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where:
•
N
s
=
L
s
for a steepest descent (gradient) method. For instance, this approach
has been used in [12];
N
s
=
L
s
+
for a Gauss-Newton (GN) method. It is the control law usually used.
Note also that the case where
N
s
=
L
s
∗
•
+
is also very used in practice [10];
N
s
=
H
+
diag(
H
)
−
1
L
s
for a Levenberg-Marquardt method.
H
=
L
s
L
s
is
an approximation of the Hessian matrix of the cost function (see Section 5.3.2).
The parameter
•
μ
makes possible to switch from a steepest descent like approach
2
to a GN one thanks to the observation of (5.26) during the minimization process;
and
μ
N
s
=(
L
s
+
L
s
∗
)
+
for the efficient second order minimization (ESM) method pro-
posed in [16]. Note that this method takes benefit of knowing the shape of the
cost function near the global minimum (through
L
s
∗
); it is thus less sensitive to
local minima than the above-mentioned methods. Its convergence domain is also
larger.
•
In practice, since the convergence of the control law (5.30) highly depends on the
cost function (5.26), we focus in the next section on its shape.
5.3.2
Shape of the Cost Function
In fact, we are interested in the shape of the cost function since we want to minimize
it. Therefore, we are interested in studying the Hessian of (5.26). It is given by
(
r
)=
∂
∂
+
i
=dim
s
i
=1
∇
2
s
i
s
i
(
r
)
s
i
(
r
∗
)
.
s
s
2
∇
C
−
(5.31)
∂
r
∂
r
However, this expression is far too complex to derive some useful results. Thus, we
study it around the desired position
r
∗
, leading to
(
r
∗
)=
∂
∂
s
s
2
∇
C
.
(5.32)
∂
r
∂
r
Moreover, since we have
s
=
∂
s
r
r
=
L
s
v
, we are interested in practice in the follow-
∂
ing matrix
H
∗
=
L
s
∗
L
s
∗
.
(5.33)
This matrix allows us to estimate the cost function around
r
∗
. Indeed, a first order
Taylor series expansion of the visual features
s
(
r
) around
r
∗
gives
s
(
r
)=
s
(
r
∗
)+
L
s
∗
Δ
r
(5.34)
2
More precisely, each component of the gradient is scaled according to the diagonal of the
Hessian, which leads to larger displacements along the direction where the gradient is low.
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