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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