Nonstationary Source Separation Research Articles

SUSHI: An algorithm for source separation of hyperspectralimages with non-stationary spectral variation. Semi-blind Unmixing with Sparsity for Hyperspectral Images

Hyperspectral images are data cubes with two spatial dimensions and a third spectral dimension, providing a spectrum for each pixel, and thus allowing the mapping of extended sources' physical properties. In this article, we present the Semi-blind Unmixing with Sparsity for Hyperspectral Images (SUSHI), an algorithm for non-stationary unmixing of hyperspectral images with spatial regularization of spectral parameters. The method allows for the disentangling of physical components without the assumption of a unique spectrum for each component. Thus, unlike most source separation methods used in astrophysics, all physical components obtained by SUSHI vary in spectral shape and in amplitude across the data cube. Non-stationary source separation is an ill-posed inverse problem that needs to be constrained. We achieve this by training a spectral model and applying a spatial regularization constraint on its parameters. For the spectral model, we used an Interpolatory Auto-Encoder, a generative model that can be trained with limited samples. For spatial regularization, we applied a sparsity constraint on the wavelet transform of the model parameter maps. We applied SUSHI to a toy model meant to resemble supernova remnants in X-ray astrophysics, though the method may be used on any extended source with any hyperspectral instrument. We compared this result to the one obtained by a classic 1D fit on each individual pixel. We find that SUSHI obtains more accurate results, particularly when it comes to reconstructing physical parameters. We then applied SUSHI to real X-ray data from the supernova remnant Cassiopeia A and to the Crab Nebula. The results obtained are realistic and in accordance with past findings but have a much better spatial resolution. Thanks to spatial regularization, SUSHI can obtain reliable physical parameters at fine scales that are out of reach for pixel-by-pixel methods.

A General Nonstationary and Time-Varying Mixed Signal Blind Source Separation Method Based on Online Gaussian Process

Most nonstationary and time-varying mixed source separation algorithms are based on the model of instantaneous mixtures. However, the observation signal is a convolutional mixed source in reverberation environment, such as mobile voice received by indoor microphone arrays. In this paper, a time-varying convolution blind source separation (BSS) algorithm for nonstationary signals is proposed, which can separate both time-varying instantaneous mixtures and time-varying convolution mixtures. We employ the variational Bayesian (VB) inference method with Gaussian process (GP) prior for separating the nonstationary source frame by frame from the time-varying convolution signal, in which the prior information of the mixing matrix and the source signal are obtained by the Gaussian autoregressive method, and the posterior distributions of parameters (source signal and mixing matrix) are obtained by the VB learning. In the learning process, the learned parameters and hyperparameters are propagated to the next frame for VB inference as the prior which is combined with the likelihood function to get the posterior distribution. The experimental results show that the proposed algorithm is effective for separating time-varying mixed speech signals.

Spectral Estimation for Multivariate Locally Time-Warped Signals

Spectral estimation generally aims at determining from a single realization of a given signal, the distribution of its power as a function of frequency. In this letter, we focus on multivariate, locally time-warped signals. We show that the spectral estimation problem can also be interpreted as a doubly nonstationary blind source separation (BSS) problem, where both the mixing matrix and the original sources contribute to nonstationarity. We then introduce a BSS algorithm for joint spectral estimation of nonstationary sources. The performance of the proposed approach is evaluated on numerical simulations, and compared with other nonstationary BSS algorithms.

Robust multisensor time–frequency signal processing: A tutorial review with illustrations of performance enhancement in selected application areas

This paper presents high-resolution multisensor time–frequency distributions (MTFDs) and their applications to the analysis of multichannel non-stationary signals. The approach combines high-resolution time–frequency analysis and array signal processing methods. The improved performance of MTFDs is demonstrated using several applications including source localization based on direction of arrival (DOA) estimation and automated component separation (ACS) of non-stationary sources, with focus on blind source separation. The MTFD approach is further illustrated by two applications to EEG signals. One specifically uses ACS and DOA estimation methods for artifacts removal and source localization. Another uses MTFDs for cross-channel coherence/causality analysis. Data and code are provided to allow readers to reproduce the results presented, and apply these methods to their own data [1].

