14th International Conference on Information Fusion Chicago, Illinois, USA, July 5-8, 2011
Particle labeling PHD filter for multi-target track-valued estimates Hongyan Zhu Chongzhao Han Yan Lin School of Electronic & Information Engineering, Xi’an Jiaotong University Xi’an, Shaanxi, 710049, P. R. China
[email protected] [email protected] [email protected] Abstract--Multi-target tracking is a difficult problem
computationally tractable alternative to the RFSs
due to the measurement origin uncertainty. Recently,
Bayesian multi-target filtering. It is a recursive
the probability hypothesis density (PHD) filter provides
mechanism which propagates the first-order statistical
a promising tool for joint estimation of target number
moment---the posterior expectation, instead of the
and multi-target states, without using data association
posterior distribution. Multi-target state estimates can be
technique. In particle implementations of the PHD
obtained by applying the peak extraction technique over
filter, clustering is used to extract the target state from
the PHD function. The PHD filter can be especially used
the particle population. This technique yields poor
in the cases with unknown target number, and gives the
performance when the estimated number of targets
estimation of target number and multi-target states
differs from the number of clusters in the particle
simultaneously. Reference [14] relaxed the Poisson
population. A particle labeling PHD filter for
assumption on the target number and derived a
multi-target track-valued estimates is developed in this
generalization of the PHD recursion known as the
paper. By implementing the efficient sampling and
Cardinalized (CPHD) recursion, by jointly propagating
particle labeling technique, the proposed method can
the posterior intensity function and the posterior
yield not only better state estimate, but also
cardinality distribution.
track-valued estimate. The multi-scan measurement
Since the PHD filter equations still include some
information is also employed to reduce the uncertainty
high-dimension integrals, there is no closed-form
of the estimates. Simulation results demonstrate the
solution in general. In [15], the Gaussian mixture PHD
efficiency of the proposed method. Keywords: Probability hypothesis density, random finite set,
(GM-PHD) filter is developed, which is a closed-form
track-valued estimate.
1 Introduction
solution to the PHD filter under linear Gaussian assumptions. In non-linear non-Gaussian cases, the Sequential Monte Carlo (SMC) techniques [16]-[19] are regarded as an efficient means to approximate the density
For the multi-target tracking (MTT) problem in the
function, by recursively propagating a group of weighted
presence of measurement origin uncertainty, data
particles. An SMC implementation of the PHD filter is
association [1-8] can be done to determine the
proposed in [20]-[21], and the analogous idea can also be
assignments between measurements and targets. The
seen in [22]-[24]. Unfortunately, since the original PHD filter [21] keeps
performance of state estimate will be seriously affected by the accuracy of data association.
no record of track identities, the estimation of individual
The random finite sets (RFSs) theory [9]-[12] provides
target can’t be obtained directly. Some approaches
a promising tool to the MTT problem. The probability
[25]-[28] have been developed in order to acquire the
hypothesis density (PHD) filter [13] is a suboptimal but
track-valued estimate. In [25]-[26], multi-target state estimates are firstly obtained by standard clustering
978-0-9824438-3-5 ©2011 ISIF
1842
techniques, and then track-valued estimates were
the weights, and the node history indicating how it is
acquired by performing track-to-estimate association
generated from the beginning. If the time index just lies
based on the MHT mechanism. Reference [27] keeps a
on the decision time, then the decision is made based on
separate tracker for each target, and constructs a
the maximal weight criteria, and the whole track-valued
two-dimensional
perform
estimate can be obtained by going back the node history.
‘peak-to-track’ association across the adjacent two scans.
In addition, we draw the samples based on a mixture
Reference [28] also introduced extra indices to each one
proposal based on the measurement set, instead of using
in the whole particle set to store the track identity. But
the transition density. In this way, the sampling efficiency
when the estimated number of target differs from the
can be improved greatly.
assignment
problem
to
number of clusters in the particle population, the original particle PHD filter fails to give good state estimates.
