Multi-target Bayesian filter for propagating marginal distribution

Signal Processing ] (]]]]) ]]]–]]]

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Contents lists available at ScienceDirect

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Signal Processing

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journal homepage: www.elsevier.com/locate/sigpro

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Multi-target Bayesian filter for propagating marginal distribution

15 Q1

Liu Zong-xiang n, Xie Wei-xin

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ATR Key Laboratory, College of Information and Engineering, Shenzhen University, Shenzhen 518060, China

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a r t i c l e i n f o

abstract

Article history: Received 13 November 2013 Received in revised form 9 May 2014 Accepted 8 June 2014

The Bayesian filter and its approximation, the probability hypothesis density (PHD) filter, propagate joint distribution of the multi-target state and the first-order moment of the joint distribution, respectively. However, these two filters fail to distinguish multiple distinct targets when these targets are closely spaced. To efficiently distinguish closely spaced targets according to a sequence of measurements, we (1) use the individual state distributions to model the uncertainties of individual target states caused by the target dynamic uncertainty and measurement uncertainty, (2) use the existence probabilities of individual targets to characterize the randomness of target appearance and disappearance, and (3) propose a novel multitarget Bayesian filter. Instead of maintaining the joint state distribution, the proposed filter jointly propagates the marginal distributions and existence probabilities of each target. An implementation of the proposed filter for linear and Gaussian models is also presented to deal with an unknown and variable number of targets. The simulation results demonstrate that the proposed filter is better at distinguishing distinct targets and tracking multiple targets than the Gaussian mixture PHD filter. & 2014 Published by Elsevier B.V.

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Keywords: Multi-target tracking Bayesian filter Probability hypothesis density filter Linear and Gaussian models Marginal distribution

33 35 37 39 41 43 45 47 49 51 53

63 1. Introduction Detection and tracking of multiple targets are crucial for distinguishing multiple distinct targets and estimating their states according to a set of uncertain measurements that consist of observations corrupted by both noise and clutter. This scenario describes a very challenging problem in a cluttered environment when targets have a small separation compared with measurement error [1]. The most commonly used multi-target tracking technique is multi-target Bayesian filter, which propagates joint posterior distribution through target dynamics, measurement likelihood, and the Bayesian rule [1–3]. However, due to the integrals of high dimensions and the requirement that

55 n

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Corresponding author. Tel.: þ86 755 26732055; fax: þ86 755 26732049. E-mail address: [email protected] (L. Zong-xiang).

the target number is known, the optimal Bayesian filter is intractable in many tracking applications. Principled approximation strategies must be developed to make the Bayesian filter practical [1,4–7]. Existing approximations of the multitarget Bayesian filter include fixed grid approximation, sequential Monte Carlo (SMC) approximation, and probability hypothesis density (PHD) approximation [1,2]. The fixed grid filter approximates the multi-target state numerically by using a fixed grid and applies numerical integration for filter recursion. However, the large associated computational cost makes this filter impractical and inherently intractable [1,2]. The SMC filter represents target distribution by using particles and importance weights, and propagates them in the filter recursion [1,2]. The SMC filter requires less computational load than the fixed grid filter. However, this difference in computational load is not enough to allow the use of the SMC filter in real-time applications [2]. Importantly, both fixed grid and SMC filters require a known number of targets. The PHD filter proposed by Mahler alleviates computational

http://dx.doi.org/10.1016/j.sigpro.2014.06.005 0165-1684/& 2014 Published by Elsevier B.V.

61 Please cite this article as: L. Zong-xiang, X. Wei-xin, Multi-target Bayesian filter for propagating marginal distribution, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.06.005i

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L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

intractability of the multi-target Bayesian filter and can estimate the instantaneous target number [2,4]. Instead of maintaining joint posterior distribution of the multi-target state, the PHD filter propagates the first-order moment of multi-target posterior density. With the development of two numerical solutions of the PHD filter, namely, SMC [8–14] and Gaussian mixtures (GM) [15–22], the PHD filter has received increasing research interest from scholars and researchers. Many extensions of the PHD filter have also been proposed to improve the performance of the PHD filter. PHD filters with observation-driven birth intensity were independently proposed in [21,23,24] to obviate the need for exact knowledge of birth intensity. Methods for maintaining track continuity were proposed in [8,25] for the SMC-PHD filter and in [26] for the GM-PHD filter. To improve the accuracy and stability of the target number estimate, the cardinalized PHD filter, which jointly propagates the moment and cardinality, was proposed in [27]. Methods for estimating an unknown clutter rate, which is an important parameter of the PHD filter, were proposed in [28,29]. In [17], the GM-PHD filter was extended to linear jump Markov multi-target models for use in tracking maneuvering targets. A GM-based PHD filter for sequentially handling the received measurement was proposed by Liu et al. [18]. The efficiency, simplicity of implementation, and success of the PHD filter in avoiding the combinatorial problem that arises from data association has made it more appealing than other filters. Given its advantages, the PHD filter is used in passive localization [13], passive radar target tracking [14], visual tracking [30], target tracking in sonar images [31], group target tracking [32], and so on. However, the PHD filter cannot distinguish multiple distinct targets when they are closely spaced. A simple example in [4] shows that when two targets are well separated, the multi-target posterior intensity of the PHD filter is bimodal, and the maxima are near the states of the two targets, but it is unimodal with a maximal value at the mean of the two target states if the two distinct targets have a sufficiently small separation. In this case, the state estimate given by the PHD filter is that of the target group, not that for either of the two targets. The same problem occurs when using the multi-target Bayesian filter, but a PHD-based multi-target tracker experiences more difficulty with closely spaced targets [2,4]. In this paper, we derive and propose a new multi-target Bayesian filter to efficiently distinguish closely spaced targets according to a sequence of measurements. The proposed filter sufficiently considers the independences of individual targets. Instead of maintaining the joint posterior density of the multi-target state, the filter propagates the marginal distributions of each target. Similar to the PHD filter, the proposed filter operates on the single-target state space, hence avoiding the combinatorial problem that arises from data association. In the proposed filter, the uncertainties of individual target states caused by the target dynamic uncertainty and measurement uncertainty are modeled by individual state distributions, and the randomness of target appearance and disappearance is characterized by the existence probabilities of individual targets. Filter recursion jointly propagates individual distributions and their existence probabilities. An implementation of the proposed filter for linear and Gaussian models is also developed. Simulation

