Extracting post-nonlinear signal with reference

Constrained independent component analysis (cICA) is an important technique which can extract the desired sources from the mixtures. The post-nonlinear (PNL) mixture model is more realistic and accurate than the linear instantaneous mixture model in many practical situations. In this paper, we address the problem of extracting the desired source as the first output from the PNL mixture. The prior knowledge about the desired source, such as its rough template (reference), is assumed to be available. Two approaches of extracting PNL signal with reference are discussed. Then a novel algorithm which alternately optimizes the contrast function and the closeness measure between the estimated output and the reference signal is proposed. The inverse of the unknown nonlinear function in the PNL mixture model is approximated by the multi-layer perception (MLP) network. The correctness and validity of the proposed algorithm are demonstrated by our experiment results.     

Analysis of sparse representation and blind source separation

In this letter, we analyze a two-stage cluster-then-l(1)-optimization approach for sparse representation of a data matrix, which is also a promising approach for blind source separation (BSS) in which fewer sensors than sources are present. First, sparse representation (factorization) of a data matrix is discussed. For a given overcomplete basis matrix, the corresponding sparse solution (coefficient matrix) with minimum l(1) norm is unique with probability one, which can be obtained using a standard linear programming algorithm. The equivalence of the l(1)-norm solution and the l(0)-norm solution is also analyzed according to a probabilistic framework. If the obtained l(1)-norm solution is sufficiently sparse, then it is equal to the l(0)-norm solution with a high probability. Furthermore, the l(1)- norm solution is robust to noise, but the l(0)-norm solution is not, showing that the l(1)-norm is a good sparsity measure. These results can be used as a recoverability analysis of BSS, as discussed. The basis matrix in this article is estimated using a clustering algorithm followed by normalization, in which the matrix columns are the cluster centers of normalized data column vectors. Zibulevsky, Pearlmutter, Boll, and Kisilev (2000) used this kind of two-stage approach in underdetermined BSS. Our recoverability analysis shows that this approach can deal with the situation in which the sources are overlapped to some degree in the analyzed domain and with the case in which the source number is unknown. It is also robust to additive noise and estimation error in the mixing matrix. Finally, four simulation examples and an EEG data analysis example are presented to illustrate the algorithm's utility and demonstrate its performance.     

基于线性阵列的欠定盲源分离几何分析

采用线性阵列对欠定盲源分离问题进行建模,研究源信号的空间分布对欠定盲源分离的影响.利用二步法和稀疏分量分析解决欠定盲源分离问题,其中,混合矩阵的估计主要利用稀疏源信号的线性混合信号沿混合矩阵列向量方向线性聚类的特性.理论分析和仿真实验结果表明,当源信号在空间处于某些特定区域时,若采用线性聚类方法,混合矩阵是不可估计的,...     

Equivalence Probability and Sparsity of Two Sparse Solutions in Sparse Representation

This paper discusses the estimation and numerical calculation of the probability that the 0-norm and 1-norm solutions of underdetermined linear equations are equivalent in the case of sparse representation. First, we define the sparsity degree of a signal. Two equivalence probability estimates are obtained when the entries of the 0-norm solution have different sparsity degrees. One is for the case in which the basis matrix is given or estimated, and the other is for the case in which the basis matrix is random. However, the computational burden to calculate these probabilities increases exponentially as the number of columns of the basis matrix increases. This computational complexity problem can be avoided through a sampling method. Next, we analyze the sparsity degree of mixtures and establish the relationship between the equivalence probability and the sparsity degree of the mixtures. This relationship can be used to analyze the performance of blind source separation (BSS). Furthermore, we extend the equivalence probability estimates to the small noise case. Finally, we illustrate how to use these theoretical results to guarantee a satisfactory performance in underdetermined BSS.      

基于广义相关系数的后非线性盲信号分离算法

基于互信息最小化的独立性测度对各分离信号间的非线性相关度度量没有归一化的问题,提出一种基于广义相关系数的肓信号分离(BSS)算法.首先选取后非线性混叠模型(PNL)分析基于广义相关系数的独立性测度;然后采用Gram-Charlier扩展形式估计输出参数并获取评价几率函数,结合最陡下降法求得分离矩阵和参数化可逆非线性映射的算法迭代公式.仿真结果表明,采用所提出的算法能够定量分析各分离信号间的非线性相关程度,有效分离后非线性混叠信号.     