Bayesian Estimation in the Presence of Deterministic Nuisance Parameters—Part II: Estimation Methods

One of the fundamental issues of estimation theory is the presence of deterministic nuisance parameters. While in the Bayesian paradigm the model parameters are random, introduction of deterministic nuisance parameters into the model exceeds the Bayesian framework to the hybrid framework. In this type of scenarios, the conventional Bayesian estimators are not valid, as they assume knowledge of the deterministic nuisance parameters. This paper is the second of a two-part study of Bayesian parameter estimation in the presence of deterministic nuisance parameters. In part I, a new Cramer–Rao (CR)-type bound on the mean-square-error (MSE) for Bayesian estimation in the presence of deterministic nuisance parameters was established based on the concept of risk-unbiasedness. The proposed bound was named risk-unbiased bound (RUB). This paper presents properties of asymptotic uniform mean- and risk-unbiasedness of some Bayesian estimators: 1) the minimum MSE (MMSE) or maximum a posteriori probability (MAP) estimators with maximum likelihood (ML) estimates substituting the deterministic parameters, named MS-ML and MAP-ML, respectively, and 2) joint MAP and ML estimator, named JMAP-ML. Furthermore, an asymptotic performance analysis of the MS-ML and MAP-ML estimators is presented. These estimators are shown to asymptotically achieve the RUB, while the existing CR-type bounds can be achieved only in distinct cases. Simulations verify these results for the problem of blind separation of nonstationary sources. It is shown that unlike existing CR-type bounds, the RUB is asymptotically tight.

A new method for blind separation of nonstationary sources

The development of nonstationary signal blind separation, especially an underdetermined mixture of nonstationary sources, has become increasingly important for mechanical multi-fault diagnosis. In this paper, a novel algorithm based on the adaptive parametric time-frequency analysis theory to blindly separate the mixtures of nonstationary sources is presented. The computer simulations illustrate the applicability of our approach, and the performance of the algorithm is successfully tested on the underdetermined blind source separation of five vibration signals of a high speed train from one instantaneous mixture of these signals.

Underdetermined DOA estimation and blind separation of non-disjoint sources in time-frequency domain based on sparse representation method

This paper deals with the blind separation of nonstation-ary sources and direction-of-arrival (DOA) estimation in the under-determined case, when there are more sources than sensors. We assume the sources to be time-frequency (TF) disjoint to a certain extent. In particular, the number of sources presented at any TF neighborhood is strictly less than that of sensors. We can identify the real number of active sources and achieve separation in any TF neighborhood by the sparse representation method. Compared with the subspace-based algorithm under the same sparseness assumption, which suffers from the extra noise effect since it cannot estimate the true number of active sources, the proposed algorithm can estimate the number of active sources and their corresponding TF values in any TF neighborhood simultaneously. Another contribution of this paper is a new estimation procedure for the DOA of sources in the underdetermined case, which combines the TF sparseness of sources and the clustering technique. Simulation results demonstrate the validity and high performance of the proposed algorithm in both blind source separation (BSS) and DOA estimation.

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.

Nonstationary source separation using sequential and variational Bayesian learning.

Independent component analysis (ICA) is a popular approach for blind source separation where the mixing process is assumed to be unchanged with a fixed set of stationary source signals. However, the mixing system and source signals are nonstationary in real-world applications, e.g., the source signals may abruptly appear or disappear, the sources may be replaced by new ones or even moving by time. This paper presents an online learning algorithm for the Gaussian process (GP) and establishes a separation procedure in the presence of nonstationary and temporally correlated mixing coefficients and source signals. In this procedure, we capture the evolved statistics from sequential signals according to online Bayesian learning. The activity of nonstationary sources is reflected by an automatic relevance determination, which is incrementally estimated at each frame and continuously propagated to the next frame. We employ the GP to characterize the temporal structures of time-varying mixing coefficients and source signals. A variational Bayesian inference is developed to approximate the true posterior for estimating the nonstationary ICA parameters and for characterizing the activity of latent sources. The differences between this ICA method and the sequential Monte Carlo ICA are illustrated. In the experiments, the proposed algorithm outperforms the other ICA methods for the separation of audio signals in the presence of different nonstationary scenarios.

Separation of stationary and non-stationary sources with a generalized eigenvalue problem

Non-stationary effects are ubiquitous in real world data. In many settings, the observed signals are a mixture of underlying stationary and non-stationary sources that cannot be measured directly. For example, in EEG analysis, electrodes on the scalp record the activity from several sources located inside the brain, which one could only measure invasively. Discerning stationary and non-stationary contributions is an important step towards uncovering the mechanisms of the data generating system. To that end, in Stationary Subspace Analysis (SSA), the observed signal is modeled as a linear superposition of stationary and non-stationary sources, where the aim is to separate the two groups in the mixture. In this paper, we propose the first SSA algorithm that has a closed form solution. The novel method, Analytic SSA (ASSA), is more than 100 times faster than the state-of-the-art, numerically stable, and guaranteed to be optimal when the covariance between stationary and non-stationary sources is time-constant. In numerical simulations on wide range of settings, we show that our method yields superior results, even for signals with time-varying group-wise covariance. In an application to geophysical data analysis, ASSA extracts meaningful components that shed new light on the Pi 2 pulsations of the geomagnetic field.

