Posterior Cramer–Rao lower bounds for complicated multi-target tracking with labeled FISST based filters

Signal Processing 127 (2016) 156–167

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Posterior Cramer–Rao lower bounds for complicated multi-target tracking with labeled FISST based filters Yunxiang Li n, Huaitie Xiao, Hao Wu, Rui Hu, Qiang Fu Science and Technology on Automatic Target Recognition Laboratory, National University of Defense Technology, Changsha 410073, China

a r t i c l e in f o

abstract

Article history: Received 28 May 2015 Received in revised form 11 January 2016 Accepted 1 March 2016 Available online 9 March 2016

As science develops, multi-target tracking technique advances towards dealing with complicated scenes, in which target number is time-varying and unknown, detection, measurement source and data association are uncertain. Among the new tracking methods, Mahler's Finite Set Statistics (FISST) based multi-target tracking technology naturally suits such complicated scenes. A performance metric with rigid theoretical explanation and clear physical connotation is the cornerstone for further improving multitarget tracking methods on precision and stability. The Posterior Cramer–Rao lower bounds (PCRLB) is widely used for assessing tracking performance. While, for the complicated multi-target tracking mentioned above, existing PCRLBs do not work well. Therefore, we derived multi-target PCRLB (MT-PCRLB) under random finite set frame as well as its iterative expression through analyzing lower bound of the FISST based filters’ performance in complicated multi-target tracking. The derived lower bound is compatible with current labeled FISST based filters. Based on multi-target tracks obtained by labeled FISST based filters and association relations between tracks and measurement set, recursive calculation of MT-PCRLB is realized. Simulation results demonstrate that the proposed methods behave in a manner consistent with our expectations. & 2016 Elsevier B.V. All rights reserved.

Keywords: Posterior Cramer–Rao lower bounds Complicated multi-target tracking Random finite set Probability hypothesis density filter Data association

1. Introduction A rule for the evaluation of target tracking algorithm's performance is vital in tracking system design, parameter tuning and performance comparison. A multiple target tracking performance metric with rigid theoretical explanation and clear physical connotation is the cornerstone for further improving multiple target tracking precision and stability. Generally speaking, current performance evaluation rules for multiple target tracking algorithm are classified into two sorts, miss-distance rule and Cramer– Rao lower bound (CRLB) rule.

n Corresponding author. Tel.: þ 86 731 8457 6401; fax: þ86 731 8451 6060. E-mail address: [email protected] (Y. Li).

http://dx.doi.org/10.1016/j.sigpro.2016.03.003 0165-1684/& 2016 Elsevier B.V. All rights reserved.

Miss-distance rule plays an important role in target tracking system. The concept of miss-distance is the foundation of optimal filtering which is based on the least square, expectation and root mean square (RMS) error. While, as to track targets of unknown and time-varying number, missdistance rule should describe the difference between two random vector sets, which makes it more difficult to define miss-distance rule. As to the case when estimated target number is equal to actual target number, Drummond has defined a miss-distance rule based on optimal allocation [1]. Referring to Drummond's subjective performance evaluation rule and concept of Wasserstein in statistics, Hoffman [2] defines a Lp type miss-distance measure for Finite set statistics (FISST) theory based multiple target tracking algorithms, abbreviated as Wasserstein-distance (WD) rule, which is of some theoretical foundation. However, this distance rule is of no reasonable physical connotation[3]. So, referring to Wasserstein concept and optimal allocation rule, Schuhmacher [3]