Blind separation of non-stationary sources using continuous density hidden Markov models

Blind source separation (BSS) has attained much attention in signal processing society due to its ‘blind’ property and wide applications. However, there are still some open problems, such as underdetermined BSS, noise BSS. In this paper, we propose a Bayesian approach to improve the separation performance of instantaneous mixtures with non-stationary sources by taking into account the internal organization of the non-stationary sources. Gaussian mixture model (GMM) is used to model the distribution of source signals and the continuous density hidden Markov model (CDHMM) is derived to track the non-stationarity inside the source signals. Source signals can switch between several states such that the separation performance can be significantly improved. An expectation-maximization (EM) algorithm is derived to estimate the mixing coefficients, the CDHMM parameters and the noise covariance. The source signals are recovered via maximum a posteriori (MAP) approach. To ensure the convergence of the proposed algorithm, the proper prior densities, conjugate prior densities, are assigned to estimation coefficients for incorporating the prior information. The initialization scheme for the estimates is also discussed. Systematic simulations are used to illustrate the performance of the proposed algorithm. Simulation results show that the proposed algorithm has more robust separation performance in terms of similarity score in noise environments in comparison with the classical BSS algorithms in determined mixture case. Additionally, since the mixing matrix and the sources are estimated jointly, the proposed EM algorithm also works well in underdetermined case. Furthermore, the proposed algorithm converges quickly with proper initialization.     

Blind spatial unmixing of multispectral images: New methods combining sparse component analysis,clustering and non-negativity constraints

Remote sensing has become an unavoidable tool for better managing our environment, generally by realizing maps of land cover using classification techniques. Traditional classification techniques assign only one class (e.g., water, soil, grass) to each pixel of remote sensing images. However, the area covered by one pixel contains more than one surface component and results in the mixture of these surface components. In such situations, classical classification is not acceptable for many major applications, such as environmental monitoring, agriculture, mineral exploration and mining, etc. Most methods proposed for treating this problem have been developed for hyperspectral images. On the contrary, there are very few automatic techniques suited to multispectral images. In this paper, we propose new unsupervised spatial methods (called 2D-Corr-NLS and 2D-Corr-NMF) in order to unmix each pixel of a multispectral image for better recognizing the surface components constituting the observed scene. These methods are related to the blind source separation (BSS) problem, and are based on sparse component analysis (SCA), clustering and non-negativity constraints. Our approach consists in first identifying the mixing matrix involved in this BSS problem, by using the first stage of a spatial correlation-based SCA method with very limited source sparsity constraints, combined with clustering. Non-negative least squares (NLS) or non-negative matrix factorization (NMF) methods are then used to extract spatial sources. An important advantage of our proposed methods is their applicability to the possibly globally underdetermined, but locally (over)determined BSS model in multispectral remote sensing images. Experiments based on realistic synthetic mixtures and real multispectral images collected by the Landsat ETM+ and the Formosat-2 sensors are performed to evaluate the performance of the proposed approach. We also show that our methods significantly outperform the sequential maximum angle convex cone (SMACC) method.     

Source number estimation and separation algorithms of underdetermined blind separation

Recently, sparse component analysis （SCA） has become a hot spot in BSS research. Instead of independent component analysis （ICA）, SCA can be used to solve underdetermined mixture efficiently. Two-step approach （TSA） is one of the typical methods to solve SCA based BSS problems. It estimates the mixing matrix before the separation of the sources. K-means clustering is often used to estimate the mixing matrix. It relies on the prior knowledge of the source number strongly. However, the estimation of the source number is an obstacle. In this paper, a fuzzy clustering method is proposed to estimate the source number and mixing matrix simultaneously. After that, the sources are recovered by the shortest path method （SPM）. Simulations show the availability and robustness of the proposed method.     

Underdetermined Convolutive Blind Source Separation via Time–Frequency Masking

In this paper, we consider the problem of separation of unknown number of sources from their underdetermined convolutive mixtures via time-frequency (TF) masking. We propose two algorithms, one for the estimation of the masks which are to be applied to the mixture in the TF domain for the separation of signals in the frequency domain, and the other for solving the permutation problem. The algorithm for mask estimation is based on the concept of angles in complex vector space. Unlike the previously reported methods, the algorithm does not require any estimation of the mixing matrix or the source positions for mask estimation. The algorithm clusters the mixture samples in the TF domain based on the Hermitian angle between the sample vector and a reference vector using the well known k -means or fuzzy c -means clustering algorithms. The membership functions so obtained from the clustering algorithms are directly used as the masks. The algorithm for solving the permutation problem clusters the estimated masks by using k-means clustering of small groups of nearby masks with overlap. The effectiveness of the algorithm in separating the sources, including collinear sources, from their underdetermined convolutive mixtures obtained in a real room environment, is demonstrated.     