2 Problem formulation
In the above methods, the clustering is done among the whole particle set to obtain multi-target state estimates. But when the targets are close, the particle cloud from different targets can overlap each other. The particles generated from one target can be partitioned wrongly into another cluster from other target, which will yield poor performance. In this paper, a particle labeling PHD filter for multi-target track-valued estimates is proposed. Each one in the whole particle set has its own label which indicates the target identity. The particles are evolved with its label,
2.1 Multi-target state model and measurement model Let X and Z denote the multi-target state space and measurement space, N k and M k be the number of targets and measurements at time k, respectively. Then the multi-target states are xk ,1 , xk ,2 ,..., xk , N k ∈ X , and the measurements are zk ,1 , zk ,2 ,..., zk , M k
The multi-target state set and measurement set at time k can be described as the finite set variables:
X k = {xk ,1 , xk ,2 ,..., xk , Nk }, X k ∈ F (X )
and the weights are predicted and updated according to
of particles with the same label, in order to avoid being
,
Z k = {zk ,1 , zk ,2 ,..., zk ,M k }, Z k ∈ F (Z ) where F (X ) and F (Z ) represent the respective
the measurement information. The estimation for individual targets will be obtained by the weighted sum
∈Z .
collections
of
all
finite
subsets
generated
k
the targets are close, high weights of some particles are
from X and Z . Let Z be the cumulative measurement set until time k，and f k |k −1 ( X k | X k −1 )
possibly due to the measurement generated by the other
and
target. So, the directly weighted combination of all the
probability and the likelihood function. The optimal multi-target Bayesian recursive formula are given by
influenced by those particles from other targets. When
particles with the same label may be wrong. For this
gk (Zk | X k )
be
the
multi-target
transition
pk|k −1 ( X k | Z k −1 )
reason, we need to develop an efficient method to extract
= ∫ fk|k −1 ( X k | X ) pk −1|k −1 ( X | Z k −1 )dX
some particles from the particle set with the same label, which can represent the target state with a high quality. Here we do clustering in those particles with the same
pk|k (Xk | Zk ) =
label firstly, and then establish a temporary node for each
gk (Zk | Xk ) fk|k−1(Xk | Zk−1) k−1 ∫ gk (Zk | X ) pk|k−1(X | Z )dX
(1)
(2)
cluster. Next, we should make the decision to tell which node will be used to update the target state. Hence, we
2.2 Probability hypothesis density filter
define a decision window with a given width by using of
The optimal multi-target Bayesian recursive formula
the multi-scan measurement information. When the time
is computationally intractable. The Finite set statistics
index is within the decision window, each temporary
(FISST) theory [13] provides a powerful tool that allows
node will be extended into several new nodes according
the extension of the Bayesian inference to multi-target
to the clustering result. Each node records all its particles,
tracking cases directly. The PHD filter is a suboptimal 1843
alternative to the multi-target Bayesian recursive formula
integrals, the SMC technique is combined with the PHD
in the FISST framework.
filter to deal with the computational intractability [21].
The probability hypothesis density Dk |k ( x | Z ) is
For simplicity, assuming Ps ( x ) = 0 . Let Lk -1 be the
the first order statistical moment of the multi-target
total number of particles at time k-1 and set L0 =0 .
k
posterior
density
k
f k |k ( X k | Z ) ， which is a
Given a particle representation of Dk −1|k −1 by the particle
non-negative function with the following property
= ∫ X ∩ S f k |k ( X | Z k )dx = ∫ Dk |k ( x | Z k )dx S
{x
(i ) k −1
set
Nˆ k ( S )
, wk(i−)1}
Lk −1
i =1
at time k-1, Lk −1
(3)
Dk −1|k −1 ( xk −1 ) = ∑ wk( i−)1δ x( i ) ( xk −1 )
S
where Nˆ k ( S ) is the expected number of targets in
(8)
k −1
i =1
Prediction step:
For i = 1,..., Lk −1 , sample xk ∼ q (. | xk −1 , Z k ) , (i )
region S . The PHD filter involves a prediction step and an
(i )
(i )
(i )
where q ( xk | xk −1 , Z k ) is the proposal distribution.
update step as follows.
