Image Processing Reference
In-Depth Information
16.5.2.2
Nonlinear Decorrelation
The data matrix
X
of dimension
N
×
M
is a set of
M
observations of
N
variables
such that it is possible to write
represents a time
index if we have
N
time series, or a spatial index if we have
N
images. The
independent components that we have to find can be written as
Each observed variable
x
i
can be considered as a linear combination of the
unknown statistically independent sources
s
i
, so that we can write
x
=
{
x
( ),
ν
…
,
x
( )},
ν
where
ν
1
N
s
=
{
s
( ),...,
ν
s
( )}.
ν
1
N
=+
…
+
11
xasas
as
(16.32)
i
i
i
2 2
iNN
or, in matrix notation
xAs
=
(16.33)
The goal is to find an unmixing matrix, such that
sWx
=
(16.34)
are an estimate of the original sources. A first general assumption is that both the
original sources and the observed variables have zero mean. Because we are
interested in signal changes, the mean value does not carry any information, and
hence we can remove it by means of a centering stage so that
x
becomes
x
−
E
{
x
}.
In compact notation
xxEx
←−
{}
(16.35)
where
E
{·} is the expectation operator. One principle that can be used to find the
independent components is nonlinear decorrelation. We can consider two statis-
tical variables
x
and
y
to be nonlinearly uncorrelated if
E
{()()}
gxhy
=
0
(16.36)
where at least one function between
g
(·) and
h
(·) is a nonlinear one. The problem
is how to choose these functions such that this condition implies statistical
independence between
x
and
y
. It can be demonstrated that if two random variables
x
and
y
are statistically independent, then we have, for any absolutely integrable
function of
x
and
y
,
g
(·) and
h
(·),
E
{ ( ) ( )}
gxhy
=
E
{ ( )}
gx
E
{ ( )}
hy
(16.37)
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