A compressed sensing approach for underdetermined blind audio source separation with sparse representation

This paper deals with the problem of blind source separation which consists of recovering a set of signals from instantaneous linear mixture of them. So far, this problem has been solved using statistical information available on the source signals. Here, we propose an approach for blind source separation based on time-frequency (t-f) signal representations. This approach is based on a `joint diagonalization' of a combined set of time frequency distribution matrices which correspond to different t-f points. It relies on the difference in the t-f signatures of the sources to be separated. In contrast to existing techniques, the proposed approach allows the separation of Gaussian sources with identical spectra shape. Because of changes incurred in the t-f signal structures due to time- delay, the new approach can be employed to separate multipath signals received by multi-sensor array. Moreover, the effects of spreading the noise power while localizing the source energy in the time frequency domain amounts to increasing the signal to noise ratio and hence improved performance. Numerical examples are provided to illustrate the effectiveness of our method.

An important application of sparse representation is underdetermined blind source separation (BSS), where the number of sources is greater than the number of observations. Within the stochastic framework, this paper discusses recoverability of underdetermined BSS based on a two-stage sparse representation approach. The two-stage approach is effective when the source matrix is sufficiently sparse. The first stage of the two-stage approach is to estimate the mixing matrix, and the second is to estimate the source matrix by minimizing the 1-norms of the source vectors subject to some constraints. After estimating the mixing matrix and fixing the number of nonzero entries of a source vector, we estimate the recoverability probability (i.e., the probability that the source vector can be recovered). A general case is then considered where the number of nonzero entries of the source vector is fixed and the mixing matrix is drawn from a specific probability distribution. The corresponding probability estimate on recoverability is also obtained. Based on this result, we further estimate the recoverability probability when the sources are also drawn from a distribution (e.g., Laplacian distribution). These probability estimates not only reflect the relationship between the recoverability and sparseness of sources, but also indicate the overall performance and confidence of the two-stage sparse representation approach for solving BSS problems. Several simulation results have demonstrated the validity of the probability estimation approach.

Since roller bearing is one of the most vulnerable components, bearing faults usually occur in an unprepared situation with multiple faults, and the quantity of sensors is limited in the real-time working environment, resulting in an underdetermined blind source separation (UBSS) problem to extract the fault features. Because the collected signals are usually not independent and not sparse enough, traditional methods of separating signals cannot perform well. In this paper, an optimized intrinsic characteristic-scale decomposition (OICD) method is proposed to solve the underdetermined problem. Meanwhile, the constraint error factor is introduced to overcome the drawback that the ideal ending condition of ICD is not proper for the vibration signal of bearing. In addition, given that non-negative matrix factorization (NMF) is not limited by the source signal independence and sparsity, an improved UBSS model is constructed, and the PCs are used as the input matrix of local NMF to obtain the separation signal. Ultimately, envelope analysis is utilized to detect the source signal feature. Both simulated and experimental vibration signals are used to verify the effectiveness of the proposed approach. Besides, the traditional method is juxtaposed with the suggested method. The results indicate that the proposed method is effective in dealing with the compound faults separation of the rotating machinery.

The problem of blind source separation (BSS) has been intensively studied by the signal processing community. The first solutions to deal with BSS were proposed in the 1980's and are founded on the concept of independent component analysis (ICA). More recently, aiming at tackling some limitations of ICA-based methods, much attention has been paid to alternative BSS approaches. In this tutorial, in addition to providing a brief review of the classical BSS framework, we present two research trends in this area, namely source separation over Galois fields and sparse component analysis. For both subjects, we provide an overview of the main criteria, highlighting scenarios that can benefit from these more recent BSS paradigms.

This paper proposes a novel method for simultaneously solving the problems of underdetermined blind source separation (BSS), source activity detection, dereverberation and direction-of-arrival (DOA) estimation by introducing an extension of the "multichannel factorial hidden Markov model (MFH-MM)." The MFHMM is an extension of the multichannel non-negative matrix factorization (NMF) modeL in which the basis spectra are allowed to vary over time according to the transitions of the hidden states. This model has allowed us to perform source separation, source activity detection and dereverberation in a unified manner. In our previous model, the spatial covariance of each source has been treated as a model parameter. This has led the entire generative model to have an unnecessarily high degree of freedom, and thus the parameter inference has been prone to getting trapped into undesired local optima. To reasonably restrict the solution space of the spatial covariance matrix of each source, we propose to describe it as a weighted sum of the fixed spatial covariance matrix corresponding to the discrete set of DOAs. Through the parameter inference, the proposed model allows us to simultaneously solve the problems of underdetermined BSS, source activity detection, dereverberation and DOA estimation. Experimental results revealed that the proposed method was superior to a previous method in terms of the signal-to-distortion ratios of separated signals.

The problem of underdetermined blind source separation is addressed. An advanced classification method based upon competitive learning is proposed for automatically determining the number of active sources over the observation. Its introduction in underdetermined blind source separation successfully overcomes the drawback of an existing method, in which the goal of separating more sources than the number of available mixtures is achieved by exploiting the sparsity of the nonstationary sources in the time-frequency domain. Simulation studies are presented to support the proposed approach.

One (of) important application of sparse component analysis (SCA) is in underdetermined blind source separation (BSS). Within a probability framework, this paper focuses on recoverability problem of underdetermined BSS based on a two-stage SCA approach. We consider a general case in which both sources and mixing matrix are randomly drawn. First, we present a recoverability probability estimate under the condition that the nonzero entry number of a source column vector is fixed. Next, we define the sparsity degree of a signal, and establish the relationship between the sparsity degree of sources and recoverability probability. Finally, we explain how to use the relationship to guarantee the performance of BSS. Several simulation results have demonstrated the validity of the probability estimation approach.

