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i
k
n
,
L
P
Fig. 5.2
Light source mounted on the camera for a planar object when the camera and the
object planes are parallel
(
r
)=
1
2
2
C
e
(5.26)
I
(
r
∗
).
Nevertheless, regardless of the complexity of the shape of (5.26), since to evalu-
ate (5.26) at a given pose a motion has to be performed, this problem becomes more
complex than a classical optimization one if we want to ensure a suitable camera
trajectory. Therefore, powerful approaches based on backstepping cannot be used.
Indeed, in practice, only differential approaches can be employed to solve this par-
ticular optimization problem. In that case, a step of the minimization scheme can be
written as follows
where
e
=
I
(
r
)
−
r
k
+1
=
r
k
⊕
t
k
d
(
r
k
)
(5.27)
where “
” denotes the operator that combines two consecutive frame transforma-
tions;
r
k
is the current pose,
t
k
is a positive scalar (the descent step) and
d
(
r
k
) a
descent direction ensuring that (5.26) decreases if
⊕
d
(
r
k
)
∇
C
(
r
k
)
<
0
.
(5.28)
Consequently, the following velocity control law can be easily derived consider-
ing that
t
k
is small enough
v
=
λ
k
d
(
r
k
)
(5.29)
where
λ
k
is a scalar that depends on
t
k
and on the sampling rate. However, here
again, since (5.26) cannot be simply evaluated or estimated, line search algorithms
cannot be used and this value is often chosen as a constant one. In the remainder of
this chapter we will omit the subscript
k
for the sake of clarity.
Several descent directions can be used, nevertheless they lead to the following
generalized expression of (5.2) (see [6] for more details)
−
λ
N
s
e
v
=
(5.30)
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