快.
5.将独立成分分析算法应用到语音与图像去噪中.通过ICA算法得到含噪信号的独立分量,并在独立分量域中使用收缩算法对其进行去噪处理,然后经反变换得到去噪的声音或图像信号.
Independent component analysis (ICA) is a new direction in the signal pro-cessing fields which was developed in the 1990’s, and it is one of the most ef-ficient method for solving blind source separation (BSS) problems. ICA is notonly widely applied to the fields of sound signal processing and image process-ing, it has also demonstrated to be successful in various fields such as electro-cardiographic (ECG) processing, electromyography (EMG) processing, magneto-encephalograms (MEG) processing. This dissertation is devoted to the study ofICA algorithm and its application. Several adaptive ICA algorithms are proposed.At the same time, ICA is applied to denoise the sound signal and image data. Themain contributions of the dissertation are as follows.
1. After the analysis of the advantage and disadvantage of some traditionalICA algorithms, we propose an ICA algorithm through solving the gradi-ent equation. To solve the gradient equation, an iterative method based onNewton’s method is proposed where no learning rate is needed. Meanwhile,this method is very simple because the iterative equation can be obtained bymeans of solving a linear equation, which makes the algorithm very ease touse. At the same time, nonparametric density method is used to estimate theprobability density functions of the sources as well as their first and secondderivatives, which makes the algorithm adaptive to the source distributions.
2. A modified kernel density method is proposed to overcome the high com-putation cost of the standard kernel density method especially when thedata size is very large. This modified kernel density method utilize the his-tograms of the sources to directly estimated the parameters of the kernel func-tion. Thus the computation speed is faster than the standard kernel densitymethod especially when the data size is very large.
3. In general cases, the demixing matrix in ICA is not in the standard Euclideanspace but in the Riemannian space. Thus the tradition gradient direction is not the speedest direction. To obtain the speedest direction in Riemannianspace, we start from the viewpoint of natural gradient (or relative gradient)to derive two forms of natural gradient by means of invariant rule of theLie groups. Two forms of estimating equation are proposed based on thetwo forms of natural gradient. The source adaptive ICA algorithm can beobtained by solving the estimating equation iteratively where the probabilitydensity function is estimated adaptively. The algorithm based on solving theestimating equation has the property of superefficiency which can reach theFisher efficiency since the steepest direction is obtained by natural gradient.
4. After introducing the whitening pre-propessing, the demixing matrix mustbe orthogonal. Under this constraint, the learned demixing matrix must bere-orthogonized after each step, which leads to a fixed-point ICA algorithm.The computation complexity of the fixed-point algorithm is low, and the con-vergence speed is fast.
5. The ICA algorithm is applied to denoise the sound signal and image data. Theindependent components of the noisy signal is obtained by ICA algorithmand the noise is removed in the independent component domain, and thedenoised signal is obtained after the inverse transform.
引文
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