Blind Separability of Reverberation and Target Echo Based on Spatial Correlation

The detection of buried targets in shallow water is a tough task in the presence of sea-bottom reverberation. Because both target echo and reverberation are caused by the transmitted signal, they are mixed together in both time domain and frequency domain, which makes traditional signal processing methods inefficient. Blind Source Separation (BSS) is expected to isolate the reverberation from the target echo. However, the feasibility should be proved before separation. In this paper, a method based on spatial correlation is proposed to determine whether reverberation and target echo can be separated as different sources. Then, considering the nonstationarity of the reverberation, SONS (Second Order Nonstationary Source Separation) is applied to separate the original received signals. The sea experiment result shows that BSS is not only feasible but also valid to separate target echo and reverberation, and the target echo after BSS is of higher SRR which makes further process more credible.

Robust spatial time-frequency distribution matrix estimation with application to direction-of-arrival estimation

We consider the problem of estimating spatial time-frequency distribution (STFD) matrices in the presence of impulsive noise. STFD matrices are widely used in sensor array processing for direction-of-arrival estimation and blind source separation of non-stationary sources. Conventional methods fail when the noise is non-Gaussian or impulsive. We propose robust techniques for STFD estimation which are based on pre-processing, robust position based estimation and robust covariance based estimation. The proposed methods are compared in terms of direction-of-arrival estimation performance.

Nonstationary Brain Source Separation for Multiclass Motor Imagery

This paper describes a method to recover task-related brain sources in the context of multiclass brain--computer interfaces (BCIs) based on noninvasive EEG. We extend the method joint approximate diagonalization (JAD) for spatial filtering using a maximum likelihood framework. This generic formulation: 1) bridges the gap between the common spatial patterns (CSPs) and blind source separation of nonstationary sources; and 2) leads to a neurophysiologically adapted version of JAD, accounting for the successive activations/deactivations of brain sources during motor imagery (MI) trials. Using dataset 2a of BCI Competition IV (2008) in which nine subjects were involved in a four-class two-session MI-based BCI experiment, a quantitative evaluation of our extension is provided by comparing its performance against JAD and CSP in the case of cross-validation, as well as session-to-session transfer. While JAD, as already proposed in other works, does not prove to be significantly better than classical one-versus-rest CSP, our extension is shown to perform significantly better than CSP for cross-validated and session-to-session performance. The extension of JAD introduced in this paper yields among the best session-to-session transfer results presented so far for this particular dataset; thus, it appears to be of great interest for real-life BCIs.

Fast Blind Separation of Long Mixture Recordings Using Multivariate Polynomial Identification

This correspondence presents new approaches for optimizing kurtosis-based separation criteria in the case of long mixture recordings. Our methods are based on a multivariate polynomial identification step that avoids the computation of signal statistics at each step of the commonly used fixed-point optimization algorithms. As compared to the well-known FastICA algorithm and to our recent DFICA algorithm intended for blind partial separation of nonstationary sources, our new methods are very computationally efficient for long recordings of a moderate number of mixed sources. They are therefore especially suited to blind image separation, because of the high number of pixels in light sensors. Our algorithms also avoid the computation and storage of the sphered observation vector, thus saving memory space.

Multiple vehicle signals separation based on particle filtering in wireless sensor network

A novel statistical method based on particle filtering is presented for multiple vehicle acoustic signals separation problem in wireless sensor network. The particle filtering method is able to deal with non-Gaussian and nonlinear models and non-stationary sources. Using some instantaneously mixed observations of several real-world vehicle acoustic signals, the proposed statistical method is compared with a conventional non-stationary Blind Source Separation algorithm and attractive simulation results are achieved. Moreover, considering the natural convenience to transmit particles between sensor nodes, the algorithm based on particle filtering is believed to have potential to enable the task of multiple vehicles recognition collaboratively performed by sensor nodes in distributed wireless sensor network.

Blind Separation of Underdetermined Convolutive Mixtures Using Their Time–Frequency Representation

This paper considers the blind separation of nonstationary sources in the underdetermined convolutive mixture case. We introduce, two methods based on the sparsity assumption of the sources in the time-frequency (TF) domain. The first one assumes that the sources are disjoint in the TF domain, i.e., there is at most one source signal present at a given point in the TF domain. In the second method, we relax this assumption by allowing the sources to be TF-nondisjoint to a certain extent. In particular, the number of sources present (active) at a TF point should be strictly less than the number of sensors. In that case, the separation can be achieved thanks to subspace projection which allows us to identify the active sources and to estimate their corresponding time-frequency distribution (TFD) values. Another contribution of this paper is a new estimation procedure for the mixing channel in the underdetermined case. Finally, numerical performance evaluations and comparisons of the proposed methods are provided highlighting their effectiveness.

