Digital Signal Processing Reference
In-Depth Information
1
2
2
q
+ 1
δ
pq
P
p
(
x
)
P
q
(
x
)
dx
=
(11)
∫
-1
The recurrence relation, which can be used for efficient computation of the Legendre polyno‐
mials, is
P
0
(
x
)
=1,
P
1
(
x
)
=
x
,
P
p
+1
(
x
)
=
2
p
+ 1
(12)
p
p
+ 1
P
p
-1
(
x
)
p
+ 1
xP
p
(
x
)
-
2.3.2. Zernike moments
Zernike moments
(ZMs) were introduced into image analysis about 30 years ago by Teague [9]
who used ZMs to construct rotation invariants. He used the fact that the ZMs keep their
magnitude under arbitrary rotation. He also showed that the Zernike invariants of the second
and third orders are equivalent to the Hu invariants when expressed in terms of geometric
moments. He presented the invariants up to the eighth order in explicit form but no general
rule about how to derive them was given. Later, Wallin [31] described an algorithm for the
formation of rotation invariants of any order. Numerical properties and possible applications
of ZMs in image processing among others. ZMs of the
n
th order with repetition
l
are defined
as
2
π
1
A
pq
=
p
+ 1
V
p
*
(
r
,
θ
)
f
(
r
,
θ
)
rdrdθ
,
p
=0, 1, 2, …
q
= -
p
, -
p
+2, …,
p
(13)
∫
0
∫
0
π
i.e. the difference
n
-
|
l
|
is always even. The asterisk means the complex conjugate. The
Zernike polynomials
are defined as products
V
pq
(
r
,
θ
)
=
R
pq
(
r
)
e
iqθ
(14)
where the radial part is
p
R
pq
(
r
)
= ∑
k
=
|
q
|
,
|
q
|
+2,
…
B
pqk
r
k
(15)
The coefficients
p
-
k
2
(
p
+
2
)
!
(
p
-
2
)
!
(
k
+
2
)
!
(
k
-
2
)
!
(
-1
)
(16)
B
pqk
=
can be used for conversion from geometric moments,
