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Biomedical Signal and Image Processing
Spring 2008
Chapter 15  BLIND SOURCE SEPARATION:
Principal & Independent Component Analysis
c G.D. Clifford 20052008
Introduction
In this chapter we will examine how we can generalize the idea of transforming a time
series into an alternative representation, such as the Fourier (frequency) domain, to facil

itate systematic methods of either removing (ﬁltering) or adding (interpolating) data.
In
particular, we will examine the techniques of
Principal Component Analysis
(PCA) using
Singular Value Decomposition
(SVD), and
Independent Component Analysis
(ICA). Both
of these techniques utilize a representation of the data in a statistical domain rather than
a time or frequency domain.
That is, the data are projected onto a new set of axes that
fulﬁll some statistical criterion, which implies independence, rather than a set of axes that
represent discrete frequencies such as with the Fourier transform, where the independence
is
assumed
.
Another important difference between these statistical techniques and Fourierbased tech

niques is that the Fourier components onto which a data segment is projected are ﬁxed,
whereas PCA or ICAbased transformations
depend
on the structure of the data being ana

lyzed. The axes onto which the data are projected are therefore
discovered
. If the structure
of the data (or rather the statistics of the underlying sources) changes over time, then the
axes onto which the data are projected will change too
1
.
Any projection onto another set of axes (or into another space) is essentially a method for
separating the data out into separate components or
sources
which will hopefully allow
us to see important structure more clearly in a particular projection. That is, the direction
of projection increases the signaltonoise ratio (SNR) for a particular signal source.
For
example, by calculating the power spectrum of a segment of data, we hope to see peaks
at certain frequencies.
The power (amplitude squared) along certain frequency vectors
is therefore high, meaning we have a strong component in the signal at that frequency.
By discarding the projections that correspond to the unwanted sources (such as the noise
or artifact sources) and inverting the transformation,
we effectively perform a ﬁltering
of the recorded observation.
This is true for both ICA and PCA as well as Fourierbased
techniques.
However, one important difference between these techniques is that Fourier
techniques
assume
that the projections onto each frequency component are independent
of the other frequency components. In PCA and ICA we attempt to
ﬁnd
a set of axes which
are independent of one another in some sense. We assume there are a set of independent
(The structure of the data can change because existing sources are nonstationary, new signal sources manifest, or
the manner in which the sources interact at the sensor changes.
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 Spring '07
 JulieGreenberg
 Linear Algebra, Image processing, MIT OpenCourseWare, Singular value decomposition, biomedical signal

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