<s>
EigenMoments	B-Algorithm
is	O
a	O
set	O
of	O
orthogonal	O
,	O
noise	O
robust	O
,	O
invariant	O
to	O
rotation	O
,	O
scaling	B-Algorithm
and	O
translation	O
and	O
distribution	O
sensitive	O
moments	O
.	O
</s>
<s>
Their	O
application	O
can	O
be	O
found	O
in	O
signal	O
processing	O
and	O
computer	B-Application
vision	I-Application
as	O
descriptors	O
of	O
the	O
signal	O
or	O
image	O
.	O
</s>
<s>
The	O
descriptors	O
can	O
later	O
be	O
used	O
for	O
classification	B-General_Concept
purposes	O
.	O
</s>
<s>
It	O
is	O
obtained	O
by	O
performing	O
orthogonalization	O
,	O
via	O
eigen	O
analysis	O
on	O
geometric	B-Algorithm
moments	I-Algorithm
.	O
</s>
<s>
EigenMoments	B-Algorithm
are	O
computed	O
by	O
performing	O
eigen	O
analysis	O
on	O
the	O
moment	O
space	O
of	O
an	O
image	O
by	O
maximizing	O
signal-to-noise	O
ratio	O
in	O
the	O
feature	B-Algorithm
space	I-Algorithm
in	O
form	O
of	O
Rayleigh	O
quotient	O
.	O
</s>
<s>
Dependency	O
of	O
moments	O
in	O
the	O
moment	O
space	O
on	O
the	O
distribution	O
of	O
the	O
images	O
being	O
transformed	O
,	O
ensures	O
decorrelation	O
of	O
the	O
final	O
feature	B-Algorithm
space	I-Algorithm
after	O
eigen	O
analysis	O
on	O
the	O
moment	O
space	O
.	O
</s>
<s>
The	O
ability	O
of	O
EigenMoments	B-Algorithm
to	O
take	O
into	O
account	O
distribution	O
of	O
the	O
image	O
makes	O
it	O
more	O
versatile	O
and	O
adaptable	O
for	O
different	O
genres	O
.	O
</s>
<s>
This	O
makes	O
EigenMoments	B-Algorithm
robust	O
for	O
classification	B-General_Concept
applications	O
.	O
</s>
<s>
Dimension	O
of	O
signal	O
space	O
,	O
n	O
,	O
is	O
often	O
too	O
large	O
to	O
be	O
useful	O
for	O
practical	O
application	O
such	O
as	O
pattern	B-General_Concept
classification	I-General_Concept
,	O
we	O
need	O
to	O
transform	O
the	O
signal	O
space	O
into	O
a	O
space	O
with	O
lower	O
dimensionality	O
.	O
</s>
<s>
where	O
is	O
the	O
transformed	O
signal	O
,	O
a	O
fixed	O
transformation	O
matrix	O
which	O
transforms	O
the	O
signal	O
into	O
the	O
moment	O
space	O
,	O
and	O
the	O
transformation	O
matrix	O
which	O
we	O
are	O
going	O
to	O
determine	O
by	O
maximizing	O
the	O
SNR	O
of	O
the	O
feature	B-Algorithm
space	I-Algorithm
resided	O
by	O
.	O
</s>
<s>
For	O
the	O
case	O
of	O
Geometric	B-Algorithm
Moments	I-Algorithm
,	O
X	O
would	O
be	O
the	O
monomials	O
.	O
</s>
<s>
Finding	O
that	O
maximizes	O
the	O
SNR	O
of	O
the	O
feature	B-Algorithm
space	I-Algorithm
:	O
</s>
<s>
and	O
and	O
,	O
both	O
are	O
symmetric	O
and	O
is	O
positive	B-Algorithm
definite	I-Algorithm
and	O
therefore	O
invertible	O
.	O
</s>
<s>
Scaling	B-Algorithm
does	O
not	O
change	O
the	O
value	O
of	O
the	O
object	O
function	O
and	O
hence	O
and	O
additional	O
scalar	O
constraint	O
can	O
be	O
imposed	O
on	O
and	O
no	O
solution	O
would	O
be	O
lost	O
when	O
the	O
objective	O
function	O
is	O
optimized	O
.	O
</s>
<s>
Dimension	B-Algorithm
reduction	I-Algorithm
can	O
be	O
performed	O
by	O
simply	O
choosing	O
the	O
first	O
components	O
,	O
,	O
with	O
the	O
highest	O
values	O
for	O
out	O
of	O
the	O
components	O
,	O
and	O
discard	O
the	O
rest	O
.	O
</s>
<s>
Interpretation	O
of	O
this	O
transformation	O
is	O
rotating	O
and	O
scaling	B-Algorithm
the	O
moment	O
space	O
,	O
transforming	O
it	O
into	O
a	O
feature	B-Algorithm
space	I-Algorithm
with	O
maximized	O
SNR	O
and	O
therefore	O
,	O
the	O
first	O
components	O
are	O
the	O
components	O
with	O
highest	O
SNR	O
values	O
.	O
</s>
<s>
Since	O
is	O
positive	B-Algorithm
definite	I-Algorithm
,	O
thus	O
.	O
</s>
<s>
Eigenmoments	B-Algorithm
are	O
derived	O
by	O
applying	O
the	O
above	O
framework	O
on	O
Geometric	B-Algorithm
Moments	I-Algorithm
.	O
</s>
<s>
the	O
monomials	O
,	O
after	O
the	O
transformation	O
we	O
obtain	O
Geometric	B-Algorithm
Moments	I-Algorithm
,	O
denoted	O
by	O
vector	O
,	O
of	O
signal	O
,	O
i.e.	O
</s>
<s>
Taking	O
,	O
the	O
dimension	O
of	O
moment	O
space	O
as	O
and	O
the	O
dimension	O
of	O
feature	B-Algorithm
space	I-Algorithm
as	O
,	O
we	O
will	O
have	O
:	O
</s>
<s>
The	O
derivation	O
for	O
2D	O
signal	O
is	O
the	O
same	O
as	O
1D	O
signal	O
except	O
that	O
conventional	O
Geometric	B-Algorithm
Moments	I-Algorithm
are	O
directly	O
employed	O
to	O
obtain	O
the	O
set	O
of	O
2D	O
EigenMoments	B-Algorithm
.	O
</s>
<s>
The	O
definition	O
of	O
Geometric	B-Algorithm
Moments	I-Algorithm
of	O
order	O
for	O
2D	O
image	O
signal	O
is	O
:	O
</s>
<s>
Then	O
the	O
set	O
of	O
2D	O
EigenMoments	B-Algorithm
are	O
:	O
</s>
<s>
where	O
is	O
a	O
matrix	O
that	O
contains	O
the	O
set	O
of	O
EigenMoments	B-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
obtain	O
a	O
set	O
of	O
moment	B-Algorithm
invariants	I-Algorithm
we	O
can	O
use	O
normalized	O
Geometric	B-Algorithm
Moments	I-Algorithm
instead	O
of	O
.	O
</s>
<s>
Normalized	O
Geometric	B-Algorithm
Moments	I-Algorithm
are	O
invariant	O
to	O
Rotation	O
,	O
Scaling	B-Algorithm
and	O
Transformation	O
and	O
defined	O
by	O
:	O
</s>
<s>
in	O
this	O
equation	O
is	O
a	O
scaling	B-Algorithm
factor	O
depending	O
on	O
the	O
image	O
.	O
</s>
