Image Processing Reference
In-Depth Information
then,
2
-
D DFT
F
(v
!
f
(
r,
uþu
0
)
,
wþu
0
)
(
2
:
63
)
This implies that rotating f
(
n, m
)
by an angle
u
0
rotates its transform F
(
k, l
)
by the same angle.
d. Convolution property of DFT: The convolution theorem for continuous
signals states that convolution in space domain is equivalent to multiplica-
tion in the Fourier domain. This theorem can be extended to the discrete
domain; however, it should be noted that ordinary convolution is replaced
by circular convolution, that is,
!
DFT
f
(
n, m
)
h
(
n, m
)
F
(
k, l
)
H
(
k, l
)
(
2
:
64
)
The circular convolution is de
ned as
X
X
f
(
n, m
)
h
(
n, m
) ¼
f
(
u, v
)
h
(
n
u, m
v
)
(
2
:
65
)
v
u
where the shift (n
u or m
v) is circular shift. The linear discrete
convolution is the type of convolution used for
filtering signals and images
and also for
finding the output of imaging systems modeled in linear shift-
invariant form. Circular convolution is a side effect of DFT.
Example 2.8
and h(n, m)
,
21
11
11
If x(n, m)
¼
¼
nd
13
a. y(n, m)
¼ x(n, m)
h(n, m), circular convolution of x(n, m) and h(n, m).
b. g(n, m)
¼ x(n, m)*h(n, m), linear convolution of x(n, m) and h(n, m).
S
OLUTION
a. Circular convolution
X
X
1
1
y(n, m)
¼ x(n, m)
h(n, m)
¼
h(u, v)x(numod 2, y v mod 2)
v¼
0
u¼
0
Expanding the above equation yields
y(n,m)
¼h(0,0)x(n,m)
þh(0,1)x(n,m
1)
þh(1,0)x(n
1,m)
þh(1,1)x(n
1,m
1)
¼x(n,m)
þx(n,m
1)
þx(n
1,m)
þx(n
1,m
1)
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