Image Processing Reference
In-Depth Information
the problem is as follows; let
˜
be the measurement history (the angles), we must now
consider the following likelihood functional [NAR 84]:
B
B
|
X
=cstexp
,
2
B
−
B
X
p
˜
1
2
Σ
−
1
˜
−
with
w
k
as a sequence of independent, identically distributed, Gaussian white noise
and:
=
β
1
,..., β
p
∗
,
β
k
=
β
k
+
w
k
,w
:
˜
B
[6.14]
N
(0
,
Σ)
.
and
there is no general explicit solution to this optimization problem. This is why we often
resort to a Gauss-Newton type algorithm [NAR 84], written as follows:
Clearly, the likelihood functional has a non-linear dependence on the state
X
∂
B
∂
∗
Σ
−
1
∂
B
∂
−
1
∂
B
∂
∗
Σ
−
1
˜
B
−
B
,
X
+1
=
X
−
ρ
[6.15]
X
X
X
where Σ = Diag(
σ
i
),
σ
i
is the noise variance of the
i
th
measurement,
is the iter-
ation index,
ρ
the stepsize and, using notations that are not completely accurate,
B
(
X
). A simple calculation [NAR 84] helps us calculate the matrix
∂
B
=
B
/∂
X
,
i.e.:
cos
β
1
r
1
sin
β
1
r
1
cos
β
1
r
1
sin
β
1
r
1
⎛
⎞
t
1
−
t
0
−
t
1
−
t
0
−
⎝
⎠
∂
B
∂
.
.
.
.
=
X
cos
β
p
r
p
sin
β
p
r
p
cos
β
p
r
p
sin
β
p
r
p
t
p
−
t
0
−
t
p
−
t
0
−
[6.16]
where
r
i
refers to the relative target-receiver distance at the time
i
.
The phrase
comprehensive target motion analysis methods
is generally used to
refer to this type of approach [NAR 84]. They are often presented in competition with
Kalman-type methods, even though the objectives are fundamentally different. The
methods are simple to implement and have reasonable calculation costs. A few iter-
ations of the algorithm are usually enough. A difficult point can be the choice of the
increment
ρ
. See [BAZ 93] for a state of the art in knowledge on efficient methods for
choosing these parameters.
Still in the context of comprehensive methods, another type stands out:
instrumen-
tal variable
methods (
IVM
). The idea is simple: to use the successive estimations of
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