Image Processing Reference
In-Depth Information
Problem formulation
Thus, in a deterministic approach, the goal is to infer from the sequence of obser-
vations
β
1
,..., β
n
}
ˆ
{
X
0
,
the value of the state estimated at a reference time, i.e.
β
1
,..., β
n
−→
ˆ
X
0
In the case of tracking, the objective is then to estimate the sequence of states:
β
1
,..., β
n
−→
X
1
,...,
X
n
We will now briefly present the elementary model for bearings-only tracking using
passive target motion analysis (also known as
BOT TMA
), by restricting ourselves first
to a target in uniform rectilinear motion in the plane. For a general representation, see
[NAR 84] and [CHA 92]. Let
X
be the state vector related to the target (
T
), defined
by:
X
T
−
X
obs
r
x
,r
y
,v
x
,v
y
∗
,
X
=
where
∗
indicates the transpose.
The state dynamics equation (time-discrete) then has the following form:
X
k
=Φ(
k, k
−
1)
X
k
−
1
+
U
k
,
[6.11]
where:
1) =
Id
2
,
αId
2
Φ(
k, k
−
0
Id
2
[6.12]
10
01
,
Id
2
α
t
k
−
t
k
−
1
.
U
k
expresses the effects of the observer's accel-
In the formula above, the vector
−
erations. The matrix Φ(
k, k
1) is the system's transition matrix, also denoted by
F
from now on. Furthermore, we assume that
α
=1. The measurement equation is, in
this case:
β
k
=
β
k
+
w
k
=tan
−
1
r
x,k
r
y,k
+
w
k
.
[6.13]
As you can see, the target's trajectory is determined by a state vector
X
, which is
X
0
). Thus, under the Gaussian hypothesis,
defined at an arbitrary reference time (i.e.
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