欠定混合信号的盲分离
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摘要
盲信号分离是指在源信号及其混合传输信道未知情况下,仅仅利用观测到的混合信号来估计源信号。由于盲信号分离具有非常广泛的应用领域,如生物医学工程、语音增强、数字通信系统、图像处理、遥感、雷达和声纳等领域,从而成为现代信号处理领域研究的热点问题。以往的研究大多是基于观测信号个数必须大于源信号个数,而且一般假设源信号个数已知。然而,在实际应用中这些条件往往不能满足。本文围绕这些问题展开研究,重点研究源信号数目未知时欠定混合信号的盲分离问题,也就是观测信号个数少于源信号个数的盲信号分离问题。论文的主要工作概括如下:
1、在信号为稀疏信号的前提下,利用稀疏混合信号聚类特点,针对欠定稀疏混合信号盲分离问题,提出了拉普拉斯势函数法。在源信号个数未知条件下,对于充分稀疏混合信号,采用拉普拉斯势函数法来估计源信号个数和混合矩阵,再用线性规划法来估计源信号。对于不充分稀疏混合信号,则利用聚类平面势函数法先估计聚类平面的法线向量,再利用势函数法来估计聚类平面的交线,得到源数和混合矩阵的估计;然后,利用其混合信号子空间聚类特点,采用部分信号分离法来恢复源信号。为了减小噪声和异常值对势函数值估计的影响,与以往直接计算势函数值的方法不同,而是采用聚类算法来估计聚类中心势函数的大小,并对算法的鲁棒特性进行了分析。
2、在有噪声和源数未知条件下,针对欠定稀疏混合信号盲分离问题,提出了基于鲁棒竞争聚类学习算法来实现欠定混合信号盲分离。在源信号个数未知的条件下,对于充分稀疏混合信号,利用竞争聚类学习算法来估计聚类中心势的相对大小,利用聚类中心势的相对大小来估计源信号的个数以及混合矩阵,最后利用基追踪法来估计源信号。将鲁棒竞争聚类算法扩展用于不充分稀疏信号混合条件下的欠定盲分离问题,先利用鲁棒平面竞争聚类算法估计聚类平面的法线向量,再利用估计出的聚类平面,采用势函数法来估计聚类平面的交线向量及其个数,得到源信号个数和混合矩阵的估计。
3、在非负信号约束下,针对适定或过定线性瞬时混合模型,提出了基于最小相关约束的NMF方法来实现盲信号分离。该方法通过使分离信号的每行之间的相关最小来提高信号分离的效果。同时将最小相关约束的NMF方法扩展用于卷积混合信号盲分离。
4、在假定源信号为非负稀疏信号的基础上,提出一种tri-NMF算法来实现欠定混合信号盲分离,而一般情况下,常用的NMF算法不能用于欠定混合信号盲分离。该算法利用压缩采样技术原理,将一般非负矩阵分解转换为tri-NMF问题,并在目标函数中引入稀疏约束,提出一种新的乘法更新算法来实现欠定混合信号盲分离。
The goal of Blind Sources Separation (BSS) is to recover the original sources given only sensor observations that are unknown linear mixtures of the unobserved source signals. There are many potential exciting applications of blind sources separation in science and technology, especially in biomedical signal processing, wireless communication, audio and acoustics, image enhancement, sonar and radar systems; it has been one of the most active research areas in the modem signal processing. In the earlier research of the BSS, generally, most of the methods to BSS assume that the number of sources is known and the number of observed signals is not less than that of source signals. In practice, however, the number of the sources does not often hold and the number of mixing signals usually is less than that of sources. In this dissertation, we investigate the problem of underdetermined blind sources separation (UBSS) with an unknown source number, i.e. the number of observed signals is less than that of source signals. The primary contributions of the dissertation are summarized blow:
1. Based on the sparse mixture models, a potential function method based on the Lapulacial potential function is proposed for UBSS. When the source signals are assumed to be sparse sufficiently, the number of sources and mixing matrix can be obtained by estimating the local maximum of the Lapulacial potential function. When the source signals are sparse insufficiently, the normal vectors of the concentration hyperplanes can be obtained by estimating the local maximum of the potential function, and then the number of sources and mixing matrix can be estimated by finding the intersection of the concentration hyperplanes. In order to increase the robustness to the noise and the outlier, the clustering algorithm is exploited to estimate the local maximum of the potential function instead of directly estimating the local maximum of the potential function.
2. With unknown the number of sources, the Robust Competitive Agglomeration (RCA) algorithm is proposed to UBSS in the presence of noise. When the source signals are assumed to be sparse sufficiently, we can select the prototype parameters of clusters with larger cardinality as the estimation of mixing matrix’s column vectors and then we can restructure the mixing matrix. The number of column of the estimated mixing matrix can be regarded as the estimated number of the source signals. When the source signals are sparse insufficiently, the RCA algorithm can be extended to estimate the normal vectors of the concentration hyperplanes, and then the number of sources and mixing matrix can be also estimated by finding the intersection of the concentration hyperplanes.
3. When the sources are nonnegative, a novel algorithm is proposed for over or well determined BSS based on the Nonnegative Matrix Factorization (NMF) methods with the least correlated component constraints. The algorithm relaxes the source independence assumption and has low-complexity algebraic computations, and thus is computationally efficient. The algorithm can be also used to blind separation of convolutive mixed sources.