We address the problem of blind source separation of non-stationary signals of which only instantaneous linear mixtures are observed. A blind source separation approach exploiting both auto-terms and cross-terms of the time-frequency (TF) distributions of the sources is considered. The approach is based on the simultaneous diagonalization and anti-diagonalization of spatial TF distribution matrices made up of, respectively, auto-terms and cross-terms. Numerical simulations are provided to demonstrate the effectiveness of the proposed approach and compare its performances with existing TF-based methods.

Recently, sparse representation based methods have proven to be successful towards solving image restoration problems. The objective of these methods is to use sparsity prior of the underlying signal in terms of some dictionary and achieve optimal performance in terms of mean-squared error, a metric that has been widely criticized in the literature due to its poor performance as a visual quality predictor. In this work, we make one of the first attempts to employ structural similarity (SSIM) index, a more accurate perceptual image measure, by incorporating it into the framework of sparse signal representation and approximation. Specifically, the proposed optimization problem solves for coefficients with minimum norm and maximum SSIM index value. Furthermore, a gradient descent algorithm is developed to achieve SSIM-optimal compromise in combining the input and sparse dictionary reconstructed images. We demonstrate the performance of the proposed method by using image denoising and super-resolution methods as examples. Our experimental results show that the proposed SSIM-based sparse representation algorithm achieves better SSIM performance and better visual quality than the corresponding least square-based method.

Underdetermined blind source separation is to recover the source signals from the observed signals without prior knowledge of the mixing channel. In signal processing, it is an open problem attracted the attention of more and more researchers. In this paper, we presents a fast and effective time-frequency algorithm to separate speech source signals in the underdetermined mixture case. In the proposed algorithm, the time-domain mixture signals are transformed to the frequency-domain by using short-time Fourier transform (STFT). Then the mixing matrix is estimated using K-means clustering, and frequency-domain sources are separated by solving a low-dimensional linear programming problem based on the estimated mixing matrix. Finally, the time-domain source signals are obtained using inverse STFT. The proposed algorithm has two advantages, one is to save time consumption, the other is to obtain better separation performance. Experimental results based on two mixtures of four speech sources demonstrate the feasibility and superiority of the proposed algorithm.

Currently, the standard event-related potentials (ERP) technique consists in averaging many on-going electroencephalogram (EEG) trials using the same stimuli. Key questions are how to extract the ERP from on-going EEG with fewer average times and how to further decompose ERP into basic components related to cognitive process. In this paper we introduce a novel Blind Source Separation (BSS) approach based on a weak exclusion principle (WEP) to solve these problems. The superior aspect of this algorithm is that it is based on a deterministic principle, which is more appropriate to analyze non-stationary EEG signals than most other BSS methods based on statistical hypotheses. The results show that our BSS algorithm can quickly and effectively extract ERPs using fewer average times than the traditional averaging methods. We show that, via BSS, we can isolate two main ERP components, which are respectively related to an exogenous process and a cognitive process, and can discriminate between the occipital lobe and the frontal lobe responses from the brain, agreeing with the classical component modeling in ERPs. Single-trial ERP separation results have demonstrated the consistency of these two main ERP components. Thus, BSS based on WEP can provide a window to better understand ERP, not only in averaging behavior, but in the complexities of moment-to-moment dynamics as well.

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

The sparsity is an attractive property that has been widely and intensively utilized in various image processing fields (e.g., robust image representation, image compression, image analysis, etc.). Its actual success owes to the exhaustive mining of the intrinsic (or homogenous) information from the whole data carrying redundant information. From the perspective of image representation, the sparsity can successfully find an underlying homogenous subspace from a collection of training data to represent a given test sample. The famous sparse representation (SR) and its variants embed the sparsity by representing the test sample using a linear combination of training samples with L0-norm regularization and L1-norm regularization. However, although these state-of-the-art methods achieve powerful and robust performances, the sparsity is not fully exploited on the image representation in the following three aspects: 1) the within-sample sparsity, 2) the between-sample sparsity, and 3) the image structural sparsity. In this paper, to make the above-mentioned multi-context sparsity properties agree and simultaneously learned in one model, we propose the concept of consensus sparsity (Con-sparsity) and correspondingly build a multi-context sparse image representation (MCSIR) framework to realize this. We theoretically prove that the consensus sparsity can be achieved by the L∞-induced matrix variate based on the Bayesian inference. Extensive experiments and comparisons with the state-of-the-art methods (including deep learning) are performed to demonstrate the promising performance and property of the proposed consensus sparsity.

In the case of multiple nonstationary independent source signals and linear instantaneous time-varying mixing systems, it is difficult to adaptively separate the multiple source signals. Therefore, the adaptive blind source separation (BSS) problem is firstly formally expressed and compared with tradition BSS problem. Then, we propose an adaptive blind identification and separation method based on the variable learning rate equivariant adaptive source separation via independence (EASI) algorithm. Furthermore, we analyze the scope and conditions of variable-learning rate EASI algorithm. The adaptive BSS simulation results also show that the variable learning rate EASI algorithm provides better separation effect than the fixed learning rate EASI and recursive least-squares algorithms.

We consider the problem of blind source separation of independent signals which passed through a multi-input multi-output linear time-invariant system. An approximate maximum likelihood estimator and its performance were studied by Yellin and Friedlander (see IEEE Trans. Signal Processing, vol.47, no.5, p.1410-14, 1999, and vol.47, no.6, p.1542-58). The results derived therein are used here to study the performance of source separation techniques based on high order statistics.