**Introduction to Blind Source Separation**

Recently, blind source separation by Independent Component Analysis (ICA) has received attention because of its potential applications in signal processing such as in speech recognition systems, telecommunications and medical signal processing. The goal of ICA is to recover independent sources given only sensor observations that are unknown linear mixtures of the unobserved independent source signals. In contrast to correlation-based transformations such as Principal Component Analysis (PCA), ICA not only decorrelates the signals (2nd-order statistics) but also reduces higher-order statistical dependencies, attempting to make the signals as independent as possible.

There have been two different fields of research considering the analysis of independent components. On one hand, the study of separating mixed sources observed in an array of sensors has been a classical and difficult signal processing problem. The seminal work on blind source separation is by Jutten, Herault and Guerin (1988) where they proposed an adaptive algorithm in a simple feedback architecture. The learning rule was based on an neuromimetic approach and was able to separate simultaneously unknown independent sources. This approach has been explained and further developed by Jutten and Herault (1991), Comon (1991), Karhunen and Joutsensalo (1993), Cichocki and Moszczynski (1992) and others. Furthermore, Comon (1994) introduced the concept of independent component analysis and proposed cost functions related to the minimization of mutual information between the sensors.

On the other hand and in parallel to blind source separation studies unsupervised learning rules based on information-theory have been proposed by Linsker (1992), Becker and Hinton (1992) and others. The idea was to maximize the mutual information between the inputs and outputs of a neural network. This approach is related to redundancy reduction which was suggested by Barlow (1961) as a coding strategy in neurons. Each neuron should encode features that are statistically independent from other neurons. This leads to the notion of factorial code that has been explored for the visual processing strategy by Attik (1992). Nadal and Parga (1994) showed that in the low-noise case, the maximum of the mutual information between the input and output of a neural processor implied that the output distribution was factorial. Roth and Baram (1996) and Bell and Sejnowski (1995) independently derived stochastic gradient learning rules for this maximization and applied them, respectively to forecasting and time series analysis, and the blind separation of sources. Bell and Sejnowski (1995) were the first explaining the blind source separation problem from an information-theoretic viewpoint and applying them to separation and deconvolution of sources. Their adaptive methods are more plausible from a neural processing perspective than the cumulant-based cost functions proposed by Comon. A similar adaptive but 'non-neural' method for source separation has been proposed by Cardoso and Laheld (1996).

Other algorithms have been proposed from different approaches: The maximum likelihood estimation approach was first proposed by Gaeta and Lacoume (1990), negentropy maximization approach by Girolami and Fyfe (1996), nonlinear PCA algorithm developed by Karhunen and Joutsensalo (1994) and Oja (1995). Lee, Girolami and Sejnowski (1997) give an unifying framework to the source separation problem by explaining the relation of the different algorithms to each other. The derived learning rule is optimized when the natural gradient (Amari, 1997) or relative gradient (Cardoso and Laheld, 1996) is used.

The originally proposed algorithm by Bell and Sejnowski (1995) was suited to separate super-Gaussian sources. To overcome this limitation other techniques have been developed that were able to simultanously separate sub- and super-Gaussian sources. Pearlmutter and Parra (1996) derive a generalized ICA learning rule from the maximum likelihood estimation where they explicitly model the underlying source distribution which was assumed to be fixed in the original algorithm. Although the density estimation was burdensome and required a sufficient amount of data, the algorithm was able to separate a fairly large number of source with a wide range of distributions. A more simple and highly effective algorithm has been proposed by Girolami and Fyfe (1996) from the negentropy maximization approach. They used an extended exploratory projection pursuit network with inhibitory lateral connections that could separate sub- and super-Gaussian sources. Lee, Girolami and Sejnowski (1997) derive the same learning rule from the infomax approach preserving the simple architecture and showing superior convergence speed.

Extensive simulations have been performed to demonstrate the power of the learning algorithm. However, instantaneous mixing and unmixing simulations are {\em toy} problems and the challenge lies in dealing with real world data. Makeig et al. (1996) applied the original infomax algorithm to EEG and ERP data showing that the algorithm can extract EEG activations and isolate artifacts. Jung et al. (1997) show that the extended infomax algorithm is able to linearly decompose EEG artifacts such as line noise, eye blinks, and cardiac noise into independent components with sub- and super-Gaussian distributions. McKeown et. al. (1997) have used the extended ICA algorithm to investigate task-related human brain activity in fMRI data. By determining the brain regions that contained significant amounts of specific temporally independent components, they were able to specify the spatial distribution of transiently task-related brain activations. Other potential applications may result from exploring independent features in natural images. Bell and Sejnowski (1997) suggest that independent components of natural scenes are edge filters. The filters are localized, mostly oriented and similar to Gabor like filters. The outputs of the ICA filters are sparsely distributed. Bartlett and Sejnowski (1997) and Gray, Movellan and Sejnowski (1997) demonstrate the successful use of the ICA filters as features in face recognition tasks and lipreading tasks respectively.

For these applications, the instantaneous mixing model may be appropriate because the propagation delays are negligible. However, in real environments substantial time-delays may occur and an architecture and algorithm is needed to account for the mixing of time-delayed sources and convolved sources. The multichannel blind source separation problem has been addressed by Yellin and Weinstein (1994) and Ngyuen and Jutten (1995) and others based on $4^{th}$-order cumulants criteria. An extension to time-delays and convolved sources from the infomax viewpoint using a feedback architecture has been developed by Torkkola (1996). Lee, Bell and Lambert (1997) have extended the blind source separation problem to a full feedback system and a full feedforward system. The feedforward architecture allows the inversion of non-minimum phase systems. In addition, the rules are extended using polynomial filter matrix algebra in the frequency domain (Lambert, 1996). The proposed method can successfully separate voices and music recorded in a real environment. Lee, Bell and Orglmeister (1997) show that the recognition rate of an automatic speech recognition system is increased after separating the speech signals.

Since ICA is restricted and relies on several assumptions researchers have started to tackle a few limitations of ICA. One obvious but non-trivial extension is the nonlinear mixing model. In (Hermann and Yang, 1996; Lin and Cowan, 1997; Pajunen, 1997) nonlinear components are extracted using self-organizing-feature-maps (SOFM). Other researchers (Burel, 1992; Lee, Koehler and Orglmeister, 1997; Taleb and Jutten, 1997; Yang, Amari and Cichocki, 1997) have used a more direct extension to the previously presented ICA models. They include certain flexible nonlinearities in the mixing model and the goal is to invert the linear mixing matrix as well as the nonlinear mixing. Other limitations such as the under-determined problem in ICA, i.e. having less sensors than sources and noise models in the ICA formulation are subject to current research efforts.

ICA is a generally applicable method to several challenges in signal processing. It reveals a diversity of theoretical questions and opens a variety of potential applications. Successful results in EEG, fMRI, speech recognition and face recognition systems indicate the power and optimistic hope in this paradigm.