Digital Signal Processing Reference
In-Depth Information
filter outputs are obtained from the respective digital filter structure. Then, these outputs are
linearly combined to compute GMs.
For a grayscale image of size
N
×
N
and GM up to order of
s
, where
s
=
(
p
+
q
)
, the number
of additions and multiplications for the proposed method, [19] and [20] are shown in Table 2.
Additions
(Digital filter stages)
Additions (Digital filter outputs
to GMs)
Multiplications (Digital filter
outputs to GMs)
Algorithm
(
s
+ 1)
(
N
+
s
+
2
)
(
N
+ 1)
s
(
s
+ 1
)(
s
+ 2
)(
s
+ 7
)
24
s
4
+ 10
s
3
+ 23
s
2
- 34
s
- 24
24
[19]
(
s
+ 1)
(
N
+
s
+
2
)
N
s
(
s
- 1
)
(
s
2
+ 3
s
+ 14
)
24
s
(
s
- 1
)
(
s
2
+ 3
s
+ 14
)
24
[20]
Proposed
method
s
(
s
- 1
)
(
s
2
+ 3
s
+ 14
)
24
s
(
s
- 1
)
(
s
2
+ 3
s
+ 14
)
24
(
s
+ 1
)
N
(
N
+ 2
)
-
(
s
+ 2
)(
s
- 3
)
6
Table 2.
Complexity analysis of GMs computation using digital filters for image of size
N
×
N
and maximum order of
s
=
p
+
q
.
It can be seen that even though the complexity of the linear combination stage for the proposed
method is the same as [20], there is saving in the number of additions at the digital filters stage.
This can be clearly shown in the example below. For
N
=
512 and
s
=45 , the number of
additions needed in the filter stage for [20] is 12612096 while the proposed method just requires
12090594 additions. The summary of the complexity comparison for all the three methods to
compute GMs up to 45
th
order is shown in Table 3.
Algorithm
A
dditions
Multiplications
[19]
1
2847524
2
10795
[20]
1
2791451
1
79355
Proposed method
1
2269949
1
79355
Table 3.
Complexity analysis of GMs computation using digital filters for image of size 512×512 and
s
=45 .
For a 128×128 grayscale image, the advantage of the proposed filter structure as compared to
[19] and [20] is clearly depicted in Figure 4.
