Mathematics in Independent Component Analysis

114 Chapter 6. LNCS 3195:726-733, 2004
Second-order blind source separation based on
multi-dimensional autocovariances
Fabian J. Theis 1,2 , Anke Meyer-Baese 2 , and Elmar W. Lang 1
1 Institute of Biophysics,
University of Regensburg, D-93040 Regensburg, Germany
2 Department of Electrical and Computer Engineering,
Florida State University, Tallahassee, FL 32310-6046, USA
fabian@theis.name
Abstract. SOBI is a blind source separation algorithm based on time
decorrelation. It uses multiple time autocovariance matrices, and performs
joint diagonalization thus being more robust than previous time
decorrelation algorithms such as AMUSE. We propose an extension called
mdSOBI by using multidimensional autocovariances, which can be calculated
for data sets with multidimensional parameterizations such as
images or fMRI scans. mdSOBI has the advantage of using the spatial
data in all directions, whereas SOBI only uses a single direction. These
findings are confirmed by simulations and an application to fMRI analysis,
where mdSOBI outperforms SOBI considerably.
Blind source separation (BSS) describes the task of recovering the unknown
mixing process and the underlying sources of an observed data set. Currently,
many BSS algorithm assume independence of the sources (ICA), see for instance
[1, 2] and references therein. In this work, we consider BSS algorithms based on
time-decorrelation. Such algorithms include AMUSE [3] and extensions such as
SOBI [4] and the similar TDSEP [5]. These algorithms rely on the fact that
the data sets have non-trivial autocorrelations. We give an extension thereof
to data sets, which have more than one direction in the parametrization, such
as images, by replacing one-dimensional autocovariances by multi-dimensional
autocovariances.
The paper is organized as follows: In section 1 we introduce the linear mixture
model; the next section 2 recalls results on time decorrelation BSS algorithms.
We then define multidimensional autocovariances and use them to propose md-
SOBI in section 3. The paper finished with both artificial and real-world results
in section 4.
1 Linear BSS
We consider the following blind source separation (BSS) problem: Let x(t) be
an (observed) stationary m-dimensional real stochastical process (with not necessarily
discrete time t) and A an invertible real matrix such that
x(t) = As(t) + n(t) (1)