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

Cesar F Caiafa

doi:10.1186/1687-6180-2012-255

Abstract

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.

Highlights

In signal processing, a generic problem is how to separate signals that are linearly combined in the measurements
Detailed analysis of Shannon entropy (SE) and Generic absolute (GA) moments In previous sections, we proved that some objective functions applied to a unit-variance mixture of sources verifying the linear conditional expectation (LCE) law, have local extrema when only one of the coefficients is non-zero, which means that we can separate those sources by searching for local extrema
This article contributes to shed light on the theoretical aspects of the separation of independent and dependent sources based on the maximization of objective functions by filling the gaps existing among previous works and giving rigorous theoretical answers to important questions

Summary

Introduction

A generic problem is how to separate signals that are linearly combined in the measurements. Local extrema of the GA moment (general case): Given two zero-mean and unit-norm source variables Si (i = 1, 2) following the LCE law with respect to S2, the GA moment of order β of the mixture variable X = α1S1 + α2S2, constrained to the unit-variance case E[ X2] = 1, has a local extremum at (α1, α2) = (0, 1). Detailed analysis of SE and GA moments In previous sections, we proved that some objective functions applied to a unit-variance mixture of sources verifying the LCE law, have local extrema when only one of the coefficients is non-zero, which means that we can separate those sources by searching for local extrema.

13: Search for second extremum

Conclusions and discussion