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Desired image
Current image
Model input
Camera velocity
Model output:
Predicted features over
Np
k
k
k+2
k+3
k+1
k+1
k+1
k+4
k+2
k+3
k+2
k+2
k+3
k+4
k+5
Time
Fig. 20.2
Principle of image prediction (
N
p
= 3
,
N
c
= 2)
20.3.1
Nonlinear Global Model
The control input of the free-flying process is the camera velocity
applied to the
camera. Here, the state of the system can be the camera pose in the target frame:
x
=(
P
x
τ
z
). The dynamic equation can be approximated by
1
,
P
y
,
P
z
,
Θ
,
Θ
,
Θ
x
y
x
(
k
+ 1)=
x
(
k
)+
Te
τ
(
k
)=
f
(
x
(
k
)
,
τ
(
k
))
.
(20.14)
The output is the visual features expressed in the image plane noted
s
m
. In the case
of a perspective camera, the output equation for one point-like feature in normalized
coordinates can be written as
s
m
(
k
)=
u
(
k
)
v
(
k
)
=
X
(
k
)
=
g
(
X
(
k
)
/
Z
(
k
)
,
Y
(
k
)
,
Z
(
k
))
,
(20.15)
/
Y
(
k
)
Z
(
k
)
where (
X
1)
R
c
are the point coordinates in the camera frame. The rigid trans-
formation between the camera frame and the target frame, noted
l
(
x
), can easily be
deduced from the knowledge of the camera pose
x
(
k
). If the point coordinates are
known in the target frame, (
X
,
Y
,
Z
,
,
,
,
Y
Z
1)
R
t
, then the point coordinates in the camera
frame, (
X
,
Y
,
Z
,
1)
R
c
are given by
⎛
⎞
⎛
⎞
X
Y
Z
1
X
Y
Z
1
=
R
(
x
)
T
(
x
)
0
1
×
3
⎝
⎠
⎝
⎠
.
=
l
(
x
(
k
))
(20.16)
1
R
c
R
t
Finally, we obtain
s
m
(
k
)=
g
◦
l
(
x
(
k
)) =
h
(
x
(
k
))
.
(20.17)
Equations 20.7 are now completely identified with (20.14) and (20.17). This
dynamic model combines 2D and 3D data and so it is appropriate to deal with
2D and/or 3D constraints. The constraints are respectively expressed on the states
and/or the outputs of the prediction model and are easily added to the optimization
1
The exponential map could be also used to better describe the camera motion.
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