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The Zernike moment of order
n
with repetition
m
for a
N
×
N
pixels size
(
,
)
continuous image function
f
x
y
, that vanishes outside the unit disk, has the form:
2
ˀ
1
n
+
1
V
nm
(
Z
nm
=
r
,ʸ)
f
(
r
,ʸ)
rdrd
ʸ
(13.5)
ˀ
0
0
where the symbol
corresponds to conjugate.
For a digital image, the integrals are replaced according to the zeroth order approx-
imation Eq. (
13.2
) by summations to get:
(
∗
)
N
−
1
N
−
1
n
+
1
V
nm
(
Z
nm
=
r
ij
,ʸ
ij
)
f
(
i
,
j
).
(13.6)
ˀ
i
=
0
j
=
0
The above transformation from continuous to discrete form adds some approxi-
mation errors. For this reason, several attempts [
32
] to decrease these approximation
errors have been reported in the literature. Moreover, significant work has been done
[
7
] in the last years towards the fast computation of the radial polynomials (Eq.
13.4
)
and the moments (Eq.
13.6
).
13.2.1.2 Legendre Moments
(
+
)
(
,
)
The
n
m
th order Legendre moment [
6
] of an intensity function
f
x
y
, is defined
in [
−
1,1] as:
1
1
L
nm
=
(
2
n
+
1
)(
2
m
+
1
)
P
n
(
x
)
P
m
(
y
)
f
(
x
,
y
)
dxdy
(13.7)
4
−
1
−
1
where
P
n
(
x
)
is the
n
th order Legendre polynomial defined as:
d
dx
n
n
1
2
n
n
0
ʱ
k
,
n
x
k
x
2
n
P
n
(
x
)
=
=
(
−
1
)
(13.8)
!
k
=
The above Legendre polynomials satisfy the following recursive equation:
)
=
(
)
/
P
n
(
x
2
n
−
1
)
xP
n
−
1
(
x
)
−
(
n
−
1
)
P
n
−
2
(
x
n
(13.9)
P
0
(
)
=
,
P
1
(
)
=
x
1
x
x
The recursive formula of Eq. (
13.10
) permits the fast computation of the Legendre
polynomials by using polynomials of lower order. In case of computing the Legendre
moments of a
N
×
N
image, Eq. (
13.7
) takes the following discrete form:
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