Y. Li et al. / Signal Processing 127 (2016) 156–167

developed the Optimal Sub-pattern Assignment (OSPA) distance, which is of clear physical connotation and overcomes the disadvantage of WD rule. Ristic [4] proposed a OSPA based performance metric which could jointly measure error in tracking label and multiple target detection and tracking error. The OSPA distance has been publicly viewed as the most suitable and most popular performance evaluation metric for FISST based multiple target tracking algorithms, for its clear physical connotation, simple calculation as well as its capability of jointly measuring multiple target state evaluation error and target number evaluation error [4]. As a fundamental tool for evaluation of deterministic system's performance, CRLB describes the theoretical lower bound of system performance. Similar to CRLB, posterior CRLB (PCRLB) describes the theoretical lower bound of system performance in which measurement and state are both random variables [5]. Applied in target tracking, PCRLB describes the highest precision which tracking algorithm would acquire in target state estimation, and plays an important role in evaluation and prediction of tracking algorithm's performance, design of tracking system and sensor management. PCRLB has been widely applied in evaluating system performance with various targets and various sensors, such as space target detection with spacebased sensors [6], track of unresolved targets with multiple input multiple output (MIMO) radar [7], identification of missile head and decoy with radar [8], detection of ground target's motion with airborne multipath exploitation radar [9], target tracking with wireless sensor network composed of randomly distributed range-only sensors [10], joint evaluation of ground extended target's motion state and morphology features with high range resolution ground moving target indicator (HRRGMTI) [11], maneuvering target motion analysis (TMA) with bearings-only measurement [12], target track and sensor management with bistatic radar [13]. To avoid calculating inverse of large matrix in computation of CRLB, Tichavshy [5] proposed the time iterative expression of CRLB for single target, which greatly simplifies the computation of CRLB. Based on that, Hue [14] generalizes it to multiple target scenes, which make it possible to evaluate tracking performance of multiple target tracking algorithms with CRLB. However, as it requires association of measurement with track and multiple numerical integration, computational complexity increase and performance decline would be significant when target number becomes relatively large. As to tracking targets whose number is fixed and known, Ristic [15] proposes an ultimate CRLB which is of simple expression and needs no multiple numerical integration. However, as it is based on unthresholded data, ultimate CRLB does not suite measurement based multiple target tracking algorithm. Besides, Tong [16] derives an iterative expression for calculation of error bounds in evaluating target tracking algorithm's performance under FISST frame. However, as such error bounds only suit multiple target tracking problems with zero false alarm rate, its actual application is apparently limited. The current trend in multiple target tracking is to deal with complicated scenes, in which target number is unknown and time-varying, detection, measurement source and data association are all uncertain. To be specially mentioned, Mahler's Finite Set Statistics (FISST) theory based multiple

157

target tracking technology naturally suits such complicated scene [17]. From above analysis, it is clear that many limits exist in miss-distance rule and PCRLB rule. Thus, we are to propose a PCRLB which is compatible with above complicated multiple target tracking. In [14], it is pointed out that association of measurement with track is the outstanding obstacle for calculation of PCRLB in multiple target tracking. Though with FISST based multiple target tracking algorithms multiple target states could be estimated without association of measurement with track, target track extraction and calculation of corresponding PCRLB could not be realized, which brings much difficulty for later process such as recognition of target motion, target identification and final state estimation. Current track extraction methods are generally classified into two sorts, one is to combine FISST based filters with traditional association algorithm and produce target track with data association during or posterior filtering [18], which significantly increases computational complexity for association process. The other sort is to add track label to target state and form target track while estimating its state [19], which is of less computational complexity and has been highly recommended by Mahler [20]. The second method has become mainstream for algorithm research and on such approach many achievements have been made, such as labeled particle PHD (L-P-PHD) filter [19], labeled Gaussian mixture PHD (L-GM-PHD) [21], labeled particle PHD smoothing [22], δ-generalized labeled multitarget multi-Bernoulli filter [23,24] as well as its approximation: labeled multi-Bernoulli filter [25]. These methods are all classified as the labeled FISST based filters. Besides, as to disadvantages of L-P-PHD filter, a data-driven multiple target state extraction algorithm is proposed in [26], with the datadriven adaptive target birth intensity PHD (ATBI-PHD) filter from [27]. The crossing entropy based PHD track extraction algorithm proposed in [28] significantly improves algorithm's performance when target tracks are cross or close. Contributions from this paper mainly include three points. Firstly, under the random set frame, this paper derives a multi-target PCRLB (MT-PCRLB) for complicated multiple target tracking problems, as well as its iterative calculation expression for lower bound of such algorithms. Secondly, the authors proposes the improved L-P-PHD (IL-P-PHD) filter by modifying labeled P-PHD filter in [29], which stably obtains target state and corresponding track label when target birth, target death, target track crossing or target approaching happens. Based on such obtained multi-target tracks, we propose a new data association method for precisely obtaining association relations between multiple target tracks and measurement set in this work. Thirdly, based on this new data association method, this paper derives the detailed expression of MTPCRLB for evaluating lower bounds of typical radar multiple target tracking problems. Besides, such lower bound is compatible with current labeled FISST based filters. Then, based on the association relations between multiple target tracks and measurement set, iteration of MT-PCRLB is accessible. Simulation experiment shows that the proposed MT-PCRLB quantitatively evaluates the lower bounds of multiple target tracking algorithm's performance in dealing with complicated multiple target tracking scenes. The paper is organized as follows. Section 2 is mainly about random set modeling for complicated multiple target

