Information Technology Reference
In-Depth Information
N
−
1
N
−
1
L
nm
=
(
2
n
+
1
)(
2
m
+
1
)
P
n
(
x
)
P
m
(
y
)
f
(
x
,
y
).
(13.10)
(
N
−
1
)(
N
−
1
)
x
=
0
y
=
0
The computation of the Legendre moments through Eq. (
13.10
) shows significant
approximation errors as discussed lately and the resulted Legendre moments do not
satisfy the properties of the theoretical ones, by affecting their ability to describe the
image in process. For this reason new algorithms ensuring the accurate computation
of the moments have been proposed [
6
].
13.2.1.3 Gaussian-Hermite Moments
Gaussian-Hermite moments are continuous moments that have been introduced in
image analysis quite recently by Yang and Dai [
33
]. The
(
n
+
m
)
th order Gaussian-
Hermite moment is defined in
(
−∞
,
+∞
)
and has the form:
∞
∞
H
n
(
;
˃)
H
m
(
GH
nm
=
x
y
;
˃)
f
(
x
,
y
)
dxdy
(13.11)
−∞
−∞
where
e
−
x
2
2
H
n
(
1
2
n
n
H
n
(
/
2
˃
x
;
˃)
=
x
;
˃)
(13.12)
!
˃
√
ˀ
is the weighted Hermite orthonormal polynomial of order
n
, derived by the ordinary
Hermite polynomial
H
n
(
x
;
˃)
, modulated by a Gaussian function with
˃
variance.
The ordinary Hermite polynomial of order
n
is defined as:
n
/
2
k
(
−
1
)
n
−
2
k
H
n
(
x
)
=
n
!
)
!
(
2
x
)
(13.13)
k
!
(
n
−
2
k
k
=
0
The recursive computation of the aboveHermite polynomials is performed accord-
ing to:
H
n
+
1
(
x
)
=
2
xH
n
(
x
)
−
2
nH
n
−
1
(
x
),
for
n
≥
1
H
0
(
x
)
=
1
,
H
1
(
x
)
=
2
x
(13.14)
The Gaussian-Hermite moments have proved to be of higher image representation
ability [
33
], compared to some traditional moment families e.g. Legendre and thus
their usage has been rapidly increased in many applications of the engineering life.
Due to this popularity, a faster and more accurate computation algorithm has been
proposed recently [
8
].
Search WWH ::
Custom Search