Digital Signal Processing Reference
In-Depth Information
4.1. Reduced digital filter structure
Unlike the previous researchers who used a 1-D Z-transform to derive a 2-D digital filter
structure by cascading the filters both in the rows and columns. However, we use a 2-D
definition of Z-transform to obtain the impulse response of the filter which led to a reduced
digital filter structure as compared with [19]. The 2-D Z-transform for the image,
f m
,
n
is
given as:
+∞
+∞
F
(
z
1
,
z
2
)
=
m
=1
f m
,
n z
1
-
m
z
2
-
n
n
=1
(54)
The impulse response for a 2-D image becomes:
h
p
,
q
m
,
n
=
(
m
+
m
)
u m
(
n
+
n
)
u n
(55)
and the transfer function
H
p
,
q
(
z
1
,
z
2
)
, in the 2-D Z-transform domain for this filter structure is
shown as
1
(
1 -
z
1
-1
)
p
+1
(
1 -
z
2
-1
)
q
+1
H
p
,
q
(
z
1
,
z
2
)
=
(56)
Using the above transfer function, the relationship between the input and the output of the
digital filter is:
X
(
z
1
,
z
2
)
(
1 -
z
1
-1
)
p
+1
(
1 -
z
2
-1
)
q
+1
Y
p
,
q
(
z
1
,
z
2
)
=
(57)
Based on (4.4), the zero order of the digital filter output is derived as
X
(
z
1
,
z
2
)
(
1 -
z
1
-1
)(
1 -
z
2
-1
)
Y
00
(
z
1
,
z
2
)
=
(58)
Thereafter, a recurrence relationship between the previous and the next outputs of each digital
filter for the row and columns as shown in Figure 8 can be obtained:
Y
p
+1,
q
(
z
1
,
z
2
)
=
Y
p
,
q
(
z
1
,
z
2
)
1 -
z
1
-1
(59)
Y
p
,
q
+1
(
z
1
,
z
2
)
=
Y
p
,
q
(
z
1
,
z
2
)
1 -
z
2
-1
(60)
By taking the inverse 2-D Z- transform of (4.6) and (4.7), we get the following:
