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projection of a 3D point
X
i
in the current desired image (at time
t
). We define vector
i
(
t
)
p
∗
i
(
t
). It is well-known that for all 3D points
h
i
=
α
i
(
t
)
p
∗
i
(
t
)=
G
(
t
)
p
∗
i
(1)+
h
i
(
t
)=
α
τ
γ
(
t
)
(7.25)
i
where
α
i
(
t
) is a positive scaling factor depending on time,
τ
i
is a constant scaling
factor null if the target point belongs to
γ(
t
)=
Kb
(
t
) represents the epipole
in the current image (that is the projection in the image at time
t
of the optical center
when the camera is in its desired position). After the initial collineation has been
estimated, the optimal path of the collineation matrix can be computed as described
previously. The initial value of the epipole,
Π
,and
γ
(0)=
γ
0
, can also be computed directly
from image data (
i.e.
,
γ
0
is independent of the
K
-matrix) [10]. Furthermore, the
epipole at time
t
can easily be computed from the state vector
x
(
t
). Note also that
the scaling factor
τ
i
is not time dependent and can be computed directly from the
initial and desired image data since (refer to (7.25))
α
i
(
t
)
p
∗
i
(0)
p
∗
i
(0)=0 =
G
(
t
)
p
∗
i
(1)
p
∗
i
(0)+
p
∗
i
(0)
∧
∧
τ
i
γ
(
t
)
∧
.
We thus obtain
3
:
(
G
0
p
∗
i
(1)
p
∗
i
(0))
1
∧
−
.
τ
i
=
p
∗
i
(0))
1
(γ
0
∧
The trajectories of the considered point in the image corresponding to an optimal
camera path can thus also be computed using
x
∗
i
(
t
)=
(
h
i
(
t
))
1
y
∗
i
(
t
)=
(
h
i
(
t
))
2
(
h
i
(
t
))
3
,
(
h
i
(
t
))
3
.
(7.26)
7.6
Example
The results have been obtained using an analog charge-coupled device (CCD) cam-
era providing 640
480 images. Only the path-planning problem is addressed. Note
however that the generated trajectories can be used in a visual servoing scheme as
the one proposed in [22]. Our approach is illustrated by two experiments. The goal
is to synthesize intermediate images between initial and desired images acquired
by a CCD camera (boxed in Figures 7.4 and 7.5) corresponding to the solution of
the variational problem previously presented. First, the optimal trajectory is com-
puted without introducing the visibility constraints (7.13). As expected the camera
follow a straight line (see Figure 7.2), however we observe that a part of the target
get out of the camera field of view (see Figure 7.4 from time
t
= 0
×
2). In the second
set of results, the visibility constraints is introduced in the variational problem. The
optimization process is initialized using the unconstrained trajectories. The camera
trajectory is no more a straight line (refer to Figure 7.3) and the target remains in
the camera field of view (see Figure 7.5).
.
3
(
v
)
j
denotes the
j
th
components of
v
.
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