Digital Signal Processing Reference
In-Depth Information
N
p
y
p
N
-
p
=
p
!
k
=0
s
1
(
p
,
i
)
x k
(
N
-
k
)
i
i
=0
(48)
where
s
1
(
p
,
i
)
are the Stirling numbers of the first kind [28], which satisfy
(
N
-
k
)!
(
N
-
k
-
p
)
!
=
i
=0
p
s
1
(
p
,
i
)(
N
-
k
)
i
(49)
Using (3.2), we can rewrite (3.15) in terms of GMs as follows:
y
p
N
-
p
=
p
!
i
=0
p
s
1
(
p
,
i
)
m
i
(50)
Now by taking the inverse of (3.16), the GMs can be obtained in terms of the digital filter
outputs thus:
p
r
!
s
2
(
p
,
r
)
y
r
N
-
r
m
p
=
r
=0
(51)
where
s
2
(
p
,
r
)
are the Stirling numbers of the second kind [28], and the Stirling numbers of the
first and second kind can be considered to be inverses of one another:
max {
i
,
r
}
(-1)
p
-
r
s
1
(
p
,
i
)
s
2
(
r
,
p
)=
δ
ir
p
=0
(52)
where
δ
ir
is the Kronecker delta.
Notice now, for the
p
order, it can be shown that the digital filter outputs are sampled at
N
-
p
, unlike the previous works which were sampled at
N
or later instances of
N
[19]-[20]-
[21]. As the order
p
2
is reached, the digital filter output values begin to decrease. This allows
the use of low value digital filter outputs for the formulation of GM. The 2D moments can be
obtained by expanding the 1D model for the digital filter outputs as follows:
p
q
r
!
s
!
s
2
(
p
,
r
)
s
2
(
q
,
s
)
y
rs
N
-
r
,
N
-
s
m
p
,
q
=
r
=0
s
=0
(53)
3.4. Experimental studies
A set of experiments were carried out to validate the theoretical framework developed in the
previous sections and to evaluate the performance of the proposed structure. This section is
divided into 3 parts. In the first subsection, an artificial image of size 4×4 is used to generate
GMs up to third order. The computational complexity of three algorithms - the algorithms of
[19], [20] and the proposed method - is then analyzed and discussed in the second subsection.
