The Gaussian Particle Multi-target Multi-Bernoulli Filter - IEEE Xplore

The Gaussian Particle Multi-target Multi-Bernoulli Filter Jianjun Yin, Jianqiu Zhang, Jin Zhao Electronic Engineering Department Fudan University Shanghai, China {yinjianjun, jqzhang01, 051021018}@fudan.edu.cn

Abstract—Multi-target multi-Bernoulli (MeMBer) filter is a new attractive approach to tracking an unknown and timevarying number of targets. In this paper, we present a new implementation of the MeMBer recursion—the Gaussian particle MeMBer (GP-MeMBer) filter—for nonlinear models. The probability density in the multi-Bernoulli is approximated by a weighted sum of Gaussians, as in the existed Gaussian mixture (GM-MeMBer) filter, but the target dynamics or observation can be nonlinear. Monte Carlo integration is applied for approximating the prediction and posterior densities in the multi-Bernoulli and the multi-Bernoulli existence probability. The simulation results verify the effectiveness of the proposed GP-MeMBer filter. Keywords-signal processing; Gaussian particle multi-target multi-Bernoulli (GS-MeMBer); simulation; random finite sets (RFSs); nonlinear; tracking

I.

INTRODUCTION

Filtering theory is concerned with estimating the state of a stochastic process recursively from the history of noisy observations. In simple single-target track filtering, where there are no appearing or disappearing targets or spurious measurements (clutter), the states and measurements are both vectors, whose dimensions will not submit to changes. However, in multi-target tracking, the number of targets and the measured tracks will be time-varying with changing dimensions of the states and measurements, because of targets disappearing, spawning, spontaneous births, and the clutter. [1 - 6] So multi-target filtering is a class of dynamic state estimation problems in which the entity of interest is a finite set that is random in the number of elements as well as the values of individual elements. [4] Mahler’s finite set statistics (FISST) provides a general systematic foundation for multi-target filtering based on the theory of random finite sets (RFSs), which performs filtering on set-valued observations and states without explicit connections among measurements and targets [1, 5], which is the bulk of the computational work in traditional multitarget tracking algorithms [7, 8]. Besides the probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters [2, 4, 9, 10], Mahler recently proposed the multi-target multi-Bernoulli (MeMBer) recursion, which is a tractable approximation to the Bayes multi-target recursion under low clutter density scenarios. [5] The MeMBer recursion propagates (approximately) the multi-target posterior density, rather than the moments and cardinality distributions in the PHD and CPHD recursions. Since the original MeMBer 978-1-4244-5848-6/10/$26.00 ©2010 IEEE 556

filter overestimates the cardinality (number of targets), a new MeMBer filter was given by Vo et al. [10] Analytic implementation for linear models are also proposed, which is called Gaussian mixture MeMBer (GM-MeMBer) filter. This paper presents a Gaussian particle implementation of the MeMBer filter, termed as the Gaussian particle MeMBer (GP-MeMBer) filter. In the proposed GP-MeMBer recursion, each probability density in the multi-Bernoulli is approximated by a weighted sum of Gaussians, as is the case in the GM-MeMBer filter, but the transition density or likelihood function can be nonlinear since Gaussian particle approximation is asymptotically consistent even for nonlinear target models. Monte Carlo integration is applied for approximating the multi-Bernoulli existence probability as well as the prediction and posterior densities in the multiBernoulli. This is calculated using a bank of Gaussian particle filters [11], similar to the procedure used with the Gaussian sum particle filter [12] and the Gaussian particle implementation of the PHD filter [13]. Simulation results are presented to demonstrate the validity of the proposed filter. RFSS IN MULTI-TARGET TRACKING

II.

