IEEE TRANS AEROSPACE AND ELECTRONIC SYSTEMS, VOL. X, NO. X, MONTH 20XX 1
Adaptive target birth intensity for PHD and CPHDﬁlters
B. Ristic
a
, D. Clark
b
, BaNgu Vo
c
, BaTuong Vo
d
Abstract
The standard formulation of the PHD and CPHD ﬁlters assumes that the target birth intensity is known a priori.In situations where the targets can appear anywhere in the surveillance volume this is clearly inefﬁcient, since thetarget birth intensity needs to cover the entire state space.This paper presents a new extension of the PHD and CPHD ﬁlters, which distinguishes between the persistentand the newborn targets. This extension enables us to adaptively design the target birth intensity at each scan usingthe received measurements. Sequential MonteCarlo implementations of the resulting PHD and CPHD ﬁlters arepresented and their performance studied numerically. The proposed measurement driven birth intensity improves theestimation accuracy of both the number of targets and their spatial distribution.
Index Terms
Bayesian multiobject ﬁltering, random ﬁnite set, probability hypothesis density (PHD), particle ﬁlter
I. I
NTRODUCTION
Mahler [1] recently proposed a systematic generalisation of the singletarget recursive Bayes ﬁlter to the multitarget case. In this formulation, the targets can appear and disappear anywhere (within the state space of interest) andanytime (within the surveillance period), while target motion can be described by a nonlinear stochastic dynamicmodel. The sequentially received measurements are uncertain both due to the imperfections of the detection process(target detections could be missing and false detections can be present) and due to the stochastic nature of the
a
ISR Division, Defence Science and Technology Organisation, Bld 94, M2.30, 506 Lorimer Street, Melbourne, VIC 3207, Australia; Tel:(+61 3) 9626 8226; Fax: +61 3 9626 8341; email:
branko.ristic@dsto.defence.gov.au
b
Joint Research Institute in Signal and Image Processing, HeriotWatt University, Riccarton, Edinburgh, EH14 4AS, United Kingdom;Tel: +44 131 449 5111; email:
D. E. Clark@hw.ac.uk
c
School of EECE, The University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia; Tel: (+61 8) 6488 1767: fax:+61 8 6488 1065; email:
bangu.vo@uwa.edu.au
d
School of EECE, The University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia; Tel: (+61 8) 6488 1767: fax:+61 8 6488 1065; email:
batuong.vo@uwa.edu.au
IEEE TRANS AEROSPACE AND ELECTRONIC SYSTEMS, VOL. X, NO. X, MONTH 20XX 2
possibly nonlinear sensor model. The multitarget Bayes ﬁler sequentially estimates the number of targets presentand their individual states.The Bayes ﬁlter propagates the posterior probability density function (pdf) through a twostep procedure: theprediction and update. In the multitarget case, the multitarget posterior pdf is formulated using the ﬁnite setstatistics (FISST) [1], a set of practical mathematical tools from point process theory. The propagation of thismultitarget posterior, however, is computationally very intensive due to the high dimensionality of the multitargetstate space. If the state space of a single target is
X
, the multitarget posterior pdf is deﬁned on
F
(
X
)
, the spaceof ﬁnite subsets of
X
. To overcome the high dimensionality of the multitarget Bayes ﬁlter, Mahler introduced theProbability Hypothesis Density (PHD) ﬁlter [2], which propagates the ﬁrst moment of the multitarget posteriorknown as the intensity function or the PHD, deﬁned on the singletarget statespace
X
. The resulting PHD ﬁltersubsequently became a very popular multitarget estimation method with applications in sonar [3], computer vision[4], [5], SLAM [6], trafﬁc monitoring [7], biology [8], etc.Since the intensity function is a very crude approximation of the multitarget pdf, Mahler subsequently introducedthe Cardinalised PHD ﬁlter [9], which propagates both the intensity function and the cardinality distribution of themultitarget pdf. The resulting estimate of the number of targets is more stable than that of the PHD ﬁlter, asconﬁrmed by numerical studies in [10]. The CPHD ﬁlter has been applied to GMTI tracking [11], tracking in theaerial videos [12], etc.The standard formulations of both the PHD and CPHD ﬁlters assume that the target birth intensity is known apriori. Typically the birth intensity has the majority of its mass distributed over small speciﬁc areas of
X
, which, forexample in the air surveillance context, can be interpreted as the regions around airports [10], [13]. Note howeverthat if a target appears in a region that is not covered by the predeﬁned birth intensity, the PHD/CPHD ﬁlter willbe completely blind to its existence. Making the target birth intensity diffuse so that it covers the entire state spaceof interest, typically results in a higher incidence of shortlived false tracks and longer conﬁrmation times. Theonly way to overcome this drawback is to create at each processing step of the ﬁlter a massive number of potential(hypothesised) newborn targets covering the entire statespace, which is clearly inefﬁcient. The described limitationaffects both methods of PHD/CPHD ﬁlter implementation: the sequential Monte Carlo (SMC) method [14], [15]and the ﬁnite Gaussian mixtures (GM) [10], [13].Starting from the standard equations of the PHD and the CPHD ﬁlter, in this paper we derive novel extensionswhich distinguish, in both the prediction and the update step, between the persistent and the newborn targets. Thisapproach allows the PHD/CPHD ﬁlter to adapt the target birth intensity at each processing step using the receivedmeasurements. The resulting measurement driven birth intensity is very important in practice because it removes theneed for the prior speciﬁcation of birth intensities and eliminates the restriction on target appearance volumes within
IEEE TRANS AEROSPACE AND ELECTRONIC SYSTEMS, VOL. X, NO. X, MONTH 20XX 3
X
. The paper presents an SMC implementation of proposed extensions of PHD and CPHD ﬁlters and demonstratestheir improvement in performance by numerical examples.Two remarks are in order here. First, we point out that the proposed measurement driven target birth intensityis complementary with the recent attempts to improve the efﬁciency of the SMCPHD ﬁlter by preselectingparticles for propagation (the socalled auxiliary particle PHD ﬁlter) presented in [15]. Second, the idea to usethe measurements to adaptively build the target birth intensity has been proposed previously [16], [17]. Our paper,however, develops this initial idea much further.II. B
ACKGROUND
Suppose that at time
k
there are
n
k
target states
x
k,
1
,...,
x
k,n
k
, each taking values in a state space
X ⊆
R
n
x
, and
m
k
measurements (detections)
z
k,
1
,...,
z
k,m
k
, each taking values in the observation space
Z ⊆
R
n
z
. A multitargetstate and a multitarget observation are then represented by the ﬁnite sets:
X
k
=
{
x
k,
1
,...,
x
k,n
k
} ∈ F
(
X
)
,
(1)
Z
k
=
{
z
k,
1
,...,
z
k,m
k
}∈ F
(
Z
)
,
(2)respectively. Here
F
(
X
)
and
F
(
Z
)
are the ﬁnite subsets of
X
and
Z
, respectively. At each time step sometargets may disappear (die), others may survive and transition into a new state, and new targets may appear. Dueto the imperfections in the detector, some of the surviving and newborn targets may not be detected, whereasthe observation set
Z
k
may include false detections (or clutter). The evolution of the targets and the originof measurements are unknown. Uncertainty in both multitarget state and multitarget measurement is naturallymodelled by random ﬁnite sets.The objective of the recursive multitarget Bayesian estimator [1] is to determine at each time step
k
theposterior probability density of multitarget state
f
k