Speech intelligibility improvement using convolutive blind source separation assisted by denoising algorithms

The present study is concerned with the blind source separation (BSS) of speech and speech-shaped noise sources. All recordings were carried out in an anechoic chamber using a dummy head (two microphones, one in each ear). The program which implements the algorithm for BSS of convolutive mixtures introduced by Parra and Spence [Parra, L., Spence, C., 2000a. Convolutive blind source separation of non-stationary sources. IEEE Trans. Speech Audio Process. 8(3), 320–327 (US Patent US6167417)] was used to separate out the signals. In the postprocessing phase two different denoising algorithms were used. The first was based on a minimum mean-square error log-spectral amplitude estimator [Ephraim, E., Malah, D., 1985. Speech enhancement using a minimum mean-square error log-spectral amplitude estimator. IEEE Trans. Speech Audio Process. ASSP-33(2), 443–445], while the second one was based on Wiener filter in which the concept of an a priori signal-to-noise estimation presented by Ephraim (as mentioned above) was applied [Scalart, P., Filho, J.V., 1996. Speech enhancement based on a priori signal to noise estimation. IEEE Internat. Conf. Acoust. Speech Signal Process. 1, 629–632]. Non-sense word tests were used as a target speech in both cases while one or two disturbing sources were used as interferences. The speech intelligibility before and after the BSS was measured for three subjects with audiologically normal hearing. Next the speech signal after BSS was denoised and presented to the same listeners. The results revealed some ambiguities caused by the insufficient number of microphones compared to the number of sound sources. For one disturbance only, the intelligibility improvement was significant. However, when there were two disturbances and the target speech, the separation was much poorer. The additional denoising, as could be expected, raises the intelligibility slightly. Although the BSS method requires more research on optimization, the results of the investigation imply that it may be applied to hearing aids in the future.

Blind-Source Separation Based on Decorrelation and Nonstationarity

In this paper, discrete-time blind-source separation (BSS) of instantaneous mixtures is studied. Decorrelation-based sufficient criteria for BSS of stationary and nonstationary sources are derived based on nonstationarity and nonwhiteness. A gradient algorithm is proposed based on these criteria. A batch-data algorithm and an on-line algorithm are developed based on the corollaries of the BSS criteria. These algorithms are especially useful for the separation of nonstationary sources. They are robust to additive white noises if the time-delayed decorrelation and the nonstationarity of the sources are considered simultaneously in the algorithms. Experiment results show the effectiveness and performance of the proposed algorithms

Bayesian nonstationary source separation

A Bayesian nonstationary source separation algorithm is proposed in this paper to recover nonstationary sources from noisy mixtures. In order to exploit the temporal structure of the data, we use a time-varying autoregressive (TVAR) process to model each source signal. Then variational Bayesian (VB) learning is adopted to integrate the source model with blind source separation (BSS) in probabilistic form. Our separation algorithm makes full use of temporally correlated prior information and avoids overfitting in separation process. Experimental results demonstrate that our vbICA-TVAR algorithm learns the temporal structure of sources and acquires cleaner source reconstruction.

Underdetermined Blind Separation of Nondisjoint Sources in the Time-Frequency Domain

This paper considers the blind separation of nonstationary sources in the underdetermined case, when there are more sources than sensors. A general framework for this problem is to work on sources that are sparse in some signal representation domain. Recently, two methods have been proposed with respect to the time-frequency (TF) domain. The first uses quadratic time-frequency distributions (TFDs) and a clustering approach, and the second uses a linear TFD. Both of these methods assume that the sources are disjoint in the TF domain; i.e., there is, at most, one source present at a point in the TF domain. In this paper, we relax this assumption by allowing the sources to be TF-nondisjoint to a certain extent. In particular, the number of sources present at a point is strictly less than the number of sensors. The separation can still be achieved due to subspace projection that allows us to identify the sources present and to estimate their corresponding TFD values. In particular, we propose two subspace-based algorithms for TF-nondisjoint sources: one uses quadratic TFDs and the other a linear TFD. Another contribution of this paper is a new estimation procedure for the mixing matrix. Finally, then numerical performance of the proposed methods are provided highlighting their performance gain compared to existing ones