And then the predicted weights can be given by
Prediction step:
wk(i|k) −1 =
Dk|k−1(xk | Z ) = γk (xk ) + ∫ϕ(xk , xk−1)Dk−1|k−1(xk−1)dxk−1 (4) k−1
ϕ ( xk(i ) , xk(i−)1 ) (i ) k
(i ) k −1
q( x | x , Z k )
wk(i−)1
(9)
where
To explore newborn targets, J k new particles are
ϕ(xk , xk −1) = Ps (xk −1) fk|k −1(xk | xk−1) + bk|k−1(xk | xk−1) (5)
generated.
The target survives with a probability and
Ps ( xk −1 ) ,
(i ) k
sample x
(i ) k |k −1
w
bk |k −1 (. | xk −1 ) be the
intensity of the target RFS spawned from previous state xk −1 , and γ k ( xk ) be the intensity function of the spontaneous birth RFS. Update step:
(10)
targets. Update step: For each measurement z ∈ Z k , compute
Ck ( z ) =
(6)
Lk −1 + J k
∑ψ i =1
k ,z
( xk( i ) ) wk(i|k) −1
(11)
whereψ k , z ( x ) = PD ( x ) g k ( z | x ) , and then update the weights
where
Ck ( z ) = ∫ PD ( xk )g k ( z | xk ) Dk |k −1 ( xk | Z k −1 )dxk
1 γ k ( xk( i ) ) = J k pk ( xk(i ) | Z k )
Where pk (. | Z k ) is the probability density for newborn
Dk|k ( xk | Z k −1 ) ⎡ P ( x ) g ( z | xk ) ⎤ = ⎢1 − PD ( xk ) + ∑ D k k ⎥ Dk|k −1 ( xk ) z∈Zk κ k ( z ) + Ck ( z ) ⎦ ⎣
,
∼ pk (. | Z k ) , and compute the weights
f k |k −1 ( xk | xk −1 ) denotes the transition probability
density for individual targets. Let
i = Lk −1 + 1,..., Lk −1 + J k
For
⎡ ψk,z (xk(i) ) ⎤ (i) (i ) w = ⎢1− PD (xk ) + ∑ ⎥ wk|k−1 z∈Zk κk (z) + Ck (z) ⎦ ⎣
(7)
(i ) k
where PD ( xk ) and g k ( z | xk ) are the detection
probability and the likelihood function for individual targets, respectively. κ k ( z ) denotes the intensity of the
(12)
Resampling step:
Firstly, the sum of weights are given by
clutter RFS. It can be also written to be rk ck (.) , and rk
Nˆ k |k =
is the average clutter number per scan assuming a Poisson distribution, ck (.) is the probability distribution
and then resample
for each clutter measurement.
{x
(i ) k
2.3 Particle PHD filter Since the PHD filter still involves high dimensional 1844
, wk(i ) Nˆ k |k
}
Lk
i =1
Lk −1 + J k
∑ i =1
{x
(i ) k
wk(i )
, wk(i ) Nˆ k |k
(13)
}
Lk −1 + J k
i =1
to get
, where Lk = Lk −1 + J k . Finally,
rescale the weights by Nˆ k |k to get the updated particle
{
}
( i ) Lk
(i )
set xk , wk
i =1
at time k.
determine which cluster will be used to compute the target state, the multi-scan measurement information is used to lessen the uncertainty.
State estimate:
In the above figure, the width of the decision window
Multi-target state estimates can be obtained by the
is set to be 2. The circles denote the nodes which are
peak extraction over the PHD function represented by a
produced according to the clustering result among the
set of weighted particles, and standard clustering
particles with the same label at each scan. If a node has
techniques [33] can be used to acquire these peaks.
no a father node, then it corresponds to a newborn target.
3 Particle labeling PHD filter 3.1 Basic idea In the original particle PHD filter described in
The label in the circle is the track identity, and it will be inherited by all its descendant nodes. At the decision moment for each target, the cluster with the maximal sum of weights (black circles in the figure) is kept, and the other ones are eliminated.
Section 2, multi-target state estimates are obtained by
How to draw the samples is a key step to implement
standard clustering technique. But if the estimated
the particle filter. In this paper, we sample from a mixture
number of targets is incorrect, multi-target state estimates
proposal, instead of the transition probability done in the
given by standard clustering technique are unreliable.
original particle PHD filter. The presented algorithm is
Actually, each one in the whole particle set is produced from different origin, either from some existing target or newborn target. This information is ignored in the
described as follows.
