
33Image Transforms
Now,
∑
∑
=
=
=
−
π
=
−
−
π
N
Fuve fuy
N
fuye
v
N
j
N
vy
u
N
j
N
ux
1
(,)(,)(Intermediateresult)
1
(,)(Finalresult)
0
1
2
0
1
2
3.4.2 Translation
A 2D image f(x,y) is translated by (x
0
,y
0
) to form the translated version f(x – x
0
,
y – y
0
). Let the Fourier transform of this version be
F
uv
tran
,
:
F
N
fx xyye
N
fx xyye e
uv
trans
x
N
y
N
j
N
ux xvyy
x
N
y
N
j
N
ux vy j
N
ux vy
1
(,)
1
(,)
,
0
1
0
1
00
2
[( )( )]
0
1
0
1
00
2
()
2
00
∑∑
∑∑
=−−
=−−
=
−
=
−
−
π
−+ −
=
−
=
−
−
π
+−
π
+
FFuve
uv
trans
j
N
ux vy
(,)
,
=
−
π
+
fx xyyFuve
j
N
ux vy
(,)(,)
00
∴− −→
−
π
+
(3.18)
For the inverse Fourier transform,
Fu uv vfxye
j
N
ux vy
()(,)
0,0
−−→
π
+
(3.19)
It can be seen from Equations (3.18) and (3.19) that translating the origin i ...