4. In generally, the problem of UBSS can not be solved by using the standard NMF methods. With the assumption of that the source signals are nonnegative and sparse; a tri-NMF algorithm is proposed to recover the source signals for UBSS. Based on the theory of Compressive Sampling (CS), the problem of NMF can be converted to a tri-NMF model. By incorporating the regularization and sparse penalty into the cost function, a novel multiplicative update rules is proposed to solve the problem of UBSS based on tri-NMF.
1、在信号为稀疏信号的前提下,利用稀疏混合信号聚类特点,针对欠定稀疏混合信号盲分离问题,提出了拉普拉斯势函数法。在源信号个数未知条件下,对于充分稀疏混合信号,采用拉普拉斯势函数法来估计源信号个数和混合矩阵,再用线性规划法来估计源信号。对于不充分稀疏混合信号,则利用聚类平面势函数法先估计聚类平面的法线向量,再利用势函数法来估计聚类平面的交线,得到源数和混合矩阵的估计;然后,利用其混合信号子空间聚类特点,采用部分信号分离法来恢复源信号。为了减小噪声和异常值对势函数值估计的影响,与以往直接计算势函数值的方法不同,而是采用聚类算法来估计聚类中心势函数的大小,并对算法的鲁棒特性进行了分析。
2、在有噪声和源数未知条件下,针对欠定稀疏混合信号盲分离问题,提出了基于鲁棒竞争聚类学习算法来实现欠定混合信号盲分离。在源信号个数未知的条件下,对于充分稀疏混合信号,利用竞争聚类学习算法来估计聚类中心势的相对大小,利用聚类中心势的相对大小来估计源信号的个数以及混合矩阵,最后利用基追踪法来估计源信号。将鲁棒竞争聚类算法扩展用于不充分稀疏信号混合条件下的欠定盲分离问题,先利用鲁棒平面竞争聚类算法估计聚类平面的法线向量,再利用估计出的聚类平面,采用势函数法来估计聚类平面的交线向量及其个数,得到源信号个数和混合矩阵的估计。
3、在非负信号约束下,针对适定或过定线性瞬时混合模型,提出了基于最小相关约束的NMF方法来实现盲信号分离。该方法通过使分离信号的每行之间的相关最小来提高信号分离的效果。同时将最小相关约束的NMF方法扩展用于卷积混合信号盲分离。
4、在假定源信号为非负稀疏信号的基础上,提出一种tri-NMF算法来实现欠定混合信号盲分离,而一般情况下,常用的NMF算法不能用于欠定混合信号盲分离。该算法利用压缩采样技术原理,将一般非负矩阵分解转换为tri-NMF问题,并在目标函数中引入稀疏约束,提出一种新的乘法更新算法来实现欠定混合信号盲分离。
The goal of Blind Sources Separation (BSS) is to recover the original sources given only sensor observations that are unknown linear mixtures of the unobserved source signals. There are many potential exciting applications of blind sources separation in science and technology, especially in biomedical signal processing, wireless communication, audio and acoustics, image enhancement, sonar and radar systems; it has been one of the most active research areas in the modem signal processing. In the earlier research of the BSS, generally, most of the methods to BSS assume that the number of sources is known and the number of observed signals is not less than that of source signals. In practice, however, the number of the sources does not often hold and the number of mixing signals usually is less than that of sources. In this dissertation, we investigate the problem of underdetermined blind sources separation (UBSS) with an unknown source number, i.e. the number of observed signals is less than that of source signals. The primary contributions of the dissertation are summarized blow:
1. Based on the sparse mixture models, a potential function method based on the Lapulacial potential function is proposed for UBSS. When the source signals are assumed to be sparse sufficiently, the number of sources and mixing matrix can be obtained by estimating the local maximum of the Lapulacial potential function. When the source signals are sparse insufficiently, the normal vectors of the concentration hyperplanes can be obtained by estimating the local maximum of the potential function, and then the number of sources and mixing matrix can be estimated by finding the intersection of the concentration hyperplanes. In order to increase the robustness to the noise and the outlier, the clustering algorithm is exploited to estimate the local maximum of the potential function instead of directly estimating the local maximum of the potential function.
2. With unknown the number of sources, the Robust Competitive Agglomeration (RCA) algorithm is proposed to UBSS in the presence of noise. When the source signals are assumed to be sparse sufficiently, we can select the prototype parameters of clusters with larger cardinality as the estimation of mixing matrix’s column vectors and then we can restructure the mixing matrix. The number of column of the estimated mixing matrix can be regarded as the estimated number of the source signals. When the source signals are sparse insufficiently, the RCA algorithm can be extended to estimate the normal vectors of the concentration hyperplanes, and then the number of sources and mixing matrix can be also estimated by finding the intersection of the concentration hyperplanes.
3. When the sources are nonnegative, a novel algorithm is proposed for over or well determined BSS based on the Nonnegative Matrix Factorization (NMF) methods with the least correlated component constraints. The algorithm relaxes the source independence assumption and has low-complexity algebraic computations, and thus is computationally efficient. The algorithm can be also used to blind separation of convolutive mixed sources.
4. In generally, the problem of UBSS can not be solved by using the standard NMF methods. With the assumption of that the source signals are nonnegative and sparse; a tri-NMF algorithm is proposed to recover the source signals for UBSS. Based on the theory of Compressive Sampling (CS), the problem of NMF can be converted to a tri-NMF model. By incorporating the regularization and sparse penalty into the cost function, a novel multiplicative update rules is proposed to solve the problem of UBSS based on tri-NMF.
引文
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