results are obtained by comparing the proposed filter and the PHD filter in terms of optimal subpattern assignment (OSPA) distance [33], which demonstrates that the proposed filter is better than the PHD filter at distinguishing distinct targets and tracking multiple targets. The similarity between the two filters ensures that some improvements to the PHD filter can be applied to the proposed filter. The joint integrated probabilistic data association filter (JIPDAF) proposed in [34] is also a multi-target filter used to propagate marginal posterior distribution. However, the proposed filter and JIPDAF are different in nature. JIPDAF handles the possible presence of multiple targets in a joint PDAF [35] manner, where joint events are formed by creating all possible combinations of track-measurement assignments. The combinatorial nature of this data association-based approach makes it computationally intensive in general [17], whereas the proposed method does not need data association. The main contributions of this paper focus on two points. First, we propose a new method for computing marginal posterior distribution and existence probability. Second, we present a novel implementation of the proposed filter for linear and Gaussian models. The rest of this paper is organized as follows. Section 2 briefly introduces the multi-target Bayesian filter and PHD filter. The multi-target Bayesian filter used to propagate marginal distribution is proposed in Section 3. The implementation of the proposed filter for linear and Gaussian models is presented in Section 4. The performance of the proposed filter is evaluated by simulations in Section 5. Conclusions are drawn in Section 6.

63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93

2. Multi-target Bayesian filter and PHD filter

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2.1. Multi-target Bayesian filter

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Given that the proposed filter is a multi-target Bayesian filter, we first describe the multi-target Bayesian filter briefly [1]. In a multi-target Bayesian filter, the distribution of interest is the joint posterior f ðxt jy1:t Þ, which is also known as the filtering distribution, where t denotes the discrete time index, xt ¼ ðx1;t ⋯xK;t Þ is the multi-target state at time t, K is the target number, and y1:t ¼ ðy1 ⋯yt Þ represents all the observations from time 1 to time t. The filtering distribution of a multi-target Bayesian filter can be computed by using two-step recursion.

99 101 103 105 107 109

Prediction step Z f ðxt jy1:t  1 Þ ¼ f ðxt jxt  1 Þf ðxt  1 jy1:t  1 Þdxt  1

ð1Þ

111 113

Update step gðyt jxt Þf ðxt jy1:t  1 Þ f ðxt jy1:t Þ ¼ f ðyt jy1:t  1 Þ

115 ð2Þ

117 119

where f ðxt jxt  1 Þ denotes the Markov transition probability from state xt  1 at time t  1 to state xt at time t, gðyt jxt Þ is the probability density that state xt at time t generates

Please cite this article as: L. Zong-xiang, X. Wei-xin, Multi-target Bayesian filter for propagating marginal distribution, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.06.005i

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L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

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measurement yt at time t, and f ðyt jy1:t  1 Þ is given by Z gðyt jxt Þf ðxt jy1:t  1 Þdxt f ðyt jy1:t  1 Þ ¼

39 41

ð3Þ

2.2. PHD filter The PHD filter is analogous to the filter proposed in this paper. Both filters consider the uncertainties of the target dynamics and measurement as well as the randomness of the appearance and disappearance of the target, and both can deal with an unknown and variable number of targets. The difference between the two filters is that the PHD filter propagates probability hypothesis density or intensity [4], whereas the proposed filter jointly propagates the marginal distributions and existence probabilities of each of the targets. For comparison, the PHD filter [2,4] is introduced briefly. The recursion of the PHD filter consists of the prediction and update steps.

vtjt  1 ðxt jy1:t  1 Þ ¼ γ ðxt Þ þ

Z ½pS;t ðxt  1 Þf tjt  1 ðxt jxt  1 Þ

þ βtjt  1 ðxt jxt  1 Þvt  1 ðxt  1 jy1:t  1 Þdxt  1

ð4Þ

35 37

of integrals. This problem is usually resolved by using the SMC approach [12] or GM approach [15].

Eq. (2) clearly shows that the new filtering distribution is obtained by means of direct application of the Bayesian rule. The multi-target Bayesian filter involves the integrals of high dimensions and is intractable in general. This intractability is usually resolved by means of fixed grid approximation or SMC approximation [1,2]. The target number is needed to construct the joint distribution of the multi-target state. Thus, the multitarget Bayesian filter cannot be applied when the number of targets is unknown and variable [1,2].

Prediction step

vt ðxt jy1:t Þ  ð1  pD;t ðxt ÞÞvtjt  1 ðxt jy1:t  1 Þ pD;t ðxt Þg t ðzjxt Þvtjt  1 ðxt jy1:t  1 Þ R þ ∑z A yt λt ðzÞ þ pD;t ðxt Þgt ðzjxt Þvtjt  1 ðxt jy1:t  1 Þdxt

43 45

49 51 53 55 57 59 61

where xt ¼ ðx1;t ⋯xK;t Þ denotes the multi-target state at time t, y1:t ¼ ðy1 ⋯yt Þ denotes all of the observations up to time t; vtjt  1 ðxt jy1:t  1 Þ and vt ðxt jy1:t Þ are the predicted posterior intensity and updated posterior intensity, respectively; f tjt  1 ð Ujxt  1 Þ, pS;t ðxt  1 Þ, β tjt  1 ð Ujxt  1 Þ, and γ t ð U Þ are the transition probability, survival probability, intensity of the spawned targets, and intensity of the birth targets, respectively; and yt , g t ð U jxt Þ, pD;t ðxt Þ, and λt ðzÞ are measurement set, measurement likelihood, detection probability, and clutter intensity at time t, respectively. As an approximation of the multi-target Bayesian filter, the PHD filter propagates the intensity of the joint multitarget posterior distribution, and the integral of the intensity is the expected number of targets in the integral space [2]. Unlike the Bayesian filter, the PHD filter alleviates intractable integrals, but still involves computation