具有相似性传播的K-均值欠定盲分离算法

对于稀疏信源的欠定盲分离问题,混合矩阵的估计是至关重要的。为了提高估计性能,提出一种组合的聚类分析算法。首先,利用短时傅里叶变换把时域中的观测信号转变成频域中的稀疏信号,并通过数据的归一化把稀疏信号在频域的线性聚类转变成致密聚类。然后,利用相似性传播AP聚类方法搜索每个观测数据的邻域自动形成数据族的数量和相对应的关键数据。最后,以AP聚类的结果作为K-均值算法的初始值,对每类(族)数据的聚类中心进一步修正。仿真结果表明,组合聚类法能有效地提高混合矩阵的估计精度。把AP聚类和K-均值算法相结合的另一个优势是,能够克服经典K-均值算法需要事先知道信源数量和对数据的初始划分非常敏感的缺陷。     

Mixing matrix estimation using discriminative clustering for blind source separation

Mixing matrix estimation in instantaneous blind source separation (BSS) can be performed by exploiting the sparsity and disjoint orthogonality of source signals. As a result, approaches for estimating the unknown mixing process typically employ clustering algorithms on the mixtures in a parametric domain, where the signals can be sparsely represented. In this paper, we propose two algorithms to perform discriminative clustering of the mixture signals for estimating the mixing matrix. For the case of overdetermined BSS, we develop an algorithm to perform linear discriminant analysis based on similarity measures and combine it with K-hyperline clustering. Furthermore, we propose to perform discriminative clustering in a high-dimensional feature space obtained by an implicit mapping, using the kernel trick, for the case of underdetermined source separation. Using simulations on synthetic data, we demonstrate the improvements in mixing matrix estimation performance obtained using the proposed algorithms in comparison to other clustering methods. Finally we perform mixing matrix estimation from speech mixtures, by clustering single source points in the time-frequency domain, and show that the proposed algorithms achieve higher signal to interference ratio when compared to other baseline algorithms.     

FastICA在MIMO信道估计中的应用研究

研究了MIMO无线通信系统的信道估计问题。通过对信道估计和盲分离问题的一致性分析,提出将盲分离的典型算法FastICA应用于MIMO信道估计问题中,并将信号表征形式从实数域拓展到复数域。针对盲分离次序的不确定性以至造成的估计信道参数不对应的问题,提出了相关系数重新定位法对所得的解混矩阵进行修正,最终用矩阵求逆的方法求得信道估计问题的对应解。仿真实验结果证明了方法的有效性。     

Searching-and-averaging method of underdetermined blind speech signal separation in time domain

Underdetermined blind signal separation (BSS) (with fewer observed mixtures than sources) is discussed. A novel searching-and-averaging method in time domain (SAMTD) is proposed. It can solve a kind of problems that are very hard to solve by using sparse representation in frequency domain. Bypassing the disadvantages of traditional clustering (e.g., K-means or potential-function clustering), the durative- sparsity of a speech signal in time domain is used. To recover the mixing matrix, our method deletes those samples, which are not in the same or inverse direction of the basis vectors. To recover the sources, an improved geometric approach to overcomplete ICA (Independent Component Analysis) is presented. Several speech signal experiments demonstrate the good performance of the proposed method.     

Efficient solution of nonlinear,underdetermined inverse problems with a generalized PDE model

Several parameter estimation problems (or “inverse” problems) such as those that occur in hydrology and geophysics are solved using partial differential equation (PDE)-based models of the physical system in question. Likewise, these problems are usually underdetermined due to the lack of enough data to constrain a unique solution. In this paper, we present a framework for the solution of underdetermined inverse problems using COMSOL Multiphysics (formerly FEMLAB) that is applicable to a broad range of physical systems governed by PDEs. We present a general adjoint state formulation which may be used in this framework and allows for faster calculation of sensitivity matrices in a variety of commonly encountered underdetermined problems. The aim of this approach is to provide a platform for the solution of inverse problems that is efficient, flexible, and not restricted to one particular scientific application.We present an example application of this framework on a synthetic underdetermined inverse problem in aquifer characterization, and present numerical results on the accuracy and efficiency of this method. Our results indicate that our COMSOL-based routines provide an accurate, flexible, and scalable method for the solution of PDE-based inverse problems.     

