Image Processing Reference

In-Depth Information

ˆ

ˆ

Y
2
as

observations and using an iterative method for estimating the complete state. We can,

however, define an admittedly sub-optimal, but explicit, fusion method as follows:

⎧

⎨

A first idea consists of considering the partial estimation vectors

Y
1
and

σ
2
θ
1
+
σ
1
θ
2
,

1

σ
1
+
σ
2

θ
=

d
sin(
θ
)

α
1
−

v
=

,

[6.52]

⎩

α
2

d
sin(
θ
)

α
1
−

cpa
1
=
α
1

.

α
2

We can then show using Taylor series expansions [DON 00, DON 02] that this

estimator of

X

asymptotically follows the law:

⎛

⎝

⎞

⎠
−→ N

⎡

⎣

⎛

⎝

⎞

⎠

⎤

⎦

cpa
1

v

θ

t
cpa
1

cpa
1

v

θ

t
cpa
1

;
M
1
Γ
1
M
1
+
M
2
Γ
2
M
2

,

[6.53]

where:

v
1

α
1
d
cos(
θ
)

α
1
−

⎛

⎝

⎞

⎠

σ
2

σ
1
+
σ
2

α
1

α
1
−

−

0

α
2

α
2

σ
2

σ
1
+
σ
2

v

α
1
−

d
cos(
θ
)

α
1
−

−

0

α
2

α
2

M
1
=

,

[6.54]

σ
2

σ
1
+
σ
2

0

1

0

0

0

and:

v
α
1

α
1
−

α
1
d
cos(
θ
)

α
1
−

⎛

⎝

⎞

⎠

σ
1

σ
1
+
σ
2

0

α
2

α
2

σ
1

σ
1
+
σ
2

v

α
1
−

d
cos(
θ
)

α
1
−

0

α
2

α
2

M
2
=

.

[6.55]

σ
1

σ
1
+
σ
2

0

0

0

0

0

In the equation above, the matrix Γ
i
, (
i
=1
,
2) is unknown, which is why we have

to replace it with its estimate. We can then show that the performances of decentralized

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