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3. Multi-target Bayesian filter for propagating marginal distribution Instead of maintaining the joint posterior distribution of the multi-target state, the proposed filter jointly propagates the marginal distributions for each of the targets and the existence probabilities of the individual marginal distributions. In this paper, existence probability is used to characterize the randomness of target appearance and disappearance. To derive the new Bayesian filter, we assume that each target evolves and generates observations independently of one another, that clutter is Poisson and independent of target-originated measurements, and that the survival and detection probabilities of each target are state independent. At the same time, we assume that the number of targets at time t 1 is K, the states of individual targets at time t  1 are represented by xk;t  1 , k ¼ 1⋯K, the uncertainties of individual target states are denoted by the probability density distributions f k ðxk;t  1 jy1:t  1 Þ, k ¼ 1⋯K, and the uncertainties of individual targets are denoted by pk;t  1 , k ¼ 1⋯K, where t is the discrete time index and y1:t  1 ¼ ðy1 ⋯yt  1 Þ represents all the observations up to time t  1. The objective is to determine the state distributions for each of the targets and their existence probabilities at time t. Proceeding independently for each target through the Bayesian recursions in (1), we can obtain the individual predicted distributions f k ðxk;t jy1:t  1 Þ, k ¼ 1⋯K. The prediction step of the new filter is given by Z f k ðxk;t jxk;t  1 Þf k ðxk;t  1 jy1:t  1 Þdxk;t  1 ; f k ðxk;t jy1:t  1 Þ ¼ k ¼ 1⋯K

Update step

ð5Þ

47

3

67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97

ð6Þ

99

The existence probability pk;tjt  1 of the predicted distribution f k ðxk;t jy1:t  1 Þ can be given by

101

pk;tjt  1 ¼ pS;t pk;t  1 ;

103

k ¼ 1⋯K

ð7Þ

where pS;t is the survival probability of the target. To enable the Bayesian filter to track the birth targets and spawned targets, we extend the predicted distributions to include the birth and spawned distributions. Assuming that the birth and spawned distributions at time t are as follows:

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k

k ¼ 1⋯K γ

ð8Þ

111

k

k ¼ 1⋯K β

ð9Þ

113

and that the existence probabilities of the birth and spawned distributions are given by

115

pkγ ;

k ¼ 1⋯K γ

ð10Þ

117

pkβ ;

k ¼ 1⋯K β

ð11Þ

119

where K γ is the number of the birth distributions and K β is the number of the spawned distributions. The extended prediction distributions and their existence probabilities

121

f γ ðxk;t Þ; f β ðxk;t Þ;

Please cite this article as: L. Zong-xiang, X. Wei-xin, Multi-target Bayesian filter for propagating marginal distribution, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.06.005i

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L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

4

1 3 5

f k ðxk;t jy1:t  1 Þ;

9

k ¼ 1⋯K p

ð13Þ

pk;tjt ¼ 1  ∏ ð1  pik;tjt Þ;

pk;tjt  1 ;

k ¼ 1⋯K p Kp ¼ K þ Kγ þ Kβ,

pk;tjt  1 ¼ pkγ  K k  K  Kγ

fβ

kK

f k ðxk;t jy1:t  1 Þ ¼ f γ

ðxk;t Þ

and

The combined distribution in (16) is obtained only if the target is detected. The target may not be detected at time t when the detection probability pD;t is smaller than 1. The updated distribution of a missing target should be the one to maintain its prediction distribution as follows:

for K o k r K þK γ , and f k ðxk;t jy1:t  1 Þ ¼ k  K  Kγ

ðxk;t Þ and pk;tjt  1 ¼ pβ

for k 4K þ K γ .

25

After we obtain the extended prediction distributions f k ðxk;t jy1:t  1 Þ, k ¼ 1⋯K p of individual target states and the corresponding existence probabilities pk;tjt  1 , k ¼ 1⋯K p , the next step is to determine the individual filtering distributions and their existence probabilities. To do this, we assume that the number of observations at time t is M and that the observations at time t is denoted by yt ¼ ðy1;t ⋯yM;t Þ. Owing to the uncertainty of the target state, the uncertainty of the measurement, and the existence of clutter, any measurement at time t may originate from one of targets at time t or from clutter. Given the extended prediction distributions f k ðxk;t jy1:t  1 Þ, k ¼ 1⋯K p of individual target states, the existence probabilities pk;tjt  1 , k ¼ 1⋯K p of individual extended prediction distributions, and measurement yi;t at time t, we can obtain the update distributions by using the Bayesian rule as

27

i f k;tjt ; ðxk;t jy1:t ; yi;t Þ

11 13 15 17 19 21 23

29

pD;t pk;tjt  1 g k ðyi;t jxk;t Þf k ðxkjt jy1:t  1 Þ ¼ ; R λc;t þ pD;t ∑kep¼ 1 pe;tjt  1 ge ðyi;t jxe;t Þf e ðye;t jy1:t  1 Þdxe;t

31

i ¼ 1⋯M

33 35

ð14Þ

where λc;t is the clutter density, pD;t is the detection probability of the target, and g k ðyi;t jxk;t Þ is the measurement likelihood. The predicted distribution f k ðxk;t jy1:t  1 Þ i

37

clearly gives rise to M updated distributions f k;tjt ðxk;t jy1:t  1 ; yi;t Þ, i ¼ 1⋯M in the update step of the new Bayesian filter. The existence probability of the updated distribution

39

f k;tjt ðxk;t jy1:t  1 ; yi;t Þ can be defined as its integral

41

pik;tjt ¼

i

f k;tjt ðxk;t jy1:t  1 ; yi;t Þdxk;t ;

i ¼ 1⋯M

ð15Þ

43 45 47 49 51 53 55 57 59 61

i¼1

k ¼ 1⋯K p

ð18Þ 67

Given that a predicted distribution generates M updated distributions in the update step, the M distributions must be combined to a distribution. The combined distribution can be given by the weighted sum of the M distributions f k;tjt ðxk;t jy1:t Þ ¼

M

1 1 i ðx jy ; y Þ ¼ ∑yi;t A yt ∑ f c i ¼ 1 k;tjt k;t 1:t  1 i;t c

pD;t pk;tjt  1 g k ðyi;t jxk;t Þf k ðxk;t jx1:t  1 Þ R λc;t þpD;t ∑Ke p¼ 1 pe;tjt  1 ge ðyi;t jxe;t Þf e ðxe;t jy1:t  1 Þdxe;t



ð16Þ where c is the normalized coefficient and is given by Z M i f k;tjt ðxk;t jy1:t  1 ; yi;t Þdxk;t ¼ ∑yi;t A yt c¼ ∑ i¼1 R pD;t pk;tjt  1 g k ðyi;t jxk;t Þf k ðxk;t jy1:t  1 Þdxk;t  R Kp λc;t þ pD;t ∑e ¼ 1 pe;tjt  1 ge ðyi;t jxe;t Þf e ðxe;t jy1:t  1 Þdxe;t ð17Þ

