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In-Depth Information
The constant Euclidean coordinates
m
∗
j
∈
R
3
of the feature points expressed in
F
c
are defined as
m
∗
j
=
x
∗
1
j
,
the camera frame
x
∗
3
j
T
x
∗
2
j
,
.
These feature points, projected on the image plane
π
i
, are given by the constant
normalized coordinates
m
∗
j
∈
R
3
as
x
∗
1
j
x
∗
3
j
,
1
T
x
∗
2
j
x
∗
3
j
,
m
∗
j
=
.
(12.1)
3
The Euclidean coordinates
m
j
(
t
)
∈
R
of the feature points expressed in the cam-
3
era frame
F
c
and the respective normalized Euclidean coordinates
m
j
(
t
)
∈
R
are
defined as
m
j
(
t
)=
x
1
j
(
t
)
x
3
j
(
t
)
T
,
x
2
j
(
t
)
,
,
(12.2)
m
j
(
t
)=
x
1
j
(
t
)
1
T
x
2
j
(
t
)
x
3
j
(
t
)
,
x
3
j
(
t
)
,
.
(12.3)
3
. To facilitate the subsequent develop-
Consider a closed and bounded set
Y ⊂
R
ment, auxiliary state vectors
y
∗
j
=[
y
∗
1
j
,
y
∗
2
j
,
y
∗
3
j
]
T
∈ Y
and
y
j
(
t
)=[
y
1
j
(
t
),
y
2
j
(
t
),
y
3
j
(
t
)]
T
∈Y
are constructed from (12.1) and (12.3) as
y
∗
j
=
x
∗
1
j
T
y
j
=
x
1
j
T
x
∗
2
j
x
∗
3
j
,
1
x
∗
3
j
x
2
j
x
3
j
,
1
x
3
j
x
∗
3
j
,
x
3
j
,
.
(12.4)
The corresponding feature points
m
∗
j
and
m
j
(
t
) viewed by the camera from two
different locations (and two different instances in time) are related by a depth ratio
α
j
(
t
)
∈
R
3
×
3
as
and a homography matrix
H
(
t
)
∈
R
R
+
n
∗
T
x
∗
3
j
x
3
j
x
f
d
∗
m
∗
j
.
m
j
=
(12.5)
α
j
(
t
)
H
Using projective geometry, the normalized Euclidean coordinates
m
∗
j
and
m
j
(
t
) can
be related to the pixel coordinates in the image space as
p
∗
j
=
Am
∗
j
p
j
=
Am
j
,
(12.6)
where
p
j
(
t
)=
u
j
v
j
1
T
is a vector of the image-space feature point coordinates
3
is
a constant, known, invertible intrinsic camera calibration matrix [29]. Since
A
is
3
,and
A
3
×
∈
R
I ⊂
R
∈
R
u
j
(
t
),
v
j
(
t
)
defined on the closed and bounded set
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