Information Technology Reference
In-Depth Information
where
t
is the vector representing the spatial translation and
R
z
is the rotation
matrix of the camera around z-axis. Matrix
R
z
is defined by
R
z
=
cos
ˈ
sin
ˈ
sin
ˈ
cos
ˈ
−
(2)
where
ˈ
is the rotation angle between images.
In the ground station, corresponding characteristics
m
a
and
m
b
must be
found in order to determine the homography matrix
H
ba
from which rotation
matrix
R
z
can be obtained. Finally, the yaw angle
ˈ
can be isolated from any
element of rotation matrix
R
z
Characteristics, also named features, are key points based on intensity changes
extracted from an image, easily recognizable in subsequent frames. Occasionally,
in the case of downward looking camera attached to a quadrotor in low altitude
hovering flight, features are hard to find because of that terrains like lawn in
outdoor or carpets in indoor are too regular for feature tracking. To deal with this
problem, the frequency domain representation, obtained by Fourier Transform
and Fourier shift theorem can be used. Fourier shift theorem claims that given
two identical images
i
a
and
i
b
displaced one of each other a distance (
u,v
)
i
a
(
x, y
)=
i
b
(
x
+
u,y
+
v
)
(3)
and the Fourier Transform of both images are related by
I
a
(
ˉ
x
,ˉ
y
)=e
j
(
uˉ
x
+
vˉ
y
)
I
b
(
ˉ
x
,ˉ
y
)
(4)
where
I
a
and
I
b
are the Fourier transform of
i
a
and
i
b
, respectively. The dis-
placement
ʔ
d
=(
u,v
) can be calculated first using the
Cross Power Spectrum
(CPS) as follows
I
a
(
ˉ
x
,ˉ
y
)
I
b
(
ˉ
x
,ˉ
y
)
=e
j
(
uˉ
x
+
vˉ
y
)
C
(
I
a
,I
b
)=
(5)
I
b
(
ˉ
x
,ˉ
y
)
|
I
a
(
ˉ
x
,ˉ
y
)
||
|
and finally using the inverse Fourier transform.
Given that, to recover the homography at least four point are required, images
are divided into patches
p
i
and the Fourier transform is calculated on each one
of them. Hence, a
ʔ
d
i
displacement is found for each patch as is shown in Fig. 2.
By adding this displacements to some point of the patches of the first image,
correspondence characteristics set between images are found. Formally,
{
m
a
i
ₔ
m
a
i
+
ʔ
d
i
=
m
b
i
}
.
(6)
Finally, from the homography matrix, rotation matrix
R
z
is obtained and
from (2) the
ˈ
angle is isolated.
Algorithm 1 summarizes the described method using Algorithm 2 for displace-
ment determination.