3.2 Algorithm (i )
(i )
(i )
L
original particle PHD filter. If we add a label to each
Let {xk −1 , wk −1 , lk −1}i =k1−1 denote the whole particle
particle which records the target identity and can be
set at time k-1, xk −1 and wk −1 represent the ith particle
inherited along with the particle evolution procedure,
and the corresponding weight, lk -1 means the target
then the state estimate for individual target can be obtained by using of the particles with the same label. If the targets are well-separated, the weighted sum of particles with the same label can give good state estimate. But the result is unsatisfactory when the targets are close together, because the high weight of some particles may be produced by measurements from other targets.
(i )
(i )
(i )
identity with lk -1 ∈ Γ k −1 = {1, 2,...Nˆ max } , and Nˆ max is (i )
the maximal target identity until time k-1. At time k ≥ 1 , 1) Sample for existing targets. For i = 1,..., Lk −1 , sample xk ∼ q (. | xk −1 , Z k ) , (i )
(i )
(i )
where q (./ xk −1 , Z k ) denotes the proposal distribution. Here, a mixture proposal is given by Mk
p(./ xk(i−)1 , Z k )= ∑ b j p(. | xk( i−)1 , zk , j )
(14)
j =0
where b j
p (θ j | Z k )( j = 0,1,..., M k ) , and θ j
( j > 0) is the association hypothesis which means that the measurement zk , j originates from the target. The hypothesis θ 0 means no measurement comes from Fig.1.
the target. The probability can be computed similar to the association probability in probabilistic data association [1]. The sample procedure is drawn from the mixture
Basic principle of the presented method
Here, we do clustering among the particles with the same label. Each cluster is a set of particles, which is a candidate to be used to compute the target state. To
proposal as follows.
1845
Sample
from
the
measurement index
m∼q1(.| xk(i−)1, Zk ) where the proposal q1 (. / .) is chosen to be discrete distribution
{(0, b ), (1, b ),..., (M 0
1
k
}
wk(i ) ) .
Pj
is
classified
into and
Nˆ e j mj
clusters. is
the
measurement number in the validation gate of target j .
where the proposal q2 (. / .) is set to be the posterior
distribution, which can be approximated easily by applying the local-linearization technique or the unscented transformation. If m = 0 , then the sample can be drawn from the transition probability. The predicted weights can be computed by the following formula
ϕ ( xk( i ) , xk( i−)1 ) p( xk(i ) / xk(i−)1 , Z k )
Nˆ a =1+Nˆ c j , where Nˆ c j means the target number which are close to target j . It can be computed as the number of targets whose validation gate intersects with the one of target j . For j = 0 , the particle set P0 is partitioned into Nˆ n clusters. 5) Track Extension and decision We set a time window using the multi-scan
wk(i−)1
(15)
measurement information. When the time index is within the time window, we establish a node for each cluster
2) Sampling for newborn targets. For i = Lk−1 +1,..., Lk−1 + Jk , Sample xk ∼ pk (. | Z k ) (i )
about the particles, the weight, and its generation history.
1 γ k (x ) J k pk ( xk( i ) | Z k ) (i ) k
(i ) L
(i )
l ( 1 ≤ l ≤ Nˆ e j ) from Pj which carries all the information
,
and compute the weights for newborn targets
(i )
i = Lk −1 +1
Where Nˆ e j = min{m j , Nˆ a } ,
xk(i ) ∼ q2 (./ xk(i−)1 , zk ,m ) ,
wk(i|k) −1 =
∑
4) Clustering In this step, clustering is done among those particles with the same label. We denote Pj as the particle set
set
state
wk(i|k) −1 =
Lk −1 + J k
with label j ( j ∈ {0, Γ k −1}) . For j ∈ Γ k −1 , the particle
, bM k ) .