k
(
X
k

Z
1:
k
)
, where
Z
1:
k
= (
Z
1
,...,
Z
k
)
denotes the accumulatedobservation sets up to time
k
. The multitarget posterior is computed sequentially via the prediction and the updatesteps, see [1, ch.14].Since
f
k

k
(
X
k

Z
1:
k
)
is deﬁned over
F
(
X
)
, practical implementation of the multitarget Bayes ﬁlter is a difﬁculttask and is usually limited to a small number of targets [18]–[20]. In order to overcome this limitation, Mahlerproposed [2] to propagate only the ﬁrstorder statistical moment of
f
k

k
(
X

Z
1:
k
)
, referred to as the intensity functionor the PHD,
D
k

k
(
x

Z
1:
k
) =
δ
X
(
x
)
f
k

k
(
X

Z
1:
k
)
δ
X
. In this deﬁnition
δ
X
(
x
) =
w
∈
X
δ
w
(
x
)
. The integral of the PHD over
X
,
X
D
k

k
(
x

Z
1:
k
)
d
x
=
ν
k

k
∈
R
,
(3)gives the (posterior) expected number of targets in the state space. The resulting PHD ﬁlter replaces the predictionand the update step of the multitarget Bayes ﬁlter with the much simpler expressions for the prediction and update
IEEE TRANS AEROSPACE AND ELECTRONIC SYSTEMS, VOL. X, NO. X, MONTH 20XX 4
of the PHD (given in the next section).Since the posterior PHD
D
k

k
(
x

Z
1:
k
)
is a very crude approximation of
f
k

k
(
X

Z
1:
k
)
, Mahler subsequentlyproposed [9] to propagate the cardinality distribution
ρ
(
n
) =
Pr
(

X

=
n
)
given by:
ρ
(
n

Z
1:
k
) = 1
n
!
f
k

k
(
{
x
1
,...,
x
n
}
Z
1:
k
)
d
x
1
...
x
n
,
(4) jointly and alongside the PHD. The cardinality distribution satisﬁes the following condition:
∞
n
=1
nρ
(
n

