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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