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tracking scenes, multiple target random set motion model and sensor measurement model are established in this section with detection, measurement source and data association all uncertain. In Section 3 is derived the MT-PCRLB for describing lower bounds in multiple target tracking as well as its iterative recursive calculation formula. In Section 4 is proposed a new data association method based on the IL-PPHD filter. In Section 5, the calculation realizing of MT-PCRLB is analyzed. Section 6 is mainly about simulation experiment and results analysis. The last part is the conclusion of this paper.

2. Random set modeling for complicated multiple target tracking Building target motion model and sensor measurement model is the prerequisite for multiple target tracking. The better models match actual scene, the better tracking algorithm's performance will be. According to sensor's resolution, target's spatial distribution relative to sensors as well as target's spatial distribution relative to each other, targets would appear in three forms, multiple point targets, multiple extended targets and multiple unresolved targets [30]. Sensor's measurement forms for such different targets are also different. For point target only one measurement will be produced, for extended target several measurements, for unresolved target only one measurement from several targets. Therefore, different measurement models are needed. For precise target tracking, the key is effective extraction of state information from measurement, which requires a good motion model. Current tracking algorithms could be classified into two kinds, single model tracking method and multiple-model tracking method [31]. In following part is to be proposed the measurement model and motion model for multiple target tracking. 2.1. Standard random set measurement model for multiple point targets The standard multiple target measurement model [30] means that one target produces no more than one measurement and one measurement is from no more than one target, and the models for extended target and unresolved target are excluded. In following part, the model for measurement process of multiple point targets is to be presented, and then a likelihood function between multiple target measurement set and state set. At time step k, N k targets exist within the scene, the
N state set composed of state vectors is X k ¼ xk;i i ¼k 1 , xk;i A Rnx the state vector for ith target. Measurement set
M obtained by sensor is Z k ¼ zk;j j ¼k 1 , zk;j A Rnz the jth measurement vector, Mk the number of measurements including target detection and false detection. As false detection is related to target state, multiple target measurement set at time step k is measurement set

Zk

¼

target detection set

γ ðX k Þ

[

false detection set

C ðX k Þ

ð1Þ

In [30], for the multiple point target measurement model in Eq. (1), Mahler has derived its corresponding likelihood function between multiple point target measurement set and state set with FISST theory, which is g ðZ k jX k Þ ¼ eλðX k Þ g C ðX k Þ ðZ k Þ Ug ð∅jX k Þ U     X pD xk;i Ug zk;θðiÞ jxk;i       ∏i:θðiÞ 4 0 1  pD xk;i UλðX k Þc zk;θðiÞ jX k θ

ð2Þ

g C ðX k Þ ðZ k Þ ¼ e  λðX k Þ ∏ λðX k ÞcðzjX k Þ

ð3Þ

Nk    g ð∅jX k Þ ¼ e  λðX k Þ ∏ 1 pD xk;i

ð4Þ

z A Zk

i¼1

λðX k Þ ¼ λ0 þλ1 þ⋯ þλn ¼ λ0 þ

Nk X   λ xk;i

ð5Þ

i¼1

cðzjX k Þ ¼

λ0 c0 ðzÞ þλ1 c1 ðzÞ þ ⋯ þ λn cn ðzÞ λ0 þ λ1 þ ⋯ þλn

ð6Þ

here, g ðzjxÞ is the single target measurement likelihood function, determined by sensor's measuring characteristic. g C ðX k Þ ðZ k Þ is the probability density function of false alarm process with mean λðX k Þ and spatial distribution cðzjX k Þ. θ is mapping from state set to measurement set, representing relations between measurement set and state set, which is

 ð7Þ θ: 1; …; N k - 0; 1; …; M k Illustrations for Eq. (7): (1) As missed detection of target might occur, it is represented by ‘0’, which means if the detection of target i is missed, θðiÞ ¼ 0. (2) Eq. (7) describes the relation between multiple target state set and measurement set, which means θðiÞ could be assigned only one value. (3) As false detections exist, θ is unnecessarily a surjection. (4) As association uncertainty exists, θ could be of several expressions, which have been considered in Eq. (2). Eq. (2) describes the multiple point target measurement likelihood function, including model for uncertain detection, uncertain measurement and uncertain data association, which makes equation suitable enough for such complicated multiple target tracking scene. 2.2. Motion model for multiple point targets Four events should be included in a complete multiple point target motion model [30]: 1) Target survival: Target xk  1 at time step k  1survives to target xk at time step k with probability pS ðxk  1 Þ, the motion model for this process is xk ¼ φk ðxk  1 ; vk  1 Þ, and the corresponding single target Markov state transition density is f ðxk jxk1 Þ, determined by target motion characteristic. 2) Target death: Target xk  1 might disappear at time step k with probability 1 pS ðxk  1 Þ. 3) Target spontaneous: Spontaneous targets set at time step k is Bk , which represents new targets appearing at time step k.

Y. Li et al. / Signal Processing 127 (2016) 156–167

4) Target spawned: At time step k new targets might be spawned from target at previous step, target set spawned from state set X k  1 is BðX k  1 Þ. Here, target spontaneous and target spawned are unified as target birth. Then, motion process for multiple point target could be modeled as Xk ¼

persisting targets

Γ ðX k  1 Þ

birth targets

[ ΒðX k  1 Þ

ð8Þ

Similarly, with FISST theory Mahler has derived the Markov motion density function of multiple point targets corresponding to Eq. (8), which is f ðX k jX k  1 Þ ¼ eμðX k  1 Þ f BðX k  1 Þ ðX k Þ U f ð∅jX k  1 Þ     X pS xk  1;i Uf xk;θ0 ðiÞ jxk  1;i       U ∏i:θ0 ðiÞ 4 0 θ0 1  pS xk  1;i UμðX k  1 Þb xk;θ0 ðiÞ jX k  1 ð9Þ f BðX k  1 Þ ðX k Þ ¼ e  μðX k1 Þ ∏ μðX k1 ÞbðxjX k  1 Þ x A Xk

f ð∅jX k  1 Þ ¼ e  μðX k  1 Þ ∏

x A Xk  1



1  pS ðxÞ



    μðX k  1 Þ ¼ μ0 þ μ1 xk  1;1 þ ⋯ þ μNk  1 xk  1;Nk  1

bðxjX k  1 Þ ¼

ð10Þ ð11Þ ð12Þ

159

multi-target motion model and multi-target measurement model for complicated multi-target tracking problem under random finite set frame.

3. PCRLB for complicated multiple target tracking problems From the analysis in Section 2, current multiple target tracking algorithms usually deal with complicated scenes in which target death and target birth occur, uncertain detection, uncertain measurement source and uncertain data association appear. Such target tracking problems are generally termed as complicated multiple target tracking problems. With the multiple target motion model and sensor measurement model from Section 2, performance evaluation for complicated multiple target tracking is to be further analyzed in this section, deriving its corresponding PCRLB. Different from traditional PCRLB which only suits tracking of single target or targets of fixed number, PCRLB proposed in this paper would suit complicated multiple target tracking and will be termed as multi-target PCRLB (MT-PCRLB). In first part of this section is to be derived the recursive expression of MT-PCRLB, and in latter part is to be analyzed the computation implementation of MTPCRLB.