Z
1:
k
) =
D
k

k
(
x

Z
1:
k
)
d
x
=
ν
k

k
. This is the basis of the CPHD ﬁlter.III. E
XTENSION OF THE
PHD
FILTER
The standard PHD ﬁlter equations are reviewed ﬁrst, using the abbreviation
D
k

k
(
x

Z
1:
k
)
abbr
=
D
k

k
(
x
)
. Theprediction equation of the PHD ﬁlter is given by
1
[2]:
D
k

k
−
1
(
x
) =
γ
k

k
−
1
(
x
) +
p
S
D
k
−
1

k
−
1
, π
k

k
−
1
(
x
·
)
(5)where
•
γ
k

k
−
1
(
x
)
is the PHD of target births between time
k
and
k
+ 1
;
•
p
S
(
x
′
)
abbr
=
p
S,k

k
−
1
(
x
′
)
is the probability that a target with state
x
′
at time
k
−
1
will survive until time
k
;
•
π
k

k
−
1
(
x

x
′
)
is the singletarget transition density from time
k
−
1
to
k
;
•
g,f
=
f
(
x
)
g
(
x
)
d
x
.The ﬁrst term on the RHS of (5) refers to the newborn targets, while the second represents the persistent targets.Upon receiving the measurement set
Z
k
at time
k
, the update step of the PHD ﬁlter is computed according to:
D
k

k
(
x
) = [1
−
p
D
(
x
)]
D
k

k
−
1
(
x
) +
z
∈
Z
k
p
D
(
x
)
g
k
(
z

x
)
D
k

k
−
1
(
x
)
κ
k
(
z
) +
p
D
g
k
(
z
·
)
,D
k

k
−
1
(6)where
•
p
D
(
x
)
abbr
=
p
D,k
(
x
)
is the probability that an observation will be collected at time
k
from a target with state
x
;
•
g
k
(
z

x
)
is the singletarget measurement likelihood at time
k
;
•
κ
k
(
z
)
is the PHD of clutter at time
k
.In the above formulation of the PHD ﬁlter, new targets are “born” in the prediction step (5). The intensity functionof the newborn targets
γ
k

k
−
1
(
x
)
is independent of measurements, and in the general case, where the targets canappear anywhere in the state space, it has to cover the entire
X
. This is signiﬁcant for both the SMC and the GMimplementation of the PHD ﬁlter, because the newborn target particles or Gaussian mixture components, need tocover the entire statespace with reasonable mass for the PHD ﬁlter to work properly. Clearly this is inefﬁcient andwasteful.
1
We do not consider target spawning in this paper.
IEEE TRANS AEROSPACE AND ELECTRONIC SYSTEMS, VOL. X, NO. X, MONTH 20XX 5
Instead we propose to design a newborn target intensity in the region of the statespace
x
∈ X
for which thelikelihood
g
k
(
z

x
)
will have high values. We show that if the birth intensity is adapted in accordance with themeasurements,
the PHD equations must be applied in a different form
.Start from (5) and (6), where the state vector
x
consists of the usual kinematic/feature component (position,velocity, amplitude, etc) which we denote by
y
and a mark or a label
β
, which distinguishes a newborn target fromthe persistent target, i.e.
x
= (
y
,β
)
where
β
=
0
for a persistent target
1
for a newborn target(7)and
y
∈ Y
. The birth PHD is then:
γ
k

k
−
1
(
x
) =
γ
k

k
−
1
(
y
,β
) =
γ
k

k
−
1
(
y
)
, β
= 10
, β
= 0
.
(8)Note a slight abuse of notation in using the same symbol
γ
k

k
−
1
for both functions of
x
and
y
. Similar abuse willbe used throughout this section, but the meaning should be clear from the context.A newborn target becomes a persisting target at the next time, but a persisting target cannot become a newborntarget. Thus the mark
β
can only change from
1
to
0
but not viceversa. The transition model is then
π
k

k
−
1
(
x

x
′
) =
π
k

k
−
1
(
y
,β

y
′
,β
′
)=
π
k

k
−
1
(
y

y
′
)
π
k

k
−
1
(
β

β
′
)
(9)with
π
k

k
−
1
(
β

β
′
) =
0
, β
= 11
, β
= 0
.
(10)The probability of survival does not depend on
β
and hence
p
S
(
x
) =
p
S
(
y
,β
) =
p
S
(
y
)
(11)The PHD ﬁlter prediction equation (5) for the augmented state vector is given by:
D
k

k
−
1
(
y
,β
) =
γ
k

k
−
1
(
y
,β
) +
1
β
′
=0
D
k
−
1

k
−
1
(
y
′
,β
′
)
p
S
(
y
′
,β
′
)
π
k

k
−
1
(
y
,β

y
′
,β
′
)
d
y
′
.
(12)Upon substitution of expressions (8)(11) into (12) we obtain the new form of the PHD ﬁlter prediction:
D
k

k
−
1
(
y
,β
) =
γ
k

k
−
1
(
y
)
, β
= 1
D
k
−
1

k
−
1
(
·
,
1) +
D
k
−
1

k
−
1
(
·
,
0)
,p
S
π
k

k
−
1
(
y
·
)
β
= 0
(13)