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In practice, the computation of the weighted Krawtchouk polynomials is not per-
formed through Eq. (
13.22
), since this is a very time consuming procedure; instead,
a recursive algorithm [
34
] is applied:
p
(
N
−
n
)
K
n
+
1
(
x
;
p
,
N
)
=
A
(
Np
−
2
np
+
n
−
x
)
K
n
(
x
;
p
,
N
)
−
Bn
(
1
−
p
)
K
n
−
1
(
x
;
p
,
N
)
(13.26)
where
p
2
p
(
N
−
n
)
(
N
−
n
)(
N
−
n
+
1
)
A
=
)
,
B
=
(13.27)
2
(
1
−
p
)(
n
+
1
(
1
−
p
)
(
n
+
1
)
n
and
w
w
1
(
x
;
p
,
N
)
x
pN
(
x
;
p
,
N
)
K
0
(
x
;
p
,
N
)
=
)
,
K
1
(
x
;
p
,
N
)
=
−
)
(13.28)
The Krawtchouk moments proved to be effective local descriptors, since they can
describe the local features of an image, unlike the other moment families, which
capture only the global features of the objects they describe. This locality property
is controlled by appropriate adjustment of the
p
1
,
p
2
parameters of Eq. (
13.21
).
ˁ (
0
;
p
,
N
ˁ (
1
;
p
,
N
13.2.2.3 Dual Hahn Moments
The
(
n
+
m
)
th order orthogonal dual Hahn moment [
37
]ofa
N
×
N
image having
intensity function
f
(
x
,
y
)
is defined as:
b
−
1
b
−
1
W
(
c
)
)
W
(
c
)
W
nm
=
(
x
,
a
,
b
(
y
,
a
,
b
)
f
(
x
,
y
) ,
n
m
x
=
a
y
=
a
n
,
m
=
0
,
1
,...,
N
−
1
(13.29)
1
2
where
−
<
a
<
b
,
|
c
|
<
1
+
a
,
b
=
a
+
N
and
x
s
ˁ(
s
)
1
2
W
(
c
)
n
W
(
c
)
n
(
s
,
a
,
b
)
=
(
s
,
a
,
b
)
ʔ
−
(13.30)
d
n
is the
n
th order weighted dual Hahn polynomial used to reduce the numerical insta-
bilities caused by the ordinary dual Hahn polynomials, defined as:
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