Image Processing Reference
In-Depth Information
3.8.1 P
ROPERTIES OF
2-D DSFT
Properties of 2-D DSFT are similar to the properties of 1-D DTFT. Here are some
important properties of 2-D DSFT:
(a)
Linearity
: 2-D DSFT is a linear transform, that is, if
DSFT
X
(
j
v
1
, j
v
2
)
x
(
n
1
, n
2
) !
(
3
:
103
)
DSFT
y
(
n
1
, n
2
) !
Y
(
j
v
1
, j
v
2
)
(
3
:
104
)
then
DSFT
ax
(
n
1
, n
2
) þ
by
(
n
1
, n
2
) !
aX
(
j
v
1
, j
v
2
) þ
bY
(
j
v
1
, j
v
2
)
(
3
:
105
)
(b)
Periodicity
: X
(
j
v
1
, j
v
2
)
is periodic with a period of 2
p
, that is,
X
(
j
v
1
, j
v
2
) ¼
X
[
j
(v
1
þ
p)
, j
(v
2
þ
p)]
(
:
)
2
2
3
106
(c)
Delay Property
:If
DSFT
X
(
j
v
1
, j
v
2
)
x
(
n
1
, n
2
)
!
(
:
)
3
107
then
DSFT
e
j
av
1
e
j
bv
2
X
(
j
v
1
, j
v
2
)
x
(
n
1
a
, n
2
b)
!
(
:
)
3
108
3.8.2 I
NVERSE
2-D DSFT
The inverse 2-D DSFT can be computed using the inversion integral given by
p
p
1
j
v
2
)
e
jn
1
v
1
e
jn
2
v
2
x
(
n
1
, n
2
) ¼
X
(
j
v
1
,
dv
1
dv
2
(
3
:
109
)
4
p
2
p
p
Example 3.29
Find the inverse DSFT of X( jv
1
, jv
2
) given by
v
1
,
v
2
)
2 D
1if (
X( jv
1
, jv
2
)
¼
0
otherwise
where D is the dashed area shown in Figure 3.12.
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