Image Processing Reference
In-Depth Information
The problem was faced
by Herault and Jutten [72] and then by Cichocki and
Unbenhaven [83]. Herault and Jutten proposed using smooth functions that can
be expanded in a Taylor series around zero, i.e.,
1
2
gx
()
=+′
g
()
0
g
()
0
x
+
g
′′
()
0
x
2
+
(16.38)
The expectation operator applied to the Taylor expansion of these func-
tions introduces higher-order moments of the random variables
E
{
x
i
},
with A sufficient condition for Equation 16.36 to hold and for the
variables to be nonlinearly uncorrelated is that
x
and
y
be independent so that
Equation 16.37 is valid and either or is zero for each
i
. For this
property to be satisfied, either
g
(
x
) or
h
(
y
) must be an odd function with zero
mean. With these assumptions, looking for a matrix
W
such that the estimated
sources
s
i
and
s
j
as well as
g
(
s
i
) and
h
(
s
j
) are uncorrelated for any
i
i
=
1, ,...,
∞
.
E
{}
x
i
E
{}
y
i
j
may
lead to finding independent components. Herault and Jutten proposed the use
of
g
(
x
)
≠
x
3
and
h
(
y
)
=
arctan(
y
). However, this algorithm showed some con-
vergence problems and cannot be used to separate many sources. Cichocki
and Unbenhaven proposed an extension of this algorithm that makes use of a
feedforward network. The criterion used is that of nonlinear decorrelation,
and it is the same as that used in the Amari natural gradient algorithm [79].
=
16.5.2.2.1 Whitening as a Preprocessing Step
It is noteworthy that nonlinear decorrelation is stronger than a linear decorrela-
tion. In fact, uncorrelation is derived from Equation 16.37, where
g
(
x
) and
h
(
y
)
are linear functions. For example, even if
x
cos
(
t
) are uncorre-
lated, it can be easily shown that
x
2
and
y
2
are correlated; in fact
x
2
+
y
2
= 1.
Although independence implies uncorrelation, the converse does not hold, and
a (linear) decorrelation method will not give independent components. However,
a decorrelation step is often used as a preprocessing stage in ICA. In order to
simplify successive algorithmic steps, a whitening operation is often performed:
whitening means that the zero mean observed variables
x
i
are transformed into
a new set of variables that are uncorrelated and have unit variance. After this
operation,
E
{
x
i
x
j
}
=
sin
(
t
) and
y
=
δ
ij
holds and the variables are said to be whitened or sphered.
After the data have been sphered, the search for the independent components is
simplified because the new mixing
=
ˆ
AWA
S
=
(16.39)
where
W
S
is the whitening or sphering matrix, becomes orthogonal and so an
estimate of
N
(
N
1)/2 elements is needed to solve the ICA problem, as against
an estimate of
N
2
elements of the matrix
A
. The whitening matrix can be computed
−
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