69 71

ð19Þ

73

The existence probability of the updated distribution u f k;tjt ðxk;t jy1:t Þ can be defined as

75

puk;tjt ¼ ð1  pD;t Þpk;tjt  1 ;

ð20Þ

77

Further combining the distribution f k;tjt ðxk;t jy1:t Þ with f k;tjt ðxk;t jy1:t Þ, we can obtain a newly combined distribution

79

u

f k;tjt ðxk;t jy1:t Þ ¼ f k ðxk;t jy1:t  1 Þ;

k ¼ 1⋯K n

k ¼ 1⋯K p u

81

u

f k ðxk;t jy1:t Þ ¼

pk;tjt f k;tjt ðxk;t jy1:t Þ þ puk;tjt f k;tjt ðxk;t jy1:t Þ ; pk;tjt þ puk;tjt

k ¼ 1⋯K p

ð21Þ

83

The existence probability of the combined distribution f k ðxk;t jy1:t Þ can be defined as

85

pk;t ¼ 1 ð1  pk;tjt Þð1  puk;tjt Þ;

ð22Þ

87

The posterior distributions f k ðxk;t jy1:t Þ, k ¼ 1⋯K p in (21) and the existence probabilities pk;t , k ¼ 1⋯K in (22) are the filtering distributions of the individual target states and the existence probabilities of the individual filtering distributions at time t, respectively.

89

4. Implementation of the proposed filter for linear and Gaussian models

95

k ¼ 1⋯K p

91 93

97 Similar to the GM-PHD filter in [15], a closed-form solution to the recursion of the proposed filter requires the following assumptions:

i

Z

63 65

M

where 7

ð12Þ

The existence probability of the combined distribution f k;tjt ðxk;t jy1:t Þ can be defined as

are given by

A.1. Each target follows a linear Gaussian dynamic model and the sensor has a linear Gaussian measurement model, i.e.,

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xk;t ¼ F t  1 xk;t  1 þ ut  1

ð23Þ

105

yk;t ¼ H t xk;t þ wt

ð24Þ

107

where F t  1 and H t are state transition and observation matrices, respectively; xk;t  1 and xk;t denote the state vectors at time t  1 and t, respectively; and ut  1 and wt are zero mean Gaussian noises with covariances Q t  1 and Rt , respectively. A.2. The birth distributions and spawned distributions are assumed to exhibit Gaussian distribution, i.e., k

k

k

k

Nðxk;t ; mγ ;t ; P γ ;t Þ; Nðxk;t ; mβ;t ; P β;t Þ;

k ¼ 1⋯K γ

ð25Þ

k ¼ 1⋯K β

ð26Þ

109 111 113 115 117 119

k

where mkγ ;t and P γ ;t are the mean and covariance of the birth distribution k, respectively, and mkβ;t and P kβ;t are the mean and covariance of the spawned distribution k, respectively.

Please cite this article as: L. Zong-xiang, X. Wei-xin, Multi-target Bayesian filter for propagating marginal distribution, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.06.005i

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L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

1 3

Based on assumptions A.1 and A.2, we present a closedform solution to the recursion of the proposed filter for linear and Gaussian models, which consists of the prediction step and the update step.

5 7 9 11 13 15 17 19 21 23 25 27

Prediction step: The Gaussian distributions at time t  1 are given as Nðxk;t  1 ; mk;t  1 ; P k;t  1 Þ;

k ¼ 1⋯K

ð27Þ

where xk;t  1 is the state vector and mk;t  1 and P k;t  1 are the mean and the covariance of the Gaussian distribution k, respectively. The existence probabilities of individual distributions are given by pk;t  1 , k ¼ 1⋯K. Similar to Proposition 1 in [15], it can be derived from Eqs. (6) and (23) using the Lemmas 1 and 2 in [15] that the individual predicted distributions are Gaussian and can be given by Nðx; mk;tjt  1 ; P k;tjt  1 Þ;

k ¼ 1⋯K

ð28Þ

where xk;tjt  1 denotes the predicted state vector and mk;tjt  1 and P k;tjt  1 are the mean and covariance of the predicted distribution k, respectively, and are given by mk;tjt  1 ¼ F t  1 mk;t  1 ;

P k;tjt  1 ¼ Q t  1 þF t  1 P k;t  1 F Tt 1 ð29Þ

where T denotes the transpose. The existence probabilities for the predicted distributions Nðxk;tjt  1 ; mk;tjt  1 ; P k;tjt  1 Þ, k ¼ 1⋯K are given by

29

pk;tjt  1 ¼ pS;t pk;t  1 ;

31

where pS;t is the survival probability of the target. We extend the predicted distributions in (28) to include the birth distributions in (25) with existence probabilities pkγ ;t , k ¼ 1⋯K γ and spawned distributions in (26) with existence probabilities pkβ;t , k ¼ 1⋯K β , the extended prediction distributions can be given by

33 35

k ¼ 1⋯K

ð30Þ

37

Nðxk;tjt  1 ; m; P k;tjt  1 Þ;

39

where K p ¼ K þ K γ þ K β , mk;tjt  1 ¼ mkγ ;t K and P k;tjt  1 ¼

41

k ¼ 1⋯K p

ð31Þ

k  K  Kγ

P kγ ;t K for K ok r K þK γ , and mk;tjt  1 ¼ mβ;t k  K  Kγ

P k;tjt  1 ¼ P β;t

47

where pk;tjt  1 ¼ pkγ ;t K for K o k rK þ K γ , and pk;tjt  1 ¼

51 53 55 57 59 61

k ¼ 1⋯K p

ð32Þ

for k 4 K þK γ .

Update step: Assuming that the extended prediction distributions at time t Nðxk;tjt  1 ; mk;tjt  1 ; P k;tjt  1 Þ;

k ¼ 1⋯K p

ð33Þ

and that the existence probabilities of the extended prediction distributions are given by pk;tjt  1 ;

k ¼ 1⋯K p

ð34Þ

A finding that can be derived from Eqs. (16), (17), and (24) using the Lemmas 1 and 2 in [15] is that the combined distribution f k;tjt ðxk;t jy1:t Þ in (16) can be given by the weighted sum of M Gaussian distributions as

67 69

where mik;tjt ¼ mk;tjt  1 þ Ak U ðyi;t  H t mk;tjt  1 Þ

ð36Þ

71

P ik;tjt ¼ ðI  Ak U H t ÞP k;tjt  1

ð37Þ

73

Ak ¼ P k;tjt  1 H Tt ðH t P k;tjt  1 H Tt þ Rt Þ  1

ð38Þ

75

The normalized coefficient c is given by

77

c ¼ ∑yi;t A yt 

79

pD;t pk;tjt  1 Nðyi;t ; H t mk;tjt  1 ; H t P k;tjt  1 H t T þ Rt Þ

λ

Kp T c;t þpD;t ∑e ¼ 1 pe;tjt  1 Nðy i;t ; H t me;tjt  1 ; H t P e;tjt  1 H t

þ Rt Þ

ð39Þ The existence probability of the combined distribution f k;tjt ðxk;t jy1:t Þ is given by

81 83 85

M

pk;tjt ¼ 1  ∏

i¼1

 1

!

pD;t pk;tjt  1 Nðyi;t ; H t mk;tjt  1 ; H t P k;tjt  1 H t T þRt Þ

λc;t þ pD;t ∑Ke p¼ 1 pe;tjt  1 Nðyi;t ; H t me;tjt  1 ; H t P e;tjt  1 H t T þ Rt Þ

87 89

ð40Þ By considering the missed detection, we obtain the combined distribution from (15) as f k ðxk;t jy1:t Þ ¼

pk;tjt f k;tjt ðxk;t jy1:t Þ þ ð1 pD;t Þpk;tjt  1 Nðxk;tjt  1 ; mk;tjt  1 ; P k;tjt  1 Þ ; pk;tjt þ ð1  pD;t Þpk;tjt  1

k ¼ 1⋯K p

91 93 95

ð41Þ

97

The existence probability of the combined distribution f k ðxk;t jy1:t Þ is given by

99

pk;t ¼ 1  ð1 pk;tjt Þð1 ð1  pD;t Þpk;tjt  1 Þ;

k ¼ 1⋯K p

101

The combined distributions f k ðxk;t jy1:t Þ, k ¼ 1⋯K p are the filtering distributions of the filter at time t, and the existence probabilities of the filtering distributions are pk;t , k ¼ 1⋯K p . An implementation of the proposed filter for linear and Gaussian models comprises, in addition to above prediction step and update step, the following filtering distribution approximation step and multi-target state extraction step. Filtering distribution approximation step: As clearly shown in (35) and (41), the filtering distribution f k ðxk;t jy1:t Þ is the weighted sum of the M þ1 Gaussian

pk;tjt  1 ;

65

ð35Þ

for k 4 K þ K γ . The existence prob-

45

49

1 f k;tjt ðxk;t jy1:t Þ ¼ ∑yi;t A yt c pD;t pk;tjt  1 Nðyi;t ; H t mk;tjt  1 ; H t P k;tjt  1 H t T þ Rt ÞNðxk;t ; mik;tjt ; P ik;tjt Þ  λc;t þ pD;t ∑Ke p¼ 1 pe;tjt  1 Nðyi;t ; H t me;tjt  1 ; H t P e;tjt  1 H t T þRt Þ

ð42Þ

abilities of the extended prediction distributions Nðxk;tjt  1 ; mk;tjt  1 ; P k;tjt  1 Þ, k ¼ 1⋯K p are given by

k  K  Kγ

63

follows:

and

43

pβ;t

5

distributions composed of Nðxk;t ; mik;tjt ; P ik;tjt Þ, i ¼ 1⋯M and Nðxk;tjt  1 ; mk;tjt  1 ; P k;tjt  1 Þ. To reduce the computational load of the filter, we use the Gaussian distribution Nðxk;t ; mk;t ; P k;t Þ to approximate the filtering distribution f k ðxk;t jy1:t Þ in (41) Nðxk;t ; mk;t ; P k;t Þ  f k ðxk;t jy1:t Þ or

103 105 107 109 111 113 115 117 119 121

ð43Þ

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123

L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

6

1 3 5 7 9 11 13 15 17 19 21 23

Nðxk;t ; mk;t ; P k;t Þ  pk;tjt f k;tjt ðxk;t jy1:t Þ þð1 pD;t Þpk;tjt  1 Nðxk;tjt  1 ; mk;tjt  1 ; P k;tjt  1 Þ  pk;tjt þð1 pD;t Þpk;tjt  1

ð44Þ This strategy finds the Gaussian distribution with the maximal existence probability from the M þ 1 Gaussian distributions, eliminates Gaussian distributions that are far from the distribution with the maximal existence probability, and then merges the remaining Gaussian distributions into a Gaussian distribution. The merged Gaussian distribution is used as the approximation of the filtering distribution. Given that the filtering distribution f k ðxk;t jy1:t Þ is formed by the Gaussian distributions Nðxk;t ; mik;tjt ; P ik;tjt Þ, i ¼ 1⋯M

and Nðxk;tjt  1 ; mk;tjt  1 ; P k;tjt  1 Þ, we

denote the M þ 1 distributions by using Nðxk;t ; mik;tjt ; P ik;tjt Þ,

i ¼ 1⋯M þ1,

where

þ1 mM ¼ mk;tjt  1 k;tjt

þ1 PM k;tjt ¼ P k;tjt  1 . The existence probabilities of the M þ1 distributions are given by

pik;tjt ¼

8 > <

pD;t pk;tjt  1 Nðyi;t ;H t mk;tjt  1 ;H t P k;tjt  1 H Tt þ Rt Þ Kp T c;t þ pD;t ∑e ¼ 1 pe;tjt  1 Nðy i;t ;H t me;tjt  1 ;H t P e;tjt  1 H t

λ

> : ð1  pD;t Þpk;tjt  1 ;

29

þ Rt Þ

i ¼ 1⋯M

ð45Þ We then find the index of the Gaussian distribution with the maximal existence probability from the M þ 1 distributions j ¼ arg max pik;tjt

31

;

i ¼ M þ1

25 27

and

ð46Þ

i A 1⋯M þ 1

The remaining Gaussian distributions and their existence probabilities after pruning are used as the inputs of the next filtering recursion. We then select the Gaussian distributions with existence probabilities pk;t 4 0:5 as the outputs of the filter. The mean of a selected Gaussian distribution is the state estimate of a target.

63 65 67 69 71

5. Simulation results We consider a scenario in which targets move at a constant velocity in a 2D region [ d (m), d (m)]  [  d (m), d (m)]h in simulations. TheiT state vector is represented by xk;t ¼ xk;t x_ k;t yk;t y_ k;t , and the state transition matrix F t and covariance matrix Q t  1 in (23) are given by 2 3 2 3 1 T 0 0 T 2 =2 0 6 7 60 1 0 07 6 T 0 7 6 7 7σ 2 Ft ¼ 6 7; Q t  1 ¼ 6 v 6 0 40 0 1 T 5 T 2 =2 7 4 5 0 0 0 1 0 T where T ¼1 s and σ v is the standard deviation of the process noise. This measurement is a noisy version of the target position. The observation matrix H t and covariance matrix Rt in (24) are     1 0 0 0 1 0 2 σ Ht ¼ ; Rt ¼ 0 0 1 0 0 1 w

where σ w is the standard deviation of the observation noise. dij ¼ ðmik;tjt  mjk;tjt ÞT ðP jk;tjt Þ  1 ðmik;t  mjk;tjt Þ; i ¼ 1⋯M þ 1; ia j To assess the performance of the proposed filter, we select the GM-PHD filter as the comparison object. The ð47Þ PHD filter is a recently proposed filter and has attracted wide attention from scholars and researchers. This filter We eliminate the Gaussian distributions with dij ZU propagates the first-order moment of multi-target posterfrom the M þ 1 distributions where U is a given threshior density in the filtering recursion [2,4]. The GM-PHD old, and then merge the remaining distributions into filter is an implementation of the PHD filter for linear and the Gaussian distribution Nðxk;t ; mk;t ; P k;t Þ. The merging Gaussian models [15]. For comparison, we use the OSPA method for multiple distributions is as follows: distance, which was recently developed by Ristic et al. 1 i i mk;t ¼ ∑ p m ð48Þ [33], as the measure. For two arbitrary finite subsets i A L k;tjt k;tjt ∑i A L pik;tjt X ¼ fx1 ⋯xm g and Y ¼ fy1 ⋯yn g, if m rn, the OSPA distance is defined as h i 1 P k;t ¼ ∑ pi P i þðmik;tjt  mk;t Þðmik;tjt  mk;t ÞT !!1=p ∑i A L pik;tjt i A L k;tjt k;tjt m 1 p p min ðX; YÞ ¼ d ðx ; y Þ þc ðn  mÞ d ∑ p;c c i π ðiÞ ð49Þ n π A ∏n i ¼ 1

73 75 77 79 81 83 85 87 89 91 93

We define the merging distance dij as

33 35 37 39 41 43 45 47 49 51

where L denotes the index set of the distributions to be merged into one. The existence probability of the merged Gaussian distribution is given by pk;t ¼ 1  ∏i A L ð1 pik;tjt Þ;

k ¼ 1⋯K p :

ð50Þ

53 55 57 59 61

Multi-target state extraction step: The filtering distributions after the filtering distribution approximation step are Nðxk;t ; mk;t ; P k;t Þ, k ¼ 1⋯K p , and the existence probabilities of individual filtering distributions are pk;t , k ¼ 1⋯K p . The Gaussian distributions whose existence probabilities are sufficiently small, that is, pk;t o τ, are eliminated where τ denotes a given threshold.

where dc ðx; yÞ ¼ minðc; dðx; yÞÞ is the cutoff distance, ∏n represents the set f1⋯ng, and 1 r p o1 is the OSPA metric parameter. If m 4 n, the definition is dp;c ðX; YÞ ¼ dp;c ðY; XÞ. In our simulations, the parameters p and c are set to p ¼ 2 and c ¼ 10 m. Example 1. This example aims to compare the tracking performances of the proposed filter (MDB filter) and GMPHD filter for the targets that move closely or cross each other's path. In this example, the distance d used to construct the 2D region is set to d ¼200. The clutter density, the survival

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95 97 99 101 103 105 107 109 111 113 115 117 119 121 123

L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

1 3 5 7 9 11 13 15

probability, and the detected probability are set to λc;t ¼ 0 m  2 (no clutter), pS;t ¼ 1:0, and pD;t ¼ 1:0, respectively. Two targets exist in Example 1. These targets appear at time t¼1 s and disappear at t¼20 s. Their initial states are as follows: x1;1 ¼ ½  200 ðmÞ; 20 ðms

19

where

25

Þ

The true trajectories of the two targets are shown in Fig. 1. Example 1 includes three cases: Case 1, Case 2, and Case 3. To deal with the simulation data in Example 1, we set the corresponding parameters of the MDB and GM-PHD filters to λc;t ¼ 0 m  2 , pS;t ¼ 1:0, pD;t ¼ 1:0, σ v ¼ 1 ms  2 , σ w ¼ 3 m, τ ¼ 10  3 , and U ¼ 4. The birth state distributions at time t are given by Nðxk;t ; mkγ ;t ; P kγ ;t Þ;

23

Þ; 10 ðmÞ;  1 ðms

1 T

x2;1 ¼ ½ 200 ðmÞ; 20 ðms  1 Þ; 10 ðmÞ; 1 ðms  1 ÞT

17

21

1

k ¼ 1⋯2

7

Case 1. In Case 1, the simulation observations are generated by using σ v ¼ 0 ms  2 and σ w ¼ 0 m. Fig. 2 shows the simulation data for Case 1. We deal with the simulation data by using the MDB and GM-PHD filters. Fig. 3 shows the result given by the MDB filter, Fig. 4 shows the result of the GM-PHD filter, and Fig. 5 shows the OSPA distances of the two filters. Figs. 3 and 4 show that the MDB filter provides two different state estimates at any time, whereas the GM-PHD filter yields two same state estimates from t¼10 s to t¼14 s. These results indicate that the GM-PHD filter cannot distinguish multiple distinct targets when they are closely spaced, and that the MDB filter has a higher resolution than the GM-PHD filter. According to the OSPA distance shown in Fig. 5, the MDB filter also provides more accurate multi-target state estimates than the GM-PHD filter.

m2γ ;t ¼ ½ 200 ðmÞ; 0 ðms  1 Þ;  10 ðmÞ; 0 ðms  1 ÞT   and P kγ ;t ¼ ðdiagð 10 20 10 20 ÞÞ2 , k ¼ 1⋯2. The existence probabilities of the two birth distributions are set to pk;t ¼ 0:05;

k ¼ 1⋯2

Case 2. In Case 2, the simulation observations are generated by using σ v ¼ 0 ms  2 and σ w ¼ 0 m for Target 1 and σ w ¼ 1 m for Target 2. The simulation data for Case 2 are shown in Fig. 6. We handle the simulation data by using the MDB filter and GM-PHD filter, respectively. The results of the MDB

27

y/m

33

y/m

Initial position Target tracjectory

Target 1

71 73 75 77 79

-150

-100

-50

0

50

100

150

200

x/m

-100

-50

0

50

100

150

97

200

Fig. 4. Result of the GM-PHD filter.

x/m

37

87

95 -200

-150

85

93

-10

Target 2

-200

83

0

-10

35

69

91

-5

0 -5

True trajectory Position estimate

5

10 5

67

89

10

31

65

81

m1γ ;t ¼ ½  200 ðmÞ; 0 ðms  1 Þ; 10 ðmÞ; 0 ðms  1 ÞT

29

63

99

Fig. 1. True target trajectories.

101

39 4

OSPA Distance/m

41 10

43 y/m

45

5 0 -5

-200

-150

-100

-50

0

50

100

150

105

1

107 2

4

6

8

10

12

14

16

18

20

t/s

200

x/m

49

103

2

0

-10

47

MDB filter GM-PHD filter

3

109 111

Fig. 5. OSPA distances of the two filters.

Fig. 2. Simulation data for Case 1.

113

51 10

53

57 59 61

0

-5

-10

-10

-150

-100

-50

0

50

x/m

Fig. 3. Result of the MDB filter.

100

150

200

117

0

-5

-200

115

5

y/m

y/m

55

10

True trajectory Position estimate

5

119 -200

-150

-100

-50

0

50

100

150

200

x/m

Fig. 6. Simulation data for Case 2.

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121 123

L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

8

1 3 5

and GM-PHD filters, as well as the OSPA distances of the two filters, are shown in Figs. 7, 8 and 9, respectively. From Figs. 7 to 9, a similar conclusion to that of Case 1 can be drawn. The MDB filter has a higher resolution and provides more accurate multi-target state estimates than the GM-PHD filter.

7 9 11 13 15 17

Case 3. In this case, the simulation observations are generated by using σ v ¼ 0 ms  2 and σ w ¼ 1 m for both targets. We use the two filters for 100 Monte Carlo runs. The average OSPA distances of the two filters are shown in Fig. 10. Fig. 10 shows that the average OSPA distances of the MDB and GM-PHD filters are consistent when the targets are well separated. Moreover, the average OSPA distance given by the GM-PHD filter is evidently larger than that given by the MDB filter as the target separation becomes

19 10

21 y/m

23

True trajectory Position estimate

5 0 -5

25

-10 -200

-150

-100

-50

27

50

100

150

200

x/m

True trajectory Position estimate

y/m

5

-5

-150

-50

0

50

100

150

200

4

47

MDB filter GM-PHD filter

3 2

59 61

Þ; 200 ðmÞ; 25 ðms

71 73 75 77 79 81 83 85 87

93

Þ

95 97 99

1 T

Þ

The true trajectories of the 10 targets are shown in Fig. 11. The measurements are generated with pS;t ¼ 1:0, pD;t ¼ 0:98, σ v ¼ 0 ms  2 , and σ w ¼ 2 m. Fig. 12 shows the simulation data for the experiment.

103 105 107 109

1000

0

2

4

6

8

10

12

14

16

18

T2

20

800

t/s

Initial position Target trajectory

T4

111

600 400

MDB filter GM-PHD filter

2.5

113 T6

0

2

115

T10

-200

T7

-400

1.5

-600

1

-800

0.5

T9

200

3

y/m

Average OSPA Distance/m

69

101

51

57

1

67

x10;15 ¼ ½  450 ðmÞ; 40 ðms  1 Þ;  200 ðmÞ; 40 ðms  1 ÞT

Fig. 9. OSPA distances of the two filters.

55

Þ

1

49

53

Þ; 860 ðmÞ; 35 ðms

1 T

x8;12 ¼ ½ 200 ðmÞ; 30 ðms  1 Þ; 900 ðmÞ; 20 ðms  1 ÞT x9;14 ¼ ½ 900 ðmÞ; 25 ðms

Fig. 8. Result of the GM-PHD filter.

OSPA Distance/m

45

-100

x/m

39

43

Þ;  860 ðmÞ; 35 ðms

65

91

x7;10 ¼ ½ 450 ðmÞ; 35 ðms  1 Þ; 200 ðmÞ; 0 ðms  1 ÞT -200

41

x4;3 ¼ ½  900 ðmÞ; 35 ðms

1

1 T

x6;8 ¼ ½  900 ðmÞ; 30 ðms  1 Þ; 200 ðmÞ; 10 ðms  1 ÞT

-10

37

x3;3 ¼ ½  900 ðmÞ; 35 ðms

1

63

89

x5;5 ¼ ½  200 ðmÞ; 25 ðms  1 Þ;  900 ðmÞ; 30 ðms  1 ÞT

0

35

In this example, the distance d is set to d ¼1000 and the clutter density is set to λc;t ¼ 5  10  6 m  2 . Ten targets exist in Example 2. Targets 1, 2, 3, 4, 5, 6, 7, and 8 appear at t ¼1 s, t¼1 s, t¼3 s, t¼3 s, t ¼5 s, t ¼8 s, t¼10 s and t ¼12 s, respectively, and continue to exist for the remaining times. Targets 9 and 10 appear at t¼14 s and t¼15 s, respectively, and disappear at t ¼40 s and t ¼30 s, respectively. The initial states of the individual targets are as follows:

x2;1 ¼ ½  900 ðmÞ; 35 ðms  1 Þ; 900 ðmÞ; 35 ðms  1 ÞT

10

31

Example 2. This example aims to compare the performances of the MDB and GM-PHD filters in dealing with an unknown and time varying number of targets in a cluttered environment.

x1; 1 ¼ ½  900 ðmÞ; 35 ðms  1 Þ;  900 ðmÞ; 35 ðms  1 ÞT

Fig. 7. Result of the MDB filter.

29

33

0

smaller. This tendency may be due to the fact that the GMPHD filter propagates the first-order moment of multitarget posterior density, and as the target distance become smaller, the interference between distinct targets becomes stronger, while the MDB filter propagates the marginal distributions for each target and enhances their independence. Thus, although the individual target states also interfere with one another as the target separation becomes smaller, this interference is effectively reduced in the MDB filter compared with the GM-PHD filter. The MDB filter has a higher resolution of distinct targets than the GM-PHD filter.

117 T3

2

4

6

8

10

12

14

16

t/s

Fig. 10. Average OSPA distances of the two filters.

18

20

-1000 -1000

T5 T1

-800

119

T8

-600

-400

-200

0

200

400

600

800

1000

121

x/m

Fig. 11. True target trajectories.

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123

L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

1 3

The corresponding parameters of the two filters are set to λc;t ¼ 5  10  6 m  2 , pS;t ¼ 1:0, pD;t ¼ 0:98, σ v ¼ 1 ms  2 , σ w ¼ 2 m, τ ¼ 10  3 , and U ¼ 4. The birth state distributions at time t are given by

where

Nðxk;t ; mkγ ;t ; P kγ ;t Þ;

m3γ ;t

5 7

k ¼ 1⋯5

800

400 200

y/m

17

0 -200 -400

19

-600

21

-800 -1000 -1000

23

-800

-600

-400

-200

0

200

400

600

800

1000

x/m

25

Fig. 12. Simulation data for the experiment.

27 29 5

31

MDB filter GM-PHD filter

35 37 39

Average OSPA Distance/m

4.5

33

4

3.5

3

2.5

41

2

5

10

15

43

47

20

25

30

35

40

45

50

t/s

Fig. 13. Average OSPA distances of the two filters.

45

m2γ ;t

¼ ½ 900 ðmÞ; 0 ðms

1

¼ ½ 200 ðmÞ; 0 ðms

1

pk;t ¼ 0:1;

600

15

m1γ ;t ¼ ½  900 ðmÞ; 0 ðms  1 Þ;  900 ðmÞ; 0 ðms Þ; 900 ðmÞ; 0 ðms

Þ

65

1 T

Þ;  900 ðmÞ; 0 ðms

Þ

1 T

Þ

m5γ ;t ¼ ½ 450 ðmÞ; 0 ðms  1 Þ;  200 ðmÞ; 0 ðms  1 ÞT   2 and P kγ ;t ¼ diag 50 25 50 25 , k ¼ 1⋯5. The existence probabilities of the five birth distributions are set to

1000

13

63 1 T

m4γ ;t ¼ ½ 900 ðmÞ; 0 ðms  1 Þ; 200 ðmÞ; 0 ðms  1 ÞT

9 11

9

k ¼ 1⋯5

67 69 71 73 75

We use the MDB and GM-PHD filters for 100 Monte Carlo runs and obtain the average OSPA distances of these filters. The results are shown in Fig. 13. Fig. 13 shows that the MDB filter offers a more reliable and more accurate estimate of the target position than the GM-PHD filter in the presence of association uncertainty, detection uncertainty, and clutter, and that the average OSPA distance of the MDB filter is smaller than that of the GM-PHD filter. The main reason for this may be that the GM-PHD filter propagates the first-order moment of multi-target posterior density, whereas the MDB filter propagates marginal density distributions of individual targets. Thus, the target state of the GM-PHD filter is more easily affected by clutter and other target states than the target state of the MDB filter. Table 1 shows the average OSPA distances of the two filters for different clutter rates in Example 2 and also shows that the average OSPA distance increases as clutter rate increases and that the average OSPA distance of the MDB filter is consistently smaller than that of the GM-PHD filter. Computational complexity: A multi-target tracking problem usually suffers from the curse of dimensionality. As the number of targets increases, the size of the joint state-space increases exponentially [1]. The MDB filter effectively combats the curse of dimensionality by recursively updating the marginal filtering distributions for each target. Similar to the PHD filter, the computational complexity of the MDB filter is OðKMÞ where K is the number of targets in the scene, and M is the number of observations in the current measurement set. Table 2

Table 1 Average OSPA distances (m) of the two filters for different clutter rates.

77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109

49

λc;t

1.25  10  6 m  2

2.5  10  6 m  2

3.75  10  6 m  2

5  10  6 m  2

111

51

MDB filter GM-PHD filter

2.6136 2.6506

2.6783 2.7397

2.7071 2.7828

2.7223 2.8125

113 115

53 55 57

117

Table 2 Average performing time (s) of a Monte Carlo run for different clutter rates.

119

λc;t

1.25  10  6 m  2

2.5  10  6 m  2

3.75  10  6 m  2

5  10  6 m  2

MDB filter GM-PHD filter

0.4795 0.4256

0.5852 0.5025

0.6897 0.5611

0.7875 0.6789

121

59 61

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123

10

1

L. Zong-xiang, X. Wei-xin / Signal Processing ] (]]]]) ]]]–]]]

shows the average performing time of a Monte Carlo run for different clutter rates in Example 2. As shown in Table 2, the performing time of the MDB filter is slightly larger than that of the GM-PHD filter because the MDB filter jointly propagates the marginal distributions for each of targets and the existence probabilities of individual distributions. This propagation requires a number of computations for the MDB filter to maintain the existence probabilities in the filtering recursion. It is also clear from Table 2 that the average performing time increases as clutter rate increases.

3 5 7 9 11 13

6. Conclusions 15 In this paper, we propose a novel multi-target Bayesian filter. Instead of maintaining the joint posterior distribution of the multi-target state in the filtering recursion, the proposed filter jointly propagates the marginal distributions for each target and their existence probabilities. By modeling the uncertainties of individual target states with the use of individual state distributions and by characterizing the randomness of target appearance and disappearance with the use of the existence probability of the state distribution, the proposed filter can deal with an unknown and variable number of targets in the presence of association uncertainty, detection uncertainty, and clutter. Moreover, the proposed filter enhances the independences of distinct targets, thereby reducing the effect of a target state on other target states when the separation between individual targets becomes small. To evaluate the performance of the novel filter, an implementation of the filter for linear and Gaussian models is developed. Based on the simulation data, the proposed filter and GM-PHD filter are tested according to the OSPA distance. The experimental results show that the proposed filter is more capable of distinguishing distinct targets when targets have a small separation as well as tracks multiple targets better than the GM-PHD filter in the presence of clutter and a variable number of targets.

17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53

Acknowledgment This work was supported by the National Natural Science Foundation of China (61271107 and 61301074), Research Fund for the Doctoral Program of Higher Education of China (20104408120001), Guangdong Natural Q3 Science Foundation (S2012010009417), and the Key Project of National Science & Technology of Pillar Program (2011BAH24B12). Q2

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