The first summation on the right-hand side of (7)
is taken over all partitions p of the measurement set
Z
k
.ThesecondsummationistakenoverallcellsW in
the current partition p.
In order to derive the measurement update of the
GM-PHD filter, six assumptions were made in [18],
which are repeated here for the sake of completeness.
Assumption 1 All of the targets evolve and
generate observations independently of one another.
Assumption 2 Clutter is Poisson and independent
of target-originated measurements.
Assumption 3 The predicted multi-target RFS is
Poisson.
Assumption 4 Each target follows a linear
Gaussian dynamical model, cf. (1), and the sensor has
alinearGaussianmeasurementmodel,cf.(4).
Assumption 5 The survival and detection
probabilities are state independent, i.e., p
S
(x)=p
S
and
p
D
(x)=p
D
.
Assumption 6 The intensities of the birth and
spawn RFS are Gaussian-mixtures.
In this paper we adopt all of the above
assumptions except that we relax the assumption on
detection probability as follows.
Assumption 7 The following approximation about
the probability of detection function p
D
(¢)holds,
p
D
(x)N (x;m
(j)
kjk¡1
,P
(j)
kjk¡1
)
¼ p
D
(m
(j)
kjk¡1
)N (x;m
(j)
kjk¡1
,P
(j)
kjk¡1
)(8)
for all x and for j =1,:::,J
kjk¡1
.
Assumption 7 is weaker than Assumption 5 in
that (8) is trivially satisfied when p
D
(¢)=p
D
,i.e.,
when p
D
(¢)isconstant.Ingeneral,Assumption7
holds approximately when the function p
D
(¢)does
not vary much in the uncertainty zone of a target
determined by the covariance P
(j)
kjk¡1
.Thisistrueeither
when p
D
(¢)isasufficientlysmoothfunctionorwhen
the signal-to-noise ratio (SNR) is high enough such
that P
(j)
kjk¡1
is sufficiently small. We still note here that
Assumption 7 is only for the sake of simplification
rather than approximation, since p
D
(x)canalways
be approximated as a mixture of exponentials of
quadratic functions (or equivalently as Gaussians)
without losing the GM structure of the corrected PHD;
see [18]. This, however, would cause a multiplicative
increase in the number of components in the corrected
PHD, which would in turn make the algorithm need
more aggressive pruning and merging operations. A
similar approach to variable probability of detection
has been taken in order to model the clutter notch in
ground moving target indicator target tracking [23].
For the expected number of measurements from
the targets, represented by °(¢), similar remarks apply
and we use the following assumption.
Assumption 8 The following approximation about
°(¢)holds,
e
¡°(x)
°
n
(x)N (x;m
(j)
kjk¡1
,P
(j)
kjk¡1
)
¼ e
¡°(m
(j)
kjk¡1
)
°
n
(m
(j)
kjk¡1
)N (x;m
(j)
kjk¡1
,P
(j)
kjk¡1
)(9)
for all x, n =1,2,::: and j =1,:::,J
kjk¡1
.
The trivial situation °(¢)=°,i.e.,when°(¢)is
constant, is again a special case where Assumption 8
is satisfied. In general, satisfying Assumption 8
is more difficult than Assumption 7 and a GM
assumption for °( ¢)wouldnotworkduetothe
exponential function. Nevertheless Assumption 8 is
expected to hold approximately either when °(¢)isa
sufficiently smooth function or when the SNR is high
enough such that P
(j)
kjk¡1
is sufficiently small.
With the assumptions presented above, the
posterior intensity at time k is a GM given by
D
kjk
(x)=D
ND
kjk
(x)+
X
p
6
Z
k
X
W2p
D
D
kjk
(x,W): (10)
The GM D
ND
kjk
(¢), handling the no detection cases, is
given by
D
ND
kjk
(x)=
J
kjk¡1
X
j=1
w
(j)
kjk
N (x;m
(j)
kjk
,P
(j)
kjk
)(11a)
w
(j)
kjk
=(1¡ (1 ¡ e
¡°
(j)
)p
(j)
D
)w
(j)
kjk¡1
(11b)
m
(j)
kjk
= m
(j)
kjk¡1
, P
(j)
kjk
= P
(j)
kjk¡1
: (11c)
where we used the shorthand notations °
(j)
and p
(j)
D
for °(m
(j)
kjk¡1
)andp
D
(m
(j)
kjk¡1
), respectively.
The GM D
D
kjk
(x,W), handling the detected target
cases, is given by
D
D
kjk
(x,W)=
J
kjk¡1
X
j=1
w
(j)
kjk
N (x;m
(j)
kjk
,P
(j)
kjk
)(12a)
w
(j)
kjk
= !
p
¡
(j)
p
(j)
D
d
W
©
(j)
W
w
(j)
kjk¡1
(12b)
¡
(j)
= e
¡°
(j)
(°
(j)
)
jWj
,(12c)
©
(j)
W
= Á
(j)
W
Y
z
k
2W
1
¸
k
c
k
(z
k
)
(12d)
where the product is over all measurements z
k
in the
cell W and jWj is the number of elements in W.The
coefficient Á
(j)
W
is given by
Á
(j)
W
= N (z
W
;H
W
m
(j)
kjk¡1
,H
W
P
(j)
kjk¡1
H
T
W
+ R
W
)
(12e)
GRANSTR
¨
OM, ET AL.: EXTENDED TARGET TRACKING USING A GAUSSIAN-MIXTURE PHD FILTER 3271
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