Digital Signal Processing Reference
In-Depth Information
|
q
|
(
k
- |
q
|
)
(
|
q
l
)
w
l
B
pqk
m
k
-2
j
-
l
,2
j
+
l
k
-|
q
|
2
A
pq
=
p
+ 1
p
2
j
(17)
∑
k
=|
q
|,|
q
|+2,
…
j
=0
l
=0
π
where
w
=
{
-
i
;
q
>0
(18)
i
;
q
≤0
The Zernike polynomials satisfy the relation of orthogonality
2
π
1
V
p
*
(
r
,
θ
)
V
kl
(
r
,
θ
)
rdrdθ
=
p
+ 1
δ
kp
δ
lq
(19)
∫
0
∫
0
The recurrence relation for the radial part is
R
pq
(
r
)
=
2
pr
p
+
q
R
p
-1,
q
-1
(
r
)
-
p
-
q
p
+
q
R
p
-2.
q
(
r
)
(20)
Computation of the Zernike polynomials by this formula must commence with
R
pp
(
r
)
=
R
p
,-
p
(
r
)
=
r
p
,
p
=0, 1, …
2.4. Discrete moments
There is a group of orthogonal polynomials defined directly on a series of points and therefore
they are especially suitable for digital images. Some of these polynomials such as Tchebichef,
Krawtchouk, Hahn, dual Hahn and Meixner polynomials.
2.4.1. Tchebichef moments
The 2D TMs of order (
p
+
q
) of an image intensity function
f
(
n
,
m
) with size
N
×
M
is defined
as
N
-1
M
-1
T
pq
=
A
(
p
,
N
)
A
(
q
,
M
)
n
=0
m
=0
t
p
(
n
)
t
q
(
m
)
f
(
n
,
m
)
(21)
where
t
p
(
n
)
is the
p
th
order orthogonal discrete Tchebichef polynomial defined by [8]
(
-1
)
p
-
k
(
N
- 1 -
k
p
-
k
)(
p
+
p
)(
k
)
p
t
p
(
n
)
=
p
!
k
=0
(22)
and
A
(
p
,
N
)
=
β
(
p
,
N
)
ρ
(
p
,
N
)
