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between the moments [20]. Due to these characteristics, Zernike moment have been
utilized as feature set in different applications such as object classification, shape
analysis, content based image retrieval etc.
Zernike moment (ZM) owns following properties [21,22]:
(i)
Zernike moments are rotation, translation and scale invariant [23].
(ii)
Zernike moments are robust to noise and minor variation in shape.
(iii)
Since the basis of Zernike moment is orthogonal, therefore they have mini-
mum information redundancy.
(iv)
Zernike moment can characterize the global shape of pattern. Lower order
moments represent the global shape pattern and higher order moment repre-
sents the detail.
(v)
An image can be better described by a small set of its Zernike moments than
any other types of moments.
Zernike moment is a set of complex polynomial which form a complete orthogonal
set over the interior of the unit circle of
x
2
+ y
2
≤
1 [24,25]. These polynomials are of
the form,
Vxy Vr
(, )
=
(, )
ʸ
=
Rr
().exp(
jn
ʸ
)
(1)
mn
mn
mn
where
m
is positive integer and
n
is integer subject to constraints
m-|n|
even and
|n|≤m
,
r
is the length of vector from the origin to pixel
(x,y)
and
is the angle be-
ʸ
tween vector
r
and
x-
axis in counter clock wise direction,
R
()
r
is the Zernike radial
mn
polynomials in (, )
r
ʸ
polar coordinates and defined as
mn
−
||
2
s
m
−
2
s
(1)(
−
msr
−
)!
Rr
()
=
(2)
(
(
)
)
mn
mn
+
||
mn
−
||
s
!
−
s
!
−
s
!
s
=
0
2
2
R
()
rRr
=
()
here
mn
,
−
mn
The above mentioned polynomial in equation (2) is orthogonal and satisfies the
othogonality principle
Zernike moments are the projection of image function
I(x,y)
onto these orthogonal
basis function. The Othogonality condition simplifies the representation of the origi-
nal image because generated moments are independent [26].
The Zernike moment of order
m
with repetition
n
for a continuous image function
I(x,y)
that vanishes outside the unit circle is
m
+
1
Z
=
Ixy V
( ,
)[
( ,
r
ʸ
)]
dxdy
(3)
mn
mn
ˀ
22
1
xy
+≤
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