Adaptive blind separation of underdetermined mixtures based on sparse component analysis

The independence priori is very often used in the conventional blind source separation （BSS）. Naturally, independent component analysis （ICA） is also employed to perform BSS very often. However, ICA is difficult to use in some challenging cases, such as underdetermined BSS or blind separation of dependent sources. Recently, sparse component analysis （SCA） has attained much attention because it is theoretically available for underdetermined BSS and even for blind dependent source separation sometimes. However, SCA has not been developed very sufficiently. Up to now, there are only few existing algorithms and they are also not perfect as well in practice. For example, although Lewicki-Sejnowski＇s natural gradient for SCA is superior to K-mean clustering, it is just an approximation without rigorously theoretical basis. To overcome these problems, a new natural gradient formula is proposed in this paper. This formula is derived directly from the cost function of SCA through matrix theory. Mathematically, it is more rigorous. In addition, a new and robust adaptive BSS algorithm is developed based on the new natural gradient. Simulations illustrate that this natural gradient formula is more robust and reliable than Lewicki-Sejnowski＇s gradient.     

Method to separate sparse components from signal mixtures

This paper addresses the blind source separation (BSS) problem where the input data are mixtures of sources that are “sparse”—that is each source has zero amplitude for some of the time. It is shown that, under certain conditions, it is possible to separate these sources by the analysis of localised segments of the phase space trajectory where one source dominates rather than applying statistical methods to the whole phase space plot. Results are presented for both simulated data, and thoracic and abdominal ECG data taken from an expectant mother. It is shown that when applied to the ECG data, the proposed technique has a comparable performance to the standard Fast ICA method. For the particular simulated data set, the proposed method gives better results for relatively low noise levels but is less robust than the FastICA for higher noise levels. Subsequently, the potential advantage of using the proposed technique for mixtures of correlated sparse sources is demonstrated. However, for more general sources that are not sparse, the proposed technique in its present form has, as expected, an inferior performance compared to the FastICA technique.     

FastlCA在MIMO信道估计中的应用研究

研究了MIMO无线通信系统的信道估计问题。通过对信道估计和盲分离问题的一致性分析,提出将盲分离的典型算法FastIcA应用于MIM0信道估计问题中,并将信号表征形式从实数域拓展到复数域。针对盲分离次序的不确定性以至造成的估计信道参数不对应的问题,提出了相关系数重新定位法对所得的解混矩阵进行修正,最终用矩阵求逆的方法求得信道估计问题的对应解。仿真实验结果证明了方法的有效性。     

欠定条件下弱稀疏源信号混合矩阵盲估计

针对源信号的稀疏性影响欠定混合矩阵的估计精度, 在源信号单源频率及非单源频率分量分析的基础上,通过对观测信号频率峰值的幅值比值所 构成的列向量聚类,提出欠定条件下弱稀疏源信号混合矩阵的盲估计方法。鉴于经典聚类算 法的局部收敛性带来聚类结果的不稳定性,采用全局收敛特性较好的遗传模拟退火聚类算法 提高聚类结果的鲁棒性。仿真实验表明,本文提出的混合矩阵估计方法及采用的聚类算法 在不同欠定条件及噪声环境下具有较强的估计性能。     

基于压缩感知的混合光谱解析算法

压缩感知利用信号的稀疏性通过求解欠定线性系统的解来有效地重建信号,其稀疏性要求信号在某个域中是稀疏的.压缩感知理论认为一般情况下,信号的相关性越小,恢复算法的性能越好.求解压缩感知问题的方法有贪婪追踪、凸松弛方法、迭代收缩等算法,以及贝叶斯框架、置信传播等.从欠定线性矩阵方程角度讨论压缩感知问题,通过两种不同量测矩阵(...     

A parametric density model for blind source separation

In this paper, a parametric mixture density model is employed to be the source prior in blind source separation (BSS). A strict lower bound on the source prior is derived by using a variational method, which naturally enables the intractable posterior to be represented as a gaussian form. An expectation-maximization (EM) algorithm in closed form is therefore derived for estimating the mixing matrix and inferring the sources. Simulation results show that the proposed variational expectation-maximization algorithm can perform blind separation of not only speech source of more sources than mixtures, but also binary source of more sources than mixtures.