A. Multi-target model with RFSs Suppose that at time k the number of the targets and measurements are M(k) and N(k) respectively. Then we can use RFSs Xk and Zk to denote the multi-target states and measurements respectively as follows: [3] (1) X k xk ,1,, xk , M k  )F , where, ^xk ,i `i

Zk

M k 1

^ ^z

k ,1

`  ` )= ,

,, zk , N k

(2)

denote the single target state, each taking

values in a state space F Ž ƒ nx , and, ^zk , i `i

N k 1

denote the

single measurement, each taking values in a state space = Ž ƒnz , ) F and ) = are the spaces of finite subsets of

F Ž ƒn x and = Ž ƒnz respectively. Given a multi-target state X k 1 at time k-1, each single target state xk 1  X k 1 either survive at time k with the probability p S ,k xk 1 and moves to a new state x k 1 with transition density f k |k 1 xk | xk 1 , or disappears with the probability 1  pS , k xk 1 . Then the target states at time k can be described as follows: [3]

Xk

ª º ª º Sk | k 1 ] »  «  Gk |k 1 ] »  Bk , «¬]  ] X k 1  X k  1 ¼ ¬ ¼

^r  , p  ` i *,k

N

i 1

¦i 1 r i M

S ^x1 , , xn ` and

i j i j p x j n!S I ¦ 3 i , j 1 1 r j 1di1 in d M

^

(11)

M * ,k

i *,k

rki 1 pki 1 , pS ,k ,

(12)

pki 1 , pS ,k ,

(13)

= parameters of the multi-Bernoulli RFS of

i 1

(15)

k

where, rLi,k

(6)



rk|ik 1 1  pki|k 1 , pD ,k pki|k 1 x 1  p D,k

p Li, k

M k |k 1

¦

(7) * U ,k

r





¦

rk|ik 1

i k |k 1

i 1

M k |k 1

rk|ik 1

¦ 1 r

i k |k 1

i 1

III.

D ,k

(16) (17)

pki|k 1 , p D ,k



2

, (18)

1  rk|ik 1 pki|k 1 , p D ,k

¦ 1 r

˜; z

D ,k

rk|ik 1 pki|k 1 x , g k z | x p D ,k x

M k |k 1

M k |k 1

p

i k |k 1

i k |k 1

i k |k 1

i 1

* U ,k

i k |k 1

1  r 

i 1

z

1  r  p , p , x 1  p  , p ,

rk|ik 1 1  rk|ik 1 pki|k 1 x , g k z | x p D ,k x

N k z 

p D ,k x pki|k 1 x g k z | x .

(19)

pki|k 1 x , p D ,k x g k z | x

GAUSSIAN PARTICLE IMPLEMENTATION OF THE MEMBER RECURSION

In this section, we give the main results of the GPMeMBer filter in proposition 1 and proposition 2. The mathematical proofs are given in appendix. Proposition 1. (GP-MeMBer Prediction): Suppose that at time k-1, the posterior multi-target density is a multi-Bernoulli with the form

, the

(8)

^r  , p  `

S k 1

i k 1

M k 1

i k 1

i 1

,

(20)

i

and each probability density p k 1 , i 1,, M k 1 is a Gaussian sum of the form

M

i

,

M

r

S I 3 1  r  j . j 1

M * ,k

i 1

S k | ^rLi,k , pLi, k `i 1k|k 1 ‰ ^rU*,k z , pU* ,k ˜; z `zZ ,

probability density is [5]

n

`

M

. And the mean

cardinality of a multi-Bernoulli RFS is

^

‰ r*i,k , p*i, k

S k|k 1 ^rk|ik 1 , pki|k 1 `i 1k|k 1 , (14) then the posterior multi-target density can be approximated by a multi-Bernoulli as follows:

B. The MeMBer filter The MeMBer recursion is an approximation to the full multi-target Bayes recursion (6) - (7) using multi-Bernoulli RFSs. [5, 10] A multi-Bernoulli RFS X is a union of a fixed number of independent Bernoulli RFSs X i with existence probability r i and probability density p i , i 1, , N . Then a multi-Bernoulli RFS is completely described by the

`

M k 1

i 1

births at time k. Update step: If at time k, the predicted multi-target density is a multi-Bernoulli of the form

where P s is an appropriate reference measure [3]. Readers may refer to e.g. [1 - 5] for the details on the formulation of the multi-target filtering in RFSs.

^

`

f k|k 1 x | ˜ , pki 1 , pS ,k

pPi, k|k 1

k | k 1

multi-Bernoulli parameter set r i , p i

, pPi, k|k 1

rPi,k |k 1

­^x `, detected . (4) 4 k  xk ® z ¯ ‡, missed Then the measurement RFS Zk is formed by ª º Z k «  4 k x »  ; k , (5) ¬ xX k ¼ where ; k denotes the false measurements or clutter. Let p k ˜ | Z 1:k represent a multi-target posterior density, then the Bayes filter propagates the multi-target posterior via

 X k | X pk 1  X | Z1:k 1 Ps dX pk  X k | Z1:k g k Z k | X k pk | k 1  X k | Z1:k 1 .    | | P g Z X p X Z dX k | k 1 1:k 1 s ³ k k

i P ,k |k 1

where,

probability, g k zk | xk is the probability density obtained by an observation zk of the state xk. Therefore, at time k, each state x k  X k generates an RFS as

³f

^r 

S k|k 1

target with previous state ] and the spontaneous birth targets respectively. Assume that p D,k x k denote the target detection

,

M k 1 i k 1 i 1

i k 1

, (10) then the predicted multi-target density is also a multiBernoulli and is given by

where, S k |k 1 ] denotes the RFS of survival targets, Gk | k 1 ] and Bk denote the RFSs of targets spawned from a

pk |k 1  X k | Z1:k 1

^r  , p `

S k 1

(3)

(9) i

`

J ki 1

wki, 1j 1 x; mki, 1j , Pki ,1j , ¦ j 1

i

M

r ,p i 1. As in [10], we abbreviate (8) by S Now we recall the prediction and update steps of the MeMBer recursion: [10] Prediction step: If at time k-1, the posterior multi-target density is a multi-Bernoulli of the form

p k 1

(21)

Then, the predicted multi-target density is also a multiBernoulli

S k |k 1 557

^r 

i P , k | k 1

, pPi, k |k 1

`

M k 1

i 1

^

 r*i, k , p*i, k

`

M * ,k

i 1

,

(22)

^r  , p  ` i *,k

where,

M * ,k

i *,k

Bernoulli RFS of births at time k and rPi, k |k 1 rki 1 pS , k , J ki 1

pP , k |k 1 x i

PUi,,kj

are the parameters of the multi-

i 1

1 M

^x ` i , j .l

k 1

M

l 1

M

¦ x



i , j .l P ,k

, PPi,k, j|k 1

i , j

i , j

l 1



M

^

i , j .l

~ 1 ˜; mk 1 , Pk 1 , xP ,k

i, j P ,k |k 1

l 1

`

M

l 1

(24)



 xPi,,kj.l mPi,,kj| k 1  x Pi,,kj .l



~ f k|k 1 ˜ | xki,1j .l



T

^r 

i k | k 1

, pki|k 1

`

M k|k 1

i 1

Consider the following multi-target tracking model in a two dimensional situation with nonlinear observations [4]:

,

º 0 » » 0 » ª w1, k º xk 2 « 'T » ¬ w2, k »¼ 2 » , (35) 'T »¼ º ª § xk ,1  s x · ¸ » ª v1, k º ªT k º « arctan¨¨ xk ,3  s y ¸¹ yk « » « »« » © ¬ rk ¼ « ¬v2, k ¼ 2 2» «¬ xk ,1  s x  xk ,3  s y »¼ where, xk and yk denote the state and measurement at time k, T respectively; xk xk ,1 , xk , 2 , xk ,3 , xk , 4 , where, x k ,1 and

(26)

wki|,kj 11 x; mki|,kj 1 , Pk|ik, j 1 . ¦ j 1

(27)

Then, the multi-Bernoulli approximation of the posterior multi-target density can be given by

S k | ^rL, k , pL, k `i i

where,

i

1



i

pLi, k x

M k |k 1

¦ i 1

i

`

i

zZ k

,



rk|k 1 1  pD,k 1  rk|k 1 pD,k ,

i

rL,k

rU ,k z

^

 rU , k z , pU , k ˜; z

M k|k 1

 1  r  p

pki|k 1 x ,

rk|ik 1 1  rk|ik 1 p D ,k ] Ui, k z 2

i k |k 1

i

D ,k

,

wki , j 1 x; mUi ,,kj , PUi,,kj , ¦ ¦ i 1 j 1

pU ,k x; z

(28)

] U , k z i

¦ j 1

U ki , j ,l z

^x ` i , j .l k

M

l 1

1 wki|,kj 1 M





M

U k z ¦ l 1



xk ,3 denote the x and y positions, respectively;

(30)

denote the x and y velocities respectively; s sx s y and ' T are the sensor position and sampling interval T T and vk v1, k v2, k are respectively; wk w1, k w2, k state and measurement noises respectively. And wk ~ 1 ˜;0,0.03 I 2 , I 2 denotes two by two identity matrix,

>

(31)



(32)



S ki , j xki , j .l | Z1:k 1 , z

~ S ki , j xki , j .l | Z1:k 1 , z

i , j

wk

i , j

wU , k

i M k |k 1 J k |k 1

i

p*

2

m*

(33)

4

m*



rk|ik 1

i , j 1 i wk |k 1 M 1  rk |k 1

mUi ,, kj

U~ki , j ,l z xki , j .l ¦ l 1

@ , T

^r  , p ` 1 x; m  , P  , S*

> 50 > 30

i *

i *

i *

4

i 1

1  50 1@

where,

m* 3

, i

m*



>0 >30

xk , 4

T

@

T

1

1 10  1@ , P* T

@

 100 @ , 'T multi-Bernoulli

[10],

*

T

>0

s is

i

>

1 [4]. with

r*i

0.03

0 0 0@

T T



, ,

0  20 2@

,

diag >10 10 10 10@ . The

probability of the survival and detection are pS ,k

T

0.99 and

pD,k 0.98 , respectively. Clutter is modeled on a Poisson RFS over the surveillance region with an average of 10 clutter points per scan. At each time step, 300 particles are used, the number of Gaussian components at each time step is capped to a maximum of 100 components, the pruning is performed with a weight threshold of 10-5, and merging is performed with a threshold of 4. Additionally, pruning of the hypothesized tracks is performed with a weight threshold of 10-3 and a maximum of 100 tracks. [10]

i , j

wU , k ¦ ¦ i 1 j 1

wUi ,, kj

>

x k , 2 and

>

@

vk ~ 1 ˜;0, diag 0.052 22 The birth process

density

.

@

(29)

i , j , l

1 z; g k xki , j .l , Rk 1 x; mki|,kj 1 , Pk|ik, j 1



>

and J ki|k 1

0 0



M k |k 1 i § r p ] i z · ¨ N k z  ¦ k |k 1 D ,k U ,k ¸ ¨ 1  rk|ik 1 p D ,k ¸¹ i 1 ©

i M k |k 1 J k |k 1

ª 'T 2 0 º « « 2 0 »» xk 1  « 'T 1 'T » « 0 » « 0 1 ¼ «¬ 0

ª1 'T «0 1 « «0 0 « ¬0 0

J ki|k 1

pk |k 1

. (34)

¦ U ki, j ,l z

NUMERICAL STUDIES

IV.

and each probability density pki|k 1 , i 1,, M k 1 is a Gaussian sum of the form i

T

We know that the GP-MeMBer filter suffers from computational troubles caused by the increasing number of Gaussian components, which is similar to the GM-MeMBer filter. [10] We use the ways given in [10] to reduce the number of components.



. (25) Proposition 2. (GSP-MeMBer Update): Suppose that at time k, the predicted multi-target density is a multi-Bernoulli with the form

S k |k 1



l 1



¦ m

1 M



 xki , j .l mUi ,,kj  xki , j .l

M

U~ki , j ,l z U ki , j ,l z

(23)

where, mPi,,kj| k 1

i, j U ,k

i , j ,l

k

l 1

¦ wki, 1j 1 x; mPi,,kj| k 1, PPi,,kj|k 1 , j 1

¦ U~  z m M

M

U ki , j ,l z ¦ l 1

M

558

Sample M particles from 1 ˜; mki, 1j , Pki ,1j , for each of the

100

components, j 1,, J ki 1 , i.e.,

50

^x `

x

-50

-100

20

30

40

50

60

70

80

90

^x `

100

i , j .l P,k

40

Then

20 0

k |k 1

20

30

40

50

60

70

80

90

100

y

time step

Fig. 1 shows the results of the proposed GP-MeMBer filter, where the true and estimated tracks in x and y coordinates are given separately with the solid line for the true and the plus signs for the estimated. It can be seen that the GP-MeMBer filter is capable of providing good tracking performances. V.





J ki 1







pP , k |k 1 x | ¦ wki, 1j 1 x; mPi,,kj| k 1 , PPi,,kj|k 1 , j 1

(A.7)

where, 1 M

mPi,,kj| k 1





By substituting (27) into (16) we have that 1  ³ pki|k 1 x p D ,k dx 1  p D ,k i i .(A.9) rL,k rk|k 1 rk|ik 1 i i 1  rk|ik 1 p D ,k 1  rk|k 1 ³ pk|k 1 x p D ,k dx Using (17), we have p Li, k x

pki|k 1 x

1  p D,k 1  pki|k 1 , p D , k

pki|k 1 x . (A.10)

Substituting (27) into (19), we have

¦ i i 1 1 r rk|ik 1

M k | k 1

k |k 1

i

i

rk |k 1

¦ ¦j 1 1  r i i 1

i M k | k 1 J k | k 1

rk|ik 1

(A.2)

k |k 1

p D ,k ³

¦j 1 wki|k, j 1 1x; mki|k, j 1 , Pk|ik,j1 1z; g k xk , Rk

J ki|k 1

¦j 1 wki|k, j 1 1x; mki|k, j 1 , Pk|ik,j1 1z; g k xk , Rk dx









wki|k, j 1 1 z; g k x k , Rk 1 x; mki|k, j 1 , Pk|ik,j1

k |k 1

¦ ¦j 1 1  r i i 1

i

p D,k

k |k 1

M k | k 1 J k | k 1

The last equation holds for ³ pki 1 x dx 1 . Substituting (21) into (13) we have

J ki|k 1

rk|ik 1

¦ i i 1 1 r

(A.1)

.



B. Proof of proposition 2

pU ,k x; z

f k |k 1 x | ˜ , pk 1

i , j .l P, k

l 1

. (A.8) 1 M i , j i , j .l i , j i , j .l T PP , k |k 1 ¦ mP,k |k 1  xP,k mP,k |k 1  xP,k M l1 Proposition 1 is thus proved by combining (A.1), (A.7) and (A.8).

APPENDIX

rki 1 pS ,k .

M

¦ x

i , j

M k | k 1

pk 1 , p S ,k



; mki,1j , Pki ,1j dxk 1



i

ACKNOWLEDGMENT The work reported in this paper was funded under the Fudan youth science foundation (09FQ29) and National Natural Science Foundation of China (60872059).

i

(A.4)

a .s .

A new multi-target filter, termed as the Gaussian particle MeMBer (GP-MeMBer) filter has been proposed. It obviates the limits of the GM-MeMBer filter and can be applied to nonlinear tracking models. The recursions for the weights, means, and covariances of the constituent Gaussian components of each probability density in the posterior multi-target multi-Bernoulli density are given in detail. Simulation results show that the proposed GP-MeMBer is capable of providing good tracking performances. Future study is expected to focus on the extension of the filter to non Gaussian noises, which is more common in practice. One probable approach is using the similar procedure in the Gaussian sum particle filter [12].

f k |k 1 x | ˜ , pki 1 p S ,k

k 1

i , j .l k 1

where, o denotes almost sure convergence. Then we have

CONCLUSION

A. Proof of proposition 1 Using (12), we have that rPi,k |k 1 rki 1 ³ pki 1 x pS ,k dx

~ f k |k 1

k





Figure 1. Results of the GP-MeMBer filter

p P ,k |k 1 x

k 1

|

-60

i

 ˜ | x

(A.3)

. (A.5) 1 M f k|k 1 x | xki,1j.l ¦ M l1 By the strong law of large numbers, we know that a.s. 1 M f k|k 1 x | xki,1j .l o ³ 1 xk ; f k|k 1 xk 1 , Qk ˜ 1 xk 1 ; mki,1j , Pki ,1j dxk 1 ¦ M l1 as M o f , (A.6)

-40

10

M

l 1

³ 1x ; f x , Q 1x k

-20

, i.e., .

i , j .l

and sample M particles from f k |k 1 ˜ | xk 1 10

time step

-80



~ 1 ˜; mki, 1j , Pki ,1j ,

i , j .l M k 1 l 1

0

wki|k, j 1 ³ 1 z; g k x k , Rk 1 x; mki|k, j 1 , Pk|ik,j1 dx

.(A.11) For each of the components, j 1,, J ki|k 1 , sample M particles from an importance density, i.e.,

J ki 1

wki,1j ³ 1 xk ; f k |k 1 xk 1 , Qk 1 xk 1 ; mki,1j , Pki ,1j dxk 1 ¦ j 1

559

^x `





~ S ki , j xki , j.l | Z1:k 1, z ,

i , j .l M k l 1

M k |k 1

(A.12)

then we have the weights U ki , j ,l z of each sample according to i , j , l

Uk







1 z; g k xki , j .l , Rk 1 x; mki|,kj 1 , Pk|ik, j 1

z

S ki , j xki , j .l | Z1:k 1 , z

,

rU ,k z

(A.13) M k |k 1

¦ i 1

since that i , j i , j ³ 1z; g k xk , Rk 1 x; mk|k 1 , Pk|k 1 dx









i , j







. (A.14)



 | 1 x; m  , P  ³ 1 z; g x , R 1 x; m  1 | ¦ U  z 1 x; m  , P  M i, j U ,k

i, j U ,k

k

k

i, j k |k 1

k

i, j U ,k

¦ ³ 1  r i i 1

k |k 1



³ pk|k 1 x pD,k x dx i

rk|ik 1 1  rk|ik 1 p D,k ³ pki|k 1 x g k z | x dx

1  r

i

p

k |k 1 D ,k

M k |k 1

¦ i 1



2

rk|ik 1 p D ,k ³ pki|k 1 x g k z | x dx 1  rk|ik 1 p D ,k

dx

dx

M k |k 1

¦ i 1

 1  r  p

rk|ik 1 1  rk|ik 1 p D ,k ] Ui, k z

N k z 

i k |k 1 D ,k M k |k 1 i k |k 1 D ,k i i 1 k |k 1

¦

r

2

p ] Ui, k z

1 r

p D ,k

³ pk|k 1 x g k z | x dx i

wk|k 11 x; mk|k 1 , Pk |k 1 1 z; g k x , Rk dx . (A.21) ³¦ j 1 J ki|k 1



i , j

i , j

1

i , j

M

wki|k, j 1 ¦ U ki , j ,l z ¦ M l1 j 1

Proposition 2 is thus proved by combining (A.9), (A.10), (A.17) - (A.21).

i, j U ,k

,(A.16)

U~ i , j ,l z x i , j .l

mU ,k

¦ l 1

PUi,,kj

T ¦ U~ki, j ,l z mUi,,kj  xki, j.l mUi,,kj  xki, j.l . (A.17)

[1]

k

k

[2]

M

l 1

[3]

M

U~ki , j ,l z U ki , j ,l z

U ki , j ,l z ¦ l 1

The first approximation in (A.16) holds for the property of Gaussian distributions, and can be found e.g. in [13]. The second approximation in (A.16) holds for (A.15). By substituting (A.16) and (A.17) into (A.11), we obtain J i

M k |k 1 k |k 1

rk|ik 1

¦ ¦ 1 r

x; z

i 1

i k |k 1

j 1

wki|k, j 1

i

M k |k 1 J k |k 1

1 M

rk|ik 1

¦ ¦ 1 r i 1

i

M k |k 1 J k |k 1

¦ ¦w

i , j

k

i 1



2

REFERENCES M

i , j

p

i

rk|ik 1 pki|k 1 x g k z | x p D,k x

M k |k 1

J ki|k 1

where,

U ,k

³ pk|k 1 x pD,k x dx

k |k 1

] Ui, k z

, Pk|k 1 dx

l 1

1  r

, (A.20)

M

i , j ,l k



i

where,



i , j



N k z 

N k z 

1z; g k xk , Rk 1 x; mk|k 1 , Pk|k 1 i , j i , j.l S k xk | Z1:k 1 , z dx S ki , j xki , j.l | Z1:k 1 , z By the strong law of large numbers, we have 1 M i , j ,l a.s. ¦ U k z o ³ 1z; g k xk , Rk 1 x; mki|k, j 1 , Pk|ik,j1 dx , M l1 as M o f . (A.15) Then we have that 1 z; g k xk , Rk 1 x; mki|k, j 1 , Pk|ik,j1

³

i , j

¦ ³ i 1

rk|ik 1 1  rk|ik 1 pki|k 1 x g k z | x p D ,k x



j 1

i , j

i , j

1 x; mU ,k , PU ,k

j 1

i k |k 1

¦ U  z 1x; m M

i, j U ,k

i , j ,l k

, PUi,,kj

l 1

wki|k, j 1

1 M

[4]

[5]



[6]

M

¦ U  z

[7]

i , j ,l k

l 1

[8]



[9]

,(A.18)

where i , j

wk

wUi ,,kj

i , j

wU ,k

rk|ik 1

i M k |k 1 J k |k 1

[10]

i , j

wU ,k ¦ ¦ i 1 j 1

i , j 1 i wk|k 1 M 1  rk|k 1

.

[11]

(A.19)

M

U ki , j ,l z ¦ l 1

[12]

Substituting (27) into (18), we have

[13]

560

I. Goodman, R. Mahler and H. Nguyen, Mathematics of data fusion. Boston: Kluwer Academic Publishers, 1997. R. Mahler, “Multitarget Bayes filtering via first-order multitarget Moments”, IEEE Transactions on Aerospace & Electronic Systems, vol. 39(4), 2003, pp: 1152-1178. B. Vo, W. K. Ma, “The Gaussian mixture probability hypothesis density filter”, IEEE Transactions on Signal Processing, vol. 54(11), 2006, pp: 4091-4014. B. Vo, S. Singh and A. Doucet, “Sequential Monte Carlo methods for multi-target filtering with random finite sets”, IEEE Transactions on Aerospace & Electronic Systems, vol. 41(4), 2005, pp: 1224-1245. R. Mahler, Statistical multisourse-multitarget information fusion. Artech House, Norwood, MA, 2007. J. J. Yin, J. Q. Zhang and Z. S. Zhuang, “Gaussian sum PHD filtering algorithm for nonlinear non-Gaussian models”, Chinese Journal of Aeronautics, vol. 21 (4), 2008, pp: 341-351. Y. Bar-Shalom, E. Tse, “Tracking in a cluttered environment with probabilistic data association”, Automatica, vol. 11, pp: 451-460. Y. Bar-Shalom, T. E. Fortmann, Tracking and data association, Boston: Academic Press, 1988. B.-T. Vo, B. Vo and A. Cantoni, “Analytic implementations of the cardinalized probability hypothesis density filter”, IEEE Transactions on Signal Processing, vol. 55(7-2), 2007, pp: 3553 – 3567. B.-T. Vo, B. Vo and A. Cantoni, “The cardinality balanced multitarget multi-Bernoulli filter and its implementations”, IEEE Transactions on Signal Processing, vol. 57(2), 2009, pp: 409-423. J. Kotecha, P. Djuric, “Gaussian particle filtering”, IEEE Transactions on Signal Processing, vol. 51(10), 2003, pp: 2592-2602. J. Kotecha, P. Djuric, “Gaussian sum particle filtering”, IEEE Transactions on Signal Processing, vol. 51(10), 2003, pp: 2603-2613. D. Clark, B.-T. Vo and B. Vo, “Gaussian particle implementations of probability hypothesis density filters”, Aerospace Conference, 2007 IEEE 3-10 March.