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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