Sample from the proposal q2 (. / .) for the target
Nˆ n = round (
q1 (. / .) for the
proposal
(16)
3) The whole particle set {xk , wk |k −1 , lk }i =k1−1
+ Jk
When the time index is just on the decision time, the final decision will be made as follows. a) we denotes
can be
obtained by combining all the particles generated from
cluster l ( 1 ≤ l ≤
(i )
Nˆ e j ) from Pj ( j ∈ Γ k −1 ), and let (19)
1≤ l ≤ N e j
can be inherited from lk −1 directly, that is lk = lk −1 . (i )
the sum of weights for each
l * = arg max {ω jl }, w* = ω jl* ˆ
the above two steps. For i = 1,..., Lk −1 , the label lk (i )
ω jl as
(i )
If
For i = Lk −1 + 1,..., Lk −1 + J k , a temporary label ‘0’ is
w* is larger than some given threshold GT , then the
l *th cluster is kept. The multi-scan target states within the
assigned, until it is finally confirmed in Step 5). Then the weights can be updated as the following formula
time window can be obtained by going back in the node
⎡ ψ k , z ( xk( i ) ) ⎤ (i ) (i ) (i ) wk = ⎢1 − PD ( xk ) + ∑ ⎥ wk |k −1 (17) z∈Z k κ k ( z ) + Ck ( z ) ⎦ ⎣
be deleted along with its ancestor nodes. If
Ck ( z ) =
Lk −1 + J k
∑ψ i =1
k ,z
(i ) k
(i ) k |k −1
( x )w
(18)
Then the resampling procedure is done as the original particle PHD filter does, and then the whole particle set (i ) k
(i ) k
( i ) Lk −1 + J k k i =1
{x , w , l }
can be obtained. The estimated
number for newborn targets is given by
history of cluster l
*th
. The other clusters with label j will
w* is less
than the threshold GT , the track will be terminated. As a result, the particle set
{xk(i ) , wk( i ) , lk(i ) }iL=k1 at time k will
be updated by all the left particles. b) For each cluster from P0 , if the sum of weights is greater than some given threshold G I , then a new target can be initiated and a new track identity is assigned to all the particles in this cluster. The maximal target identity is also modified by Nˆ max = Nˆ max + 1 . Otherwise, the 1846
cluster will be deleted.
and ∏
if m ≤ n ,
For simplicity, we only consider linear Gaussian model in 2-D plane. The target moves in the surveillance region [ -1000， 1000] m × [ -1000， 1000] m . The target
d p(c) ( X , Y )
T
⎛1⎛ = ⎜ ⎜ min ⎝ n ⎝ π∈∏n if m > n ,
state takes the form x( k )=[x(k ), x(k ), y(k ), y( k )] ,
x(k ) and y(k ) are the position components, x(k ) and y(k ) denote the velocity components of
where
the target state at time k. The target dynamics is given by
T 0 0⎤ ⎡T 2 / 2 0 ⎤ ⎢ ⎥ 1 0 0⎥⎥ 0 ⎥ T ⎢ x(k) + v(k) (20) ⎢ 0 T 2 / 2⎥ 0 1 T⎥ ⎢ ⎥ ⎥ 0 0 1⎦ T ⎦ ⎣ 0 where the process noise v( k ) ∼ N (.;0, Qk ) with
⎡1 ⎢0 x(k +1) = ⎢ ⎢0 ⎢ ⎣0
Qk = diag([σ , σ ] ) . The measurement equation is 2 T vy
⎡1 0 0 0 ⎤ z (k ) = ⎢ ⎥ x(k ) + w(k ) ⎣0 0 1 0⎦
(21)
where the measurement noise w( k ) ∼ N (.;0, Rk ) 2 2 T with Rk = diag([σ wx , σ wy ] ),
{1, 2,..., k} for
any k ∈ N = {1, 2,...} . For 1 ≤ p < ∞, c > 0 , The OSPA metric of order p with cut-off c is defined as :
4 Simulation results
2 vx
be the permutations on the set
k
σ wx =2m and σ wy =1.5m .
The process noise is assumed independent of the measurement noise. We set the sample interval T = 1s . The clutter
dp(c) (X,Y) = dp(c) (Y, X)
(22)
(23)
d ( c ) ( x, y ) = min(c, d ( x, y )) denotes the distance between x, y with cut off c > 0 . We define d ( x, y ) as the Euclidean distance, and set c = 100 and p = 2 . The thresholds for track where
initiation
and
track
termination are respectively.
