Sensors 2019, 19, 5140 4 of 18
Λ
(j,W)
k
=
1
(π
|W|
|W|S
(j,W)
k|k−1
)
d
2
|V
(j)
k|k−1
|
v
(j)
k|k−1
2
Γ
d
(
v
(j,W)
k|k
2
)
|V
(j,W)
k|k−1
|
v
(j,W)
k|k
2
Γ
d
(
v
(j)
k|k−1
2
)
(7)
ω
p
=
∏
W∈P
d
W
∑
P
0
∠Z
k
∏
W
0
∈P
0
d
W
0
. (8)
Λ
(j,W)
k
presents the likelihood of the
jth
GIW component given the measurements of the
Wth
cell,
ω
p
is the weight of
pth
partition,
p
(j)
D
is the detection probability of
jth
GIW component,
γ
(j)
is
the expected number of measurements generated by
jth
GIW component,
λ
c
is the mean number of
clutter measurements,
c
k
is the spatial distribution of the clutter over the surveillance volume,
δ
i,j
is
the Kronecker delta,
|W|
is the the number of measurements in the
Wth
cell,
S
(j,W)
k|k−1
is innovation factor,
Γ
d
(·) is the multivariate Gamma function.
3. Analysis of ET-GIW-PHD
In ET-GIW-PHD, the calculation of
w
(j,W)
k|k
is important. If the measurements in
Wth
cell are
generated by clutter,
w
(j,W)
k|k
is expected to be smaller than the pruning threshold, then the corresponding
component will be eliminated and the clutter will be eliminated.
In Equation (5),
w
(j,W)
k|k
contains two parts, one is the weight of the
pth
partition, denoted by
ω
p
, the other is the weight of
Wth
cell in partition. Without loss of generality, only one partition is
considered for clarity, therefore ω
p
= 1. Substituting Equation (6) into Equation (5), we arrive at
w
(j,W)
k|k
=
e
−γ
(j)
γ
(j)
λ
c
c
k
|W|
p
(j)
D
Λ
(j,W)
k
w
(j)
k|k−1
δ
|W|,1
+
J
k|k−1
∑
l=1
e
−γ(l)
γ
(l)
λ
c
c
k
|W|
p
(l)
D
Λ
(l,W)
k
w
(l)
k|k−1
. (9)
From Equation (9), the numerator is a part of denominator, the measurements of
Wth
cell is used
to correct each GIW component, then
Λ
(j,W)
k
can be obtained,
w
(j,W)
k|k
can be given based on some prior
parameters, such as p
D
, γ, λ
c
and c
k
(for brevity, the subscript and superscript are omitted here).
If the measurements in the cell are generated by clutter, the likelihood
Λ
(j,W)
k
of each GIW
component will be very small since clutter does not obey the kinematic and extent model of target.
If the number of clutter measurements in the cell is equal to one, then
|W| =
1,
δ
|W|,1
=
1,
J
k|k−1
∑
l=1
e
−γ(l)
γ
(l)
λ
c
c
k
|W|
p
(l)
D
Λ
(l,W)
k
w
(l)
k|k−1
will be much smaller than 1 because the likelihood
Λ
(j,W)
k
achieves
a small value mentioned above and other parameters can be considered as constants, the value of
w
(j,W)
k|k
will be close to 0 and is smaller than the pruning threshold, then the corresponding component
will be eliminated and the clutter is eliminated. However, if the number of clutter measurements in
the cell is more than one, then
|W| 6=
1,
δ
|W|,1
=
0, Equation (9) is the normalization process. Although
Λ
(j,W)
k
is close to zero,
w
(j,W)
k|k
can still take a large value. In this case, ghost targets will emerge and the
number of targets will be overestimated. Further details on numerical implementation can be found in
Section 5.
According to the analysis above, if the measurement in the cell is clutter,
J
k|k−1
∑
l=1
e
−γ(l)
γ
(l)
λ
c
c
k
|W|
p
(l)
D
Λ
(l,W)
k
w
(l)
k|k−1
(denoted by
J
k|k−1
∑
l=1
ψ
l,W
) in Equation (9) should be added by
1. Otherwise, it should be added by 0 and the clutter can be eliminated. However, from Equation (9), if
the cell contains only one measurement,
J
k|k−1
∑
l=1
ψ
l,W
is added by 1, it means that the cell contains only
one measurement is considered as clutter in ET-GIW-PHD. Otherwise, it is considered as a target if the