Jian Jun Yin, Jian Qiu Zhang, Bo Hu, Qi Yong Lu. Electronic Engineering Department, Fudan University, Shanghai, 200433, China. {yinjianjun, jqzhang01, bohu ...
2010 IEEE Sensor Array and Multichannel Signal Processing Workshop
The Polynomial Predictive Gaussian Mixture MeMBer Filter Jian Jun Yin, Jian Qiu Zhang, Bo Hu, Qi Yong Lu Electronic Engineering Department, Fudan University, Shanghai, 200433, China {yinjianjun, jqzhang01, bohu, lqyong}@fudan.edu.cn Abstract—We propose a novel multi-target tracking algorithm, called the polynomial predictive Gaussian mixture Multi-target Multi-Bernoulli filter (PPGM-MeMBer) filter. We firstly present a unified state space model where the state equation may describe any dynamics of the true targets, no matter linear or nonlinear and no matter we know them well or not, which is more common in practice. Then we apply the Gaussian mixture MeMBer (GMMeMBer) filter to the unified model. The analysis results show that the proposed PPGM-MeMBer filter can deal with situations when we do not know the targets dynamics well. The multi-target tracking simulation results verify the effectiveness of the proposed method.
I.
INTRODUCTION
The main objective of multi-target tracking is to jointly estimate the unknown and time-varying number of targets as well as their individual states from the series of noisy and cluttered observation sets. Traditional approaches to this problem involve data association techniques such as Multiple Hypothesis Tracking, Joint Probabilistic Data Association, and Nearest Neighbor [1, 2], which are so complicated thus constitute the bulk of the computational work in multi-target tracking algorithms. The random finite set (RFS) method to multi-target tracking is an emerging and promising alternative [3, 4], it performs filtering on set-valued observations and states without explicit connections among measurements and targets. The probability hypothesis density (PHD) filter, the cardinalized PHD (CPHD) filter and the Multi-target Multi-Bernoulli (MeMBer) filter are very popular algorithms in RFS [5 - 7]. Note that the existing RFS filters have a preliminary to work well, that is, the target dynamics, i.e., the state equation is assumed to be known. For example, a linear Gaussian dynamical model and a nonlinear nearly-constant turn model were used in the GMPHD filter and its extensions [5]. However, this is not the case in practice because we may not know the target dynamics in advance, especially in maneuver target tracking. When the transition density is not modeled well, the filters may not provide correct results, sometimes even be divergent. Then how can we model the target dynamics correctly? We know that the polynomial predictive filter is an effective way to predict future signal values [8]. Future estimates of the signal are produced by extrapolating and it
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turns out that the coefficients of the filter are not dependent on the signal, but only on the values assigned to the degree of polynomial model, the number of latest samples and the forward prediction step [8]. In this paper, we use polynomial predictive filter to construct the dynamic state space model by extending the dimension of the state. In this way, we may obtain the extended state dynamics whether the original state transition density is known or not. The GM-MeMBer filter is then applied to the proposed extended state space model, we call this new method the polynomial predictive Gaussian mixture MeMBer (PPGMMeMBer) filter. Finally we obtain the estimate of the original state by wiping off the components corresponding to the backward time steps. The simulation results show that the proposed method works effectively even the state dynamics is not known well. II.
BACKGROUND
A. The MeMBer Filter The MeMBer filter is a tractable approximation to the Bayes multi-target recursion under low clutter density scenarios using multi-Bernoulli RFSs. A multi-Bernoulli RFS X is a union of a fixed number of independent Bernoulli RFSs X i with existence probability r i 0,1 and probability density p i , i 1, , M , where M is the number of Bernoulli. Thus a multi-Bernoulli RFS is completely described by the multi-Bernoulli parameter set
r , p i
RFS is
i
M
i 1 M
i 1
. The mean cardinality of a multi-Bernoulli r i and the probability density is [6]
x1 , , xn n!
1i1in M
where,
1 r j . M
j 1
n
j 1
r
i j i j p x j i , 1 r j
(1)
(2)
We abbreviate a probability density of the form (1) by
r i , p i i1 as in [6]. M
B. The Polynomial Predictive Filter We now recall the polynomial predictive filter [8]. Suppose that the signal xk can be modeled as a polynomial of the degree L, then we have
233
(3)
l 0
where pl is the polynomial coefficients. We may have L
xk pl k l wk ,
(4)
l 0
when signal xk does not fit the polynomial model exactly, where wk is an error term. In turns out that predictions using polynomial models can be done by taking a weighted average of the past few values of the signal and that the weighted coefficients are not dependent on the signal, but only on the values assigned to polynomial degree, number of prediction steps and the length of the filter [8], i.e., for exactly polynomial signals, a future value xk N can be obtained as M 1
h
xk N
m0
x
m k m
,
(5)
where, N and M denote the number of prediction steps and the length of the filter respectively, hm are the filter coefficients, which can be obtained by using the method of Lagrange multipliers to minimize the noise gain [8]. For example it is given in [8] that when N 1, L 1 , (6) hm 4 M 6m 4 M ( M 1) , when N 1, L 2 , hm
III.
9M 2 ( 27 36m) M 30m 2 42m 18 M 3 3M 2 2 M .
h
m0
x
m k m
wk 1 ,
z k H kVX k vk
where,
Xk
,
(19)
and zk are the state and measurement
respectively, Wk and vk are process and measurement noises respectively. B. The Polynomial Predictive Gaussian Mixture MeMBer Recursion By applying the GM-MeMBer filter [6] to the constructed unified model (19), we have the following three steps, which constitute the main part of the proposed PPGM-MeMBer filter. Prediction: Suppose that at time k-1 the multiBernoulli posterior multi-target density
k 1 rki 1 , pki1 i1k 1 is given and each p ki1 , i 1, , M k 1 , is comprised of Gaussian mixtures of the M
(9)
where, wk 1 is a noise term and it is zero if xk is a strictly polynomial signal. Then we obtain the extended form of (9) as (10) X k 1 FX k Wk 1 , where, T X k xk , xk 1 , , xk M 1 , (11) h0 h1 hM 1 1 0 0 , F 0 0 1
Q blkdiag ([Q, , Q]) . (14) Note that model (10) does not depend on the original state transition density f k , it may describe any dynamics of the targets. We now consider the correspondent measurement model. Suppose that the original measurement model is as follows: z k H k xk vk , (15) where H k is a linear function, vk is a Gaussian white noise with mean zero and covariance R . Rewrite (15) according to the extended state, i.e., (16) z k H k VX k v k , where, (17) V 1,0, ,0 , thus we have (18) VX k xk , Now we have the unified dynamic state space model, which is comprised of (15) and (16), i.e., X k 1 FX k Wk 1
A. Unified Model Construction Firstly we consider the process model. Suppose that the true model is as follows: xk 1 f k xk nk , (8) where, nk is the process noise and f k is a linear or nonlinear function. Note that (8) is reasonable because f k may not be known. We rewrite (8) as follows according to the polynomial predictive filter, M 1
covariance Q and
(7)
THE POLYNOMIAL PREDICTIVE GAUSSIAN MIXTURE MEMBER FILTER
x k 1
Wk wk , wk 1 ,, wk M 1 . (13) Suppose that wk is white Gaussian with mean zero and covariance Q, and it is independent at different time steps. Then Wk is also white Gaussian with mean zero and T
L
xk pl k l ,
form
J ki1
(20)
j 1
then the predicted multi-target density at time k is also a multi-Bernoulli
k|k 1 rPi,k |k 1 , pPi,k|k 1 i 1k 1 ri,k , pi,k i1 ,k , M
(12)
pki1 wki,1j X ; M ki,1j , Pki1, j ,
and
234
M
(21)
J ki1
w
rPi,k |k 1 rki 1 p S ,k , p Pi,k|k 1 X i , j
i , j
j 1
i, j k 1
X ; M Pi,,kj|k 1 , PP,ik, |jk1
. (22)
i , j
M P ,k|k 1 FM k 1 , PP ,k|k 1 Q FPki1, j F T
k
where as in [6] the survival probability are state independent, i.e., pS ,k X pS ,k and the birth model is a
multi-Bernoulli with parameter set ri,k , pi,k
M ,k
where
i 1
pi,k , i 1, , M ,k 1 are Gaussian mixtures of the form J i,k
i , j
i , j
i , j
pi,k X wi,,kj X ; M i,,kj , P,ik, j , with w ,k , M ,k , P,k j 1
denoting the weights, means and covariances of the jth component.
Update: Suppose that at time k the predicted multi-
Bernoulli multi-target density k|k 1 rk|ik1 , pki|k 1
M k |k 1
i 1
is
given and each pki|k 1 , i 1, , M k|k 1 , is comprised of Gaussian mixtures of the form J ki|k 1
w
i
pk|k 1 X
j 1
i, j k |k 1
X ; M ki|k, j 1 , Pk|ki ,j1 ,
(23)
then the updated multi-target density at time k is also a multi-Bernoulli
k rLi,k , pLi,k i1k|k 1 rU ,k z , pU ,k ; z zZ , (24) M
k
where rLi,k rk|ik1
1 PD,k
1 rk|ik1PD,k
, pLi,k X pki|k 1 X
M k|k 1 r i 1 r i i z k|k 1 U ,k rU ,k z k|k 1 2 i i1 1 r P k|k 1 D,k
pU ,k X ; z
i
M k|k 1 J k|k 1
M k|k 1 i r i z , (25) k z k|k 1 U ,k i i 1 1 rk|k 1 PD,k
w~Ui,,kj X ; MUi,,kj , PUi,k, j
i 1
U ,k z p D ,k
~ i , j w i , j w U ,k U ,k wUi ,,kj
J ki|k 1
w j 1
i , j
k |k 1
i , j
M k |k 1 J ki|k 1
i
rk |k 1
1 rk|ik1
i 1
i , j
z; H k VM k |k 1 , H k VPk |k 1V H Rk
w
i, j U ,k
j 1
H V P H VP V
i , j
i , j
K U ,k Pk |k 1 H
T k
k
k
i, j k |k 1
i, j k |k 1
T
H kT Rk
1
C. Implementation Issues As in the GM-MeMBer filter [6], at each time step, pruning of hypothesized tracks is performed by discarding those with existence probabilities below a threshold P. For each of the remaining racks, we eliminate components with weights below a threshold T, and merge components within a distance U of each other. Further a maximum of Jmax components per hypothesized track is imposed, see [6] for more details. IV.
DEMONSTRATION EXAMPLE
Consider a bearing and range tracking example. The measurement model is as follows [5]: v1,k 1 0 0 0 T zk xk ,1 xk , 2 xk ,3 xk , 4 v 0 0 1 0 2,k
where, xk ,1 and xk ,3 denote the x and y position respectively;
p D ,k wki|k, j1 z; H k VM ki|k, j 1 , H k VPk|ki ,j1V T H kT Rk
i , j
PU ,k I K U ,k
T k
x k , 2 and xk , 4 denote the x and y velocity, vk is the
, (26)
M Ui,,kj M ki|k, j 1 K Ui ,,kj z H k VM ki|k, j 1 , i , j
T
corresponding posterior densities. According to (11), we ˆ ˆ T may rewritten X k as X k xˆ k , xˆ k 1 , , xˆk M 1 , then we may obtain the original state estimate at time k, xˆ k , just by ˆ reserving the first component of X k and discarding the others, i.e., xˆk 1 , , xˆ k M 1 . Note that the recursions above is similar to those of SMC-MeMBer except the following differences (1) The process model is always linear here, see (12), so we may do the optimal Kalman filtering no matter the original process model is linear or nonlinear; (2) The states in the recursions here are not the original target states, but the extended state, so we have to recover the original state. (3) Since we use the proposed unified model, we may not need to suppose the target motion, which is more practical. The proofs of the prediction and update are similar to those in the GM-MeMBer filter thus omitted here.