GI = 0.8 and GT = 0.5 ,
set The
measurements are supported by 50 MC runs performed on the same target trajectory but with independently generated measurements for each trial. z Example 1 In this scenario, the average number of clutter measurement per scan is set r = 10 , and the standard derivation of process noise is σ vx
= σ vy = 2 . The width
of decision window is set to be 1. The multi-target trajectories and state estimates against time are plotted in Fig.2. In Fig.3, the OSPA error is shown.
measurements are uniformly distributed over the surveillance region with an average rate
1
⎞⎞ p d (c) ( xi , yπ (i ) ) p + c p (n − m) ⎟ ⎟ ∑ i =1 ⎠⎠ m
200
r points per 150
y,m
scan. New-born targets can appear according to a Poisson point process with the intensity function
real track presented method original particle phd
100
γ k = 0.5Ν (.; x1 , Qn ) + 0.5Ν (.; x2 , Qn )
50
with x1 = [-200 10 200 -2] , x2 = [80 -2 70 5]T , T
and
0 -300
Qn = diag ([4 , 0.01,1, 0.04]T ) .The particle
number for each existing target is set to be 500, and 1000 for newborn targets. For simplicity, we only consider the unity detection probability and survival probability. The Optimal Sub-pattern Assignment (OSPA) metric
-100
0 x,m
100
200
300
Fig.2. Multi-Target Trajectories and State Estimates 14 presented method original particle phd
12 10 OSPA,m
presented in [29] is employed to evaluate the performance. It is a mathematically and intuitively
8 6
consistent metric, which tries to capture the estimate
4
quality both the cardinality and multi-target states.
2
Let
-200
X = {x1 ,..., xm } and Y = { y1 ,..., yn } represent
0 0
the real multi-target state set and the estimated one with the cardinality m and n ( m, n ∈ N 0 = {1, 2,...} ),
5
10
15 time,s
20
Fig.3. OSPA Error
1847
25
30
z
Example 2
80
In this example, the expected number of clutter measurement per scan is set r = 60 , and the standard
60
= σ vy = 10 . The
OSPA,m
derivation of process noise is σ vx
presented method with single scan original particle phd presented method with multiple scan
70
width of decision window is set to be 1 and 3,
50 40 30 20
respectively. The multi-target trajectories and state
10
estimates against time are illustrated by the Fig.4-6.
0
0
2
4
6
8
y,m
16
18
20
PHD filter and the presented method with single-scan measurements, track loss phenomena happened. But if
200
we set the width of decision window to be 3, good
0
tracking performance can be achieved. It shows that
-200
multi-scan measurement information is needed to -200
-100
0
100
200
300
400
500
600
700
x,m
Fig.4. State Estimates by the Original Particle PHD Filter
improve the performance, in cases with large process noise and high clutter density
5 Discussions and conclusions
800
A Particle labeling PHD filter for multi-target
600 real track presented method with single-scan
y,m
14
In this scenario, when we apply the original particle
real track original particle phd
400
400
track-valued estimates is developed in this paper. There
200
are some possible future research directions. To reduce the
0
-400 -300
uncertainty
of
the
problem,
the
multi-scan
measurement is employed in the presented method. As a
-200
result, the decision delay and high computational burden -200
-100
0
100
200
300
400
500
600
700
x,m
is inevitable. So a variable width of the decision window may be more economic and can keep the balance
Fig.5. State Estimates by the Presented Method with Single-Scan
between
800
performance, according to the current measurement
burden
and
algorithm
Acknowledgement
400
The work is sponsored by the national key fundamental research & development programs (973) of P.R. China (2007CB311006) and National Natural Science Foundation of China (61004087/F030119).
200
0
-200
-400 -300
computational
distribution and the target maneuvering level.
real track presented method with multi-scan
600
y,m
12
Fig.7. OSPA Error
600
-400 -300
10
time,s
800
-200
-100
0
100
200
300
400
500
600
700
References
x,m
Fig.6. State Estimates by the Presented Method with Multi-Scan
The OSPA error is shown Fig. 7.
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