Image Processing Reference
In-Depth Information
where:
O
2
=Δ
T
x
,
V
y
,
⎧
⎨
O
1
=(
T
y
,
V
x
,
T
x
,
T
y
,
V
x
,
V
y
) and:
Δ = diag
cos
θ
0
cos
θ
0
= diag
r
0
r
0
,
,...,
cos
θ
k
cos
θ
k
,...,
r
k
r
k
[6.36]
⎩
r
x
(0)
β
−
r
x
(0)
.
O
. Let ker
O
1
denote the kernel of
Let us now examine the properties of the matrix
O
1
and
R
its supplementary subspace in
4
, i.e.
4
=ker
O
1
⊕
R
R
R
where
⊕
represents
X
R
4
,
X
the direct sum. Then, let
be any vector in
can be decomposed in a unique
O
1
and
R
, meaning that
X
K
Y
K
∈
O
1
way as a sum of two vectors of ker
=
+
(
ker
and
Y
∈
R
) and we have the following implications:
O
X
-if
Y
=
0
, then
=
0
,
O
X
-if
Y
=
0
, then
=
β
Δ(
x
1
1
+
αx
3
Z
)
.
Therefore, we only have to examine the second hypothesis (namely
Y
=0) and
we then have:
O
X
=0=
⇒
x
1
=
x
3
=0
(
1
and
Z
are linearly independent)
,
X
∈
O
1
,wealsohave
x
2
T
y
+
x
4
V
y
and therefore, since
ker
=0and therefore
T
y
in the end,
x
2
=
x
4
=0. As a result, except for the specific case where
=
O
is simply the zero vector in the multi-platform case. Even if this is
a purely algebraic result, we begin to perceive the advantage of fusing the outputs of
the platforms.
V
y
=
O
, ker
As for the filtering aspect, particle filtering methods can be used to avoid any
linearization. Briefly, its general form is as follows:
- initialization:
s
o
∼
0
p
(
X
0
)
,
=1
/N
;
n
=1
,...,N
;
-for
t
=1
,...,T
:
- prediction:
s
t
∼
X
t
|
X
t
−
1
=
s
t
−
1
, β
t
);
n
=1
,...,N
,
- calculation of the weights:
f
(
p
s
t
|
s
t
−
1
l
t
β
t
;
s
t
f
s
t
|
q
t
=
q
t
−
1
s
t
−
1
, β
t
for
n
=1
,...,N
, (normalization step)
-
(
X
t
)=
n
=1
q
t
s
t
,
E
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