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

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 measures, kurtosis, etc. It was also suggested in [1], and verified in practice in [2,4], 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 paper, 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 work, a theoretical novel approach is used to analyze Shannon entropy (SE), Non-Gaussianity (NG) measure and absolute moments of arbitrarily order, i.e. Generic Absolute (GA) 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 (pdfs) 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 (BCA). It is shown that, for dependent sources verifying the LCE law, the NG measure provides the best separation results.