<s>
A	O
whitening	B-Algorithm
transformation	I-Algorithm
or	O
sphering	B-Algorithm
transformation	I-Algorithm
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
that	O
transforms	O
a	O
vector	O
of	O
random	O
variables	O
with	O
a	O
known	O
covariance	O
matrix	O
into	O
a	O
set	O
of	O
new	O
variables	O
whose	O
covariance	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
meaning	O
that	O
they	O
are	O
uncorrelated	O
and	O
each	O
have	O
variance	O
1	O
.	O
</s>
<s>
the	O
decorrelation	B-Algorithm
transform	O
removes	O
only	O
the	O
correlations	O
but	O
leaves	O
variances	O
intact	O
,	O
</s>
<s>
a	O
coloring	O
transformation	O
transforms	O
a	O
vector	O
of	O
white	O
random	O
variables	O
into	O
a	O
random	B-General_Concept
vector	I-General_Concept
with	O
a	O
specified	O
covariance	O
matrix	O
.	O
</s>
<s>
a	O
whitening	O
matrix	O
satisfying	O
the	O
condition	O
yields	O
the	O
whitened	O
random	B-General_Concept
vector	I-General_Concept
with	O
unit	O
diagonal	O
covariance	O
.	O
</s>
<s>
Optimal	O
whitening	B-Algorithm
transforms	I-Algorithm
can	O
be	O
singled	O
out	O
by	O
investigating	O
the	O
cross-covariance	O
and	O
cross-correlation	O
of	O
and	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
unique	O
optimal	O
whitening	B-Algorithm
transformation	I-Algorithm
achieving	O
maximal	O
component-wise	O
correlation	O
between	O
original	O
and	O
whitened	O
is	O
produced	O
by	O
the	O
whitening	O
matrix	O
where	O
is	O
the	O
correlation	O
matrix	O
and	O
the	O
variance	O
matrix	O
.	O
</s>
<s>
An	O
empirical	O
whitening	B-Algorithm
transform	I-Algorithm
is	O
obtained	O
by	O
estimating	B-General_Concept
the	I-General_Concept
covariance	I-General_Concept
(	O
e.g.	O
</s>
<s>
The	O
advantage	O
of	O
these	O
whitening	B-Algorithm
transformations	I-Algorithm
is	O
that	O
they	O
can	O
be	O
optimized	O
according	O
to	O
the	O
underlying	O
topological	O
properties	O
of	O
the	O
data	O
(	O
smoothness	O
,	O
continuity	O
and	O
contiguity	O
)	O
,	O
thus	O
producing	O
more	O
robust	O
whitening	O
representations	O
.	O
</s>
<s>
An	O
implementation	O
of	O
several	O
whitening	O
procedures	O
in	O
R	B-Language
,	O
including	O
ZCA-whitening	O
and	O
PCA	O
whitening	O
but	O
also	O
CCA	O
whitening	O
,	O
is	O
available	O
in	O
the	O
"	O
whitening	O
"	O
R	B-Language
package	O
published	O
on	O
CRAN	O
.	O
</s>
<s>
The	O
R	B-Language
package	O
"	O
pfica	O
"	O
allows	O
the	O
computation	O
of	O
high-dimensional	O
whitening	O
representations	O
using	O
basis	O
function	O
systems	O
(	O
B-splines	B-Algorithm
,	O
Fourier	O
basis	O
,	O
etc	O
.	O
</s>
