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The computation speed of Tchebichef moments can be accelerated by using the
following recursive formula:
1
2
−
(
n
−
1
)
nt
n
(
)
t
n
−
1
(
t
n
−
2
(
x
)
=
(
2
n
−
1
x
)
−
(
n
−
1
)
x
)
N
2
(13.20)
2
x
+
1
−
N
t
0
(
=
,
t
1
(
)
=
x
)
1
x
N
Tchebichefmoments have proved to be superior toZernike andLegendremoments
in describing objects, while their robustness in the presence of high noise levelsmakes
them appropriate to real-time pattern classification and computer vision applications.
13.2.2.2 Krawtchouk Moments
The Krawtchouk orthogonal moments are the second type of discrete moments intro-
duced in image analysis by Yap et al. [
34
]. The
(
n
+
m
)
th order orthogonal discrete
Krawtchouk moment of a
N
×
N
image having intensity function
f
(
x
,
y
)
is defined
as:
N
−
1
N
−
1
K
nm
=
K
n
(
x
;
p
1
,
N
−
1
)
K
m
(
y
;
p
2
,
N
−
1
)
f
(
x
,
y
)
(13.21)
x
=
y
=
0
0
where
w
(
x
;
p
,
N
)
K
n
(
x
;
p
,
N
)
=
K
n
(
x
;
p
,
N
)
(13.22)
ˁ (
n
;
p
,
N
)
is the weighted Krawtchouk polynomial of
n
order, used to reduce the numerical
fluctuations presented in the ordinary Krawtchouk polynomials, defined as:
)
=
2
F
1
N
1
p
0
ʱ
k
,
n
,
p
x
k
K
n
(
x
;
p
,
N
−
n
,
−
x
;−
N
;
=
.
(13.23)
k
=
In Eq. (
13.22
)
ˁ (
n
;
p
,
N
)
is the norm of the Krawtchouk polynomials having the
form:
n
1
n
−
p
n
!
ˁ
(
;
,
)
=
(
−
)
n
,
=
,...,
n
p
N
1
)
n
1
N
(13.24)
p
(
−
N
and
w
(
x
;
p
,
N
)
is the weight function of the Krawtchouk moments,
N
x
p
x
N
−
x
w
(
x
;
p
,
N
)
=
(
1
−
p
)
(13.25)
In Eq. (
13.24
) the symbol
(
·
)
n
is the Pochhammer symbol, which for the general
case is defined as
(ʱ)
k
=
ʱ (ʱ
+
1
) ...(ʱ
+
k
+
1
)
.
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