Database Reference
In-Depth Information
[
,
]=
[
−
,
−
]
Substituting
g
i
k
f
i
a
k
b
in (
6.5
) yields:
M
1
i
=
0
−
N
1
k
=
0
−
1
MN
e
−
j
2ˀ
(
M
+
n
N
i
)
m
[
,
]=
[
−
,
−
]
G
m
n
f
i
a
k
b
(6.6)
M
−
−
a
∑
u
=
−
a
1
N
−
−
b
∑
v
=
−
b
1
1
MN
e
−
j
2ˀ
(
m
u
+
M
+
n
v
+
b
N
)
=
f
[
u
,
v
]
(6.7)
e
−
j
2ˀ
(
m
M
+
n
N
)
F
=
[
m
,
n
]
(6.8)
The relation between the first line and the second line is obtained by substituting
index
i
and
k
with
i
=
u
+
a
and
k
=
v
+
b
, and
F
[
m
,
n
]
is the Fourier transform of
. Taking the complex conjugate
F
∗
[
image
{
f
[
i
,
k
]
}
m
,
n
]
to multiply the relation in
Eq. (
6.8
) produces:
e
−
j
2ˀ
(
m
M
+
n
N
)
F
F
∗
[
F
∗
[
G
[
m
,
n
]
m
,
n
]=
[
m
,
n
]
m
,
n
]
(6.9)
e
−
j
2ˀ
(
m
M
+
n
N
)
|
2
=
F
[
m
,
n
]
|
(6.10)
e
−
j
2ˀ
(
a
M
+
n
N
)
|
m
=
G
[
m
,
n
]
||
F
[
m
,
n
]
|
(6.11)
The relation from the second line to the third line is obtained by substitution of
|
F
[
m
,
n
]
|
with
|
F
[
m
,
n
]
|
=
|
G
[
m
,
n
]
|
. Equation (
6.11
) can be written in the form of a
cross-power spectrum as:
e
−
j
2ˀ
(
)
m
M
+
n
N
F
∗
[
G
[
m
,
n
]
m
,
n
]
]
|
=
(6.12)
|
G
[
m
,
n
]
||
F
[
m
,
n
According to Gonzalez and Woods [
173
], we can obtain the inverse Fourier
transform of the cross-power spectrum using:
MN
M
−
1
m
=
0
N
−
1
n
=
0
e
j
2ˀ
(
m
i
−
M
+
n
k
−
b
,
i
=
a
,
k
=
b
)
=
(6.13)
N
0
,
i
=
a
,
k
=
b
=
MN
·
ʴ
[
i
−
a
,
k
−
b
]
(6.14)
where
ʴ
[
i
−
a
,
k
−
b
]
is equal to zero at index
i
=
a
and
k
=
b
, whereas
ʴ
[
i
−
a
,
k
−
b
]
is equal to one at index
i
b
. At this position, we can find the parameters
for image translation between the two images in the row direction by
a
pixels and
in the column direction by
b
pixels.
=
a
and
k
=
6.2.2
Parameter Estimation for Rotation
Let
0
and translates by
a
pixels in the row direction and
b
pixels in the column direction. The relation of the
gray scale corresponding image can be written as:
{
g
[
i
,
k
]
}
be the reference image
{
f
[
i
,
k
]
}
that rotates by
ʸ
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