Bounded Component Analysis Research Articles

On the conditions for valid objective functions in blind separation of independent and dependent sources

It is well known that independent sources can be blindly detected and separated, one by one, from linear mixtures by identifying local extrema of certain objective functions (contrasts), like negentropy, non-Gaussianity (NG) measures, kurtosis, etc. It was also suggested by Donoho in 1981, and verified in practice by Caiafa et al., that some of these measures remain useful for particular cases with dependent sources, but not much work has been done in this respect and a rigorous theoretical ground still lacks. In this article, it is shown that, if a specific type of pairwise dependence among sources exists, called linear conditional expectation (LCE) law, then a family of objective functions are valid for their separation. Interestingly, this particular type of dependence arises in modeling material abundances in the spectral unmixing problem of remote sensed images. In this study, a theoretical novel approach is used to analyze Shannon entropy (SE), NG measure and absolute smoments of arbitrarily order β, i.e. generic absolute moments for the separation of sources allowing them to be dependent. We provide theoretical results that show the conditions under which sources are isolated by searching for a maximum or a minimum. Also, simple and efficient algorithms based on Parzen windows estimations of probability density functions and Newton–Raphson iterations are proposed for the separation of dependent or independent sources. A set of simulation results on synthetic data and an application to the blind spectral unmixing problem are provided in order to validate our theoretical results and compare these algorithms against FastICA and a very recently proposed algorithm for dependent sources, the bounded component analysis algorithm. It is shown that, for dependent sources verifying the LCE law, the NG measure provides the best separation results.

On the Convergence of Symmetrically Orthogonalized Bounded Component Analysis Algorithms for Uncorrelated Source Separation

Bounded Component Analysis (BCA) has recently been introduced as an alternative linear decomposition scheme. In this approach the boundedness property of sources is exploited to replace the usual independence assumption with a weaker assumption, which enables development of methods to separate both independent and dependent components from their mixtures. This paper positions total output range minimization based blind source separation approach as a BCA method for the separation of uncorrelated sources. It is shown that the global minimizers of the corresponding optimization problem are the perfect separators. Furthermore, a stationary point analysis for the corresponding algorithms based on symmetrical orthogonalization is provided. The main result of this analysis is that the range minimization based parallel BCA algorithm and the kurtosis maximization based Independent Component Analysis algorithm have related set of identified stationary points.

Criterion for signal extraction in underdetermined mixtures of bounded support

This work studies the problem of recovering a complex signal (source) from an underdetermined linear mixture of bounded sources. We assume some a priori information of the desired signal in the form of a training sequence and complete absence of knowledge from the other sources, except for their bounded character. The main contribution of this letter is the proposal of a bounded component analysis of the training error that tries to condense the relevant information of the observations in a linear estimate of the desired signal. This subspace can be later used for subsequent refined estimation of the signal of interest. Simulations corroborate the good performance of the proposed method in high SNR scenarios.

Bounded Component Analysis of Linear Mixtures: A Criterion of Minimum Convex Perimeter

This study presents a blind and geometric technique which pursues the linear decomposition of the observations in bounded component signals. The bounded component analysis of the observations relies on the hypotheses of compactness and Cartesian decomposition of the convex support of the vector of component signals, and in the invertibility of the mixture. Assumptions, which in absence of noise, are able to guarantee the identifiability of the mixture and separability of the components, up to permutation, scaling, and phase ambiguities. Under these conditions, the convex perimeter of the normalized linear combination of the observations is shown to be a global contrast function whose minima correspond with the extraction of bounded components of the observations. Practical extraction and separation algorithms based on the minimization of this criterion are given. The experimental results with communications signals serve to illustrate the good performance of the proposed method in high SNR scenarios, even for a small number of samples.