j 1
and i
Estimate: We first get the estimated number of targets, which is the cardinality mean or mode. Then the extended state estimates at time k, Xˆ , are the means of the
Gaussian measurement noise and vk ~ 0, R , R R1 0; 0 R2 . Assume that we do not know the target dynamics, this is particularly true in applications such as maneuver target tracking. As in the previous discussion, we construct the extended process model according to X k 1 FX k Wk 1 . Suppose that the parameter we use is N 1, L 1, M 2 , then according to (6), we obtain h0 2, h1 1 ,
thus X k xk xk 1 , F 2 I l , 1 I l ; I l 0l , where, I l is the l l identity matrix and 0l the l l zero T
235
matrix, l is the dimension of the target state. Also we T have Wk 1 wk 1 wk , and the covariance of the
ACKNOWLEDGMENT
extended process noise Q QE , 0; 0, QE . Suppose that the true process model is the WNA model as follows: [5]
x coordinate
1000
T 2 0 0 0 2 1 0 0 0 n xk 1 T 2 k 0 1 T 0 T 2 0 0 1 0 T where, T denotes the sampling interval, nk is the process noise and, nk ~ 0, Q . T
50
60
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80
90
100
60
70
80
90
100
0
-500
10
20
30
40
50
Fig.1 Target trakcs
The work reported in this paper was funded under the Fudan youth science foundation (09FQ29) and National Natural Science Foundation of China (60872059). CONFERENCES [1]
[2]
[3] [4]
[5]
[6]
CONCLUSIONS
We proposed a new multi-target tracking method called the polynomial predictive Gaussian mixture MeMBer (PPGM-MeMBer) filter, where the system state dimension is firstly extended by the combination of the original state at current time step and several backward time steps and finally reduced by wiping off the components corresponding to the backward time steps. The PPGMMeMBer filter may work well in the situation that the state dynamics is not known well. The simulation results demonstrate the good performance of the proposed PPGMMeMBer filter when the state dynamics is not known.
40
time step
P diag10,10,10,10 , m1 0;0;0;0 , m2 400;1;600;1 ,
V.
30
500
-1000
4
modeled on a Poisson RFS over the surveillance region 1000,1000m 1000,1000m with an average of 10 clutter points per scan. The probability of the survival and detection are pS ,k 0.99 and p D ,k 0.98 , respectively. Clutter is modeled on a Poisson RFS over the surveillance region with an average of 10 clutter points per scan. Fig. 1 separately denotes the tracks of x and y coordinates, where the solid lines represent the true ones and the marks ‘o’ represent the estimated ones. Fig. 1 shows that the proposed PPGM-MeMBer filter is capable of providing accurate tracking performances.
20
1000
r , pi i1 where r 0.03 , pi x x; mi , P , 4 m3 800;0;200;2 , m 200;0;800;0 . Clutter is
10
time step
In the simulation, T 1 , R1 R2 10 2 , R2 12 , Q 52 , QE 32 . And as in [6], at each time step pruning and merging of Gaussian components are performed for each hypothesized track using a weight threshold of T 10 5 , a merging threshold of U = 4m, and a maximum of Jmax = 100 components. Furthermore, pruning of hypothesized tracks is performed with a weight threshold of T 10 3 and a maximum of Tmax = 100 tracks. Also the birth process is multi-Bernoulli with the density
0
-500
-1000
y coordinate
1 0 xk 0 0
PPGM-MeMBer filter estimates True tracks
500
[7]
[8]
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