        μ0 b0 ðxÞ þ μ1 xk  1;1 b1 xjxk  1;1 þ⋯ þ μNk  1 xk  1;Nk  1 bNk  1 xjxk  1;Nk  1     μ0 þ μ1 xk  1;1 þ ⋯ þ μNk  1 xk  1;Nk  1

ð13Þ

3.1. Recursive computation expression of MT-PCRLB in which, f BðX k  1 Þ ðX k Þ is the probability density function of birth target process with mean μðX k  1 Þ and spatial distribution bðxjX k  1 Þ. The θ0 is to describe the relations between state set at time step k and state set at time step k 1, representing change of multiple target tracks, which is

 θ0 : 1; …; N k  1 - 0; 1; …; Nk ð14Þ Illustrations for Eq. (14): (1) ‘0’ represents target death, which means θ0 ðiÞ ¼ 0 if target i dies. (2) As track is for single target motion, θ0 ðiÞ could only be of one value. (3) As target birth exists, θ0 is unnecessarily a surjection. (4) As motion process of multiple targets is quite complicated, θ0 could of several expressions, which have all been considered in Eq. (9). In Eq. (9) is described the multiple target motion function, including modeling of target survival, target birth and target death, which makes Eq. (9) suitable enough for such complicated multiple target motion scene. In this paper, we refer complicated multiple target tracking problem as the case in which target number is unknown and time-varying, detection is uncertain (detection probability less than 1), and data association is uncertain. Technically speaking, for each time step, the multi-target state set X k is a random finite set, which means the number of elements and motion states of elements in X k are unknown and time-varying, the multitarget measurement set Z k includes miss detection and false detection. The two models above are respectively

As to single target tracking, assuming target state as x A Rnx , its corresponding target measurement is z A Rnz . With tracking or filtering of z, the estimated target state x^ ðzÞ comes out. Then, the algorithm's estimate error satisfies [5] n    T o var x^ ðzÞ ¼ E x^ ðzÞ  x x^ ðzÞ  x ZJ  1 ð15Þ here, J  1 is the theoretical lower bounds of estimation x^ ðzÞ, J is a nx  nx Fisher information matrix, its calculation formula is
 ð16Þ J ¼ E Δxx log pðz; xÞ in which, pðz; xÞ is the joint probability density function of measurement and target state, operator Δyx is defined as Δyx ¼ ∇x ∇Ty ∇x ¼



ð17Þ

  T ∂=∂x1 ; ⋯; ∂=∂xnx

ð18Þ

As to multiple target tracking, the vector form of state
N
M set X k ¼ xk;i i ¼k 1 and measurement set Z k ¼ zk;i i ¼k 1 at time step k is h iT ðNk nx  1Þ ð19Þ X k ¼ xTk;i ; ⋯; xTk;Nk h iT Z k ¼ zTk;i ; ⋯; zTk;Mk

ðM k nz  1Þ

ð20Þ

160

Y. Li et al. / Signal Processing 127 (2016) 156–167

here, X k is the multiple target state vector at time step k, Z k the multiple target measurement vector at time step k. Based on this, cascading multiple target state vectors from initial step to step k and multiple target measurement vectors from initial step to step k separately, the high-dimensional vector could be obtained ! k h iT X X ðkÞ ¼ X T0 ; ⋯; X Tk N i nx  1 ð21Þ

n  o  Xk þ 1 D12 log f X k þ 1 X k k ¼ E ΔX k

ð32Þ

n  o h 12 iT  Xk  D21 ¼ Dk k ¼ E ΔX k þ 1 log f X k þ 1 X k

ð33Þ

( ) n  o     Xk þ 1 X k þ E  ΔX k þ 1 log g Z k þ 1 X k þ 1 D22 ¼ E  Δ log f X k þ 1 k Xk þ 1 Xk þ 1

ð34Þ

i¼0

h iT Z ðkÞ ¼ Z T1 ; ⋯; Z Tk

k X

D11 k

! M i nz  1

ð22Þ

k¼1

Joint probability density function for X ðkÞ and Z ðkÞ is k     k pðkÞ ¼ p X ðