Database Reference
In-Depth Information
From the relations in Eqs. (
6.17
)-(
6.22
), we can write the relationship between the
magnitudes of the Fourier transform of the two images in polar coordinates as:
M
G
[
r
,
ʸ
]=
M
F
[
r
,
ʸ
−
ʸ
0
]
(6.23)
From the relationship between the magnitudes in Eq. (
6.23
), it can be viewed as a
function having an independent variable in rectangular coordinates with
r
=
i
and
ʸ
=
k
, the same as for Eq. (
6.4
). Thus, we can obtain the cross-power spectrum as
in Eq. (
6.12
) and the inverse Fourier transform of the cross-power spectrum as in
Eq. (
6.14
). This yields the result in the form of a delta function, that is,
N
2
ʴ
[
r
,
ʸ
−
ʸ
0
]
(6.24)
It can be observed that the maximum peak value of this function is attained at
ʸ
=
ʸ
0
, which is the parameter for the rotation of the images.
6.2.3
Parameter Estimation for Scaling
If
{
g
[
i
,
k
]
}
is scale replica of
{
f
[
i
,
k
]
}
with scale factor
ʱ
, we can write the relation
{
g
[
i
,
k
]
}
=
{
f
[
ʱ
i
,
ʱ
k
]
}
. According to the Fourier scale property, the discrete-time
Fourier transform of
{
g
[
i
,
k
]
}
and
{
f
[
i
,
k
]
}
are related by:
F
m
n
ʱ
[
,
]=
ʱ
,
G
m
n
(6.25)
By converting the axes to logarithmic scale, scaling can be reduced to a translation
movement,
G
[
log
m
,
log
n
]=
F
[
log
m
−
log
ʱ
,
log
n
−
log
ʱ
]
(6.26)
That is,
G
m
,
n
=
m
−
ʱ
,
n
−
ʱ
]
F
[
(6.27)
where
m
=
n
=
ʱ
=
log
m
,
log
n
,
and
log
ʱ
.
According to Eqs. (
6.23
) and (
6.25
), if
{
g
[
i
,
k
]
}
is translated, rotated, and scaled
to replicate
{
f
[
i
,
k
]
}
, their Fourier magnitude spectra in polar representation are
related by:
M
F
r
0
M
G
[
r
,
ʸ
]=
ʱ
,
ʸ
−
ʸ
(6.28)
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