Objective: Mixtures of temporally nonstationary signals are very common in biomedical applications. The nonstationarity of the source signals can be used as a discriminative property for signal separation. Herein, a semi-blind source separation algor
ithm is proposed for the extraction of temporally nonstationary components from linear multichannel mixtures of signals and noises. Methods: A hypothesis test is proposed for the detection and fusion of temporally nonstationary events, by using ad hoc indexes for monitoring the first and second order statistics of the innovation process. As proof of concept, the general framework is customized and tested over noninvasive fetal cardiac recordings acquired from the maternal abdomen, over publicly available datasets, using two types of nonstationarity detectors: 1) a local power variations detector, and 2) a model-deviations detector using the innovation process properties of an extended Kalman filter. Results: The performance of the proposed method is assessed in presence of white and colored noise, in different signal-to-noise ratios. Conclusion and Significance: The proposed scheme is general and it can be used for the extraction of nonstationary events and sample deviations from a presumed model in multivariate data, which is a recurrent problem in many machine learning applications.
The extraction of nonstationary signals from blind and semi-blind multivariate observations is a recurrent problem. Numerous algorithms have been developed for this problem, which are based on the exact or approximate joint diagonalization of second
or higher order cumulant matrices/tensors of multichannel data. While a great body of research has been dedicated to joint diagonalization algorithms, the selection of the diagonalized matrix/tensor set remains highly problem-specific. Herein, various methods for nonstationarity identification are reviewed and a new general framework based on hypothesis testing is proposed, which results in a classification/clustering perspective to semi-blind source separation of nonstationary components. The proposed method is applied to noninvasive fetal ECG extraction, as case study.
Let A be an n by m matrix with m>n, and suppose that the underdetermined linear system As=x admits a sparse solution s0 for which ||s0||_0 < 1/2 spark(A). Such a sparse solution is unique due to a well-known uniqueness theorem. Suppose now that we ha
ve somehow a solution s_hat as an estimation of s0, and suppose that s_hat is only `approximately sparse, that is, many of its components are very small and nearly zero, but not mathematically equal to zero. Is such a solution necessarily close to the true sparsest solution? More generally, is it possible to construct an upper bound on the estimation error ||s_hat-s0||_2 without knowing s0? The answer is positive, and in this paper we construct such a bound based on minimal singular values of submatrices of A. We will also state a tight bound, which is more complicated, but besides being tight, enables us to study the case of random dictionaries and obtain probabilistic upper bounds. We will also study the noisy case, that is, where x=As+n. Moreover, we will see that where ||s0||_0 grows, to obtain a predetermined guaranty on the maximum of ||s_hat-s0||_2, s_hat is needed to be sparse with a better approximation. This can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where ||s0||_0 grows.
Finding sparse solutions of underdetermined systems of linear equations is a fundamental problem in signal processing and statistics which has become a subject of interest in recent years. In general, these systems have infinitely many solutions. How
ever, it may be shown that sufficiently sparse solutions may be identified uniquely. In other words, the corresponding linear transformation will be invertible if we restrict its domain to sufficiently sparse vectors. This property may be used, for example, to solve the underdetermined Blind Source Separation (BSS) problem, or to find sparse representation of a signal in an `overcomplete dictionary of primitive elements (i.e., the so-called atomic decomposition). The main drawback of current methods of finding sparse solutions is their computational complexity. In this paper, we will show that by detecting `active components of the (potential) solution, i.e., those components having a considerable value, a framework for fast solution of the problem may be devised. The idea leads to a family of algorithms, called `Iterative Detection-Estimation (IDE), which converge to the solution by successive detection and estimation of its active part. Comparing the performance of IDE(s) with one of the most successful method to date, which is based on Linear Programming (LP), an improvement in speed of about two to three orders of magnitude is observed.
In this paper, we propose a Bayesian Hypothesis Testing Algorithm (BHTA) for sparse representation. It uses the Bayesian framework to determine active atoms in sparse representation of a signal. The Bayesian hypothesis testing based on three assump
tions, determines the active atoms from the correlations and leads to the activity measure as proposed in Iterative Detection Estimation (IDE) algorithm. In fact, IDE uses an arbitrary decreasing sequence of thresholds while the proposed algorithm is based on a sequence which derived from hypothesis testing. So, Bayesian hypothesis testing framework leads to an improved version of the IDE algorithm. The simulations show that Hard-version of our suggested algorithm achieves one of the best results in terms of estimation accuracy among the algorithms which have been implemented in our simulations, while it has the greatest complexity in terms of simulation time.
Let x be a signal to be sparsely decomposed over a redundant dictionary A, i.e., a sparse coefficient vector s has to be found such that x=As. It is known that this problem is inherently unstable against noise, and to overcome this instability, the a
uthors of [Stable Recovery; Donoho et.al., 2006] have proposed to use an approximate decomposition, that is, a decomposition satisfying ||x - A s|| < delta, rather than satisfying the exact equality x = As. Then, they have shown that if there is a decomposition with ||s||_0 < (1+M^{-1})/2, where M denotes the coherence of the dictionary, this decomposition would be stable against noise. On the other hand, it is known that a sparse decomposition with ||s||_0 < spark(A)/2 is unique. In other words, although a decomposition with ||s||_0 < spark(A)/2 is unique, its stability against noise has been proved only for highly more restrictive decompositions satisfying ||s||_0 < (1+M^{-1})/2, because usually (1+M^{-1})/2 << spark(A)/2. This limitation maybe had not been very important before, because ||s||_0 < (1+M^{-1})/2 is also the bound which guaranties that the sparse decomposition can be found via minimizing the L1 norm, a classic approach for sparse decomposition. However, with the availability of new algorithms for sparse decomposition, namely SL0 and Robust-SL0, it would be important to know whether or not unique sparse decompositions with (1+M^{-1})/2 < ||s||_0 < spark(A)/2 are stable. In this paper, we show that such decompositions are indeed stable. In other words, we extend the stability bound from ||s||_0 < (1+M^{-1})/2 to the whole uniqueness range ||s||_0 < spark(A)/2. In summary, we show that all unique sparse decompositions are stably recoverable. Moreover, we see that sparser decompositions are more stable.
Over the last ten years blind source separation (BSS) has become a prominent processing tool in the study of human electroencephalography (EEG). Without relying on head modeling BSS aims at estimating both the waveform and the scalp spatial pattern o
f the intracranial dipolar current responsible of the observed EEG. In this review we begin by placing the BSS linear instantaneous model of EEG within the framework of brain volume conduction theory. We then review the concept and current practice of BSS based on second-order statistics (SOS) and on higher-order statistics (HOS), the latter better known as independent component analysis (ICA). Using neurophysiological knowledge we consider the fitness of SOS-based and HOS-based methods for the extraction of spontaneous and induced EEG and their separation from extra-cranial artifacts. We then illustrate a general BSS scheme operating in the time-frequency domain using SOS only. The scheme readily extends to further data expansions in order to capture experimental source of variations as well. A simple and efficient implementation based on the approximate joint diagonalization of Fourier cospectral matrices is described (AJDC). We conclude discussing useful aspects of BSS analysis of EEG, including its assumptions and limitations.
