3786 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 66, NO. 14, JULY 15, 2018
where
Υ
u
k
[v, Z](n)=
min(|Z |,n)
j=0
(|Z|−j)!ρ
K,k
(|Z|−j)
· P
n
j+u
q
D,k
,v
n−(j+u)
1,v
n
e
j
(Ξ
k
(v, Z)) (9)
Ξ
k
(v, Z)={v, ψ
k,z
: z ∈ Z} (10)
ψ
k,z
(x)=
1,κ
k
κ
k
(z)
g
k
(z|x)p
D,k
(x) (11)
q
D,k
(x)=1− p
D,k
(x) (12)
where at time k, p
D,k
(x) is the target detection probability,
g
k
(z|x) is the single target measurement likelihood, κ
k
(·) and
ρ
K,k
(·) are the clutter RFS intensity function and cardinality
distribution, respectively, P
l
j
=
l!
(l−j)!
is the permutation coef-
ficient with P
l
j
=0when j>lby convention, e
j
(·) denotes
the elementary symmetric function [25] of order j defined for a
finite set Z of real numbers as
e
j
(Z)=
S ⊆Z,|S|=j
⎛
⎝
ζ ∈S
ζ
⎞
⎠
with e
0
(Z)=1by convention, and |S| is the cardinality of a
set S.
For the CPHD filter, the cardinality estimate
ˆ
N
k
can be ob-
tained by the cardinality function
ˆ
N
k
= arg max
n
ρ
k
(n). (13)
When the cardinalities of the RFS are Poisson-distributed, the
PHD recursion can be derived from (4) and (7) (without target
spawning):
v
k|k−1
(x)=γ
k
(x)+
p
S,k
(x
)f
k|k−1
(x|x
)v
k−1
(x
)dx
(14)
v
k
(x)=v
k|k−1
(x)
×
q
D,k
(x)+
z∈Z
k
g
k
(z|x)p
D,k
(x)
κ
k
(z)+
p
D,k
g
k
(z|·),v
k|k−1
.
(15)
The cardinality estimate
ˆ
N
k
is as follows:
ˆ
N
k
= v
k
, 1 . (16)
The implementation method based on the GM model needs
pruning and merging to prevent an unbounded increase of the
Gaussian components [4], [5].
B. The Inverse Gamma Distribution and Gamma Distribution
The probability density of the inverse gamma (IG) distribution
is defined over the support x>0 as
IG(x; α; β)=
β
α
Γ(α)
x
−α−1
exp
−
β
x
(17)
with shape parameter α>0 and scale parameter β>0, and
where Γ(·) denotes the gamma function. The mode at which the
probability density function is the maximum is β/(α +1), and
the mean value is β/(α − 1). The variance of the IG distribution
is β
2
/[(α − 1)
2
(α − 2)].
The probability density of the gamma distribution G(x; α; β)
is
G(x; α; β)=
β
α
Γ(α)
x
α−1
exp (−βx) (18)
with shape parameter α>0 and rate parameter β>0. Its mode
and mean are (α − 1)/β and α/β, respectively.
III. A
NALYTIC IMPLEMENTATION OF THE IGGM
CPHD A
ND PHD FILTERS
In this section, the inverse gamma Gaussian mixture (IGGM)
CPHD filter and PHD filter will be derived in detail. The non-
negative feature determining the detection probability will be
introduced to help the IGGM CPHD and IGGM PHD filters
track multiple targets without the detection probability being
known a priori. The closed-form solution to the CPHD recur-
sion will be discussed in Section III-A. Section III-B will show
the pruning and merging procedure for the IGGM components.
A. IGGM CPHD Filter and IGGM PHD Filter
1) Augmented Target State Transition and Observation Mod-
els: In the following, first, the mixture target state and the cor-
responding measurement are introduced. Then, we present the
transition and observation processes of the mixture target state
in detail.
Single-target state and measurement: Let the joint single-
target state x contain the positions and velocities ˜x, and the
nonnegative feature d, which denotes the SNR throughout, i.e.,
x =[˜x,d]
T
. (19)
We assume that the feature state d and the kinematic state ˜x
are statistically independent and that the single-target detection
probability is determined by the feature d, i.e.,
p
D,k
(x)=p
D,k
(d). (20)
The validity of this assumption mainly comes from the fact that
the factor determining the feature is relatively stable in a short
time window. For instance, the SNR and echo amplitude for
radar and infrared sensors mainly depend on the changes of
radar cross section (RCS) and target temperature rather than the
small change in target location. Each measurement z consists
of the position measurement ˜z and the feature measurement h,
i.e.,
z =[˜z,h]
T
. (21)
Mixture target intensity: By convention, N (·; m; P ) denotes
a Gaussian density with mean m and covariance P . So, at time
k the t arget density can be denoted by the IGGM as
v
k
(˜x,d)=
J
k
i=1
w
(i)
k
N (˜x; m
(i)
k
; P
(i)
k
)IG(d; α
(i)
k
; β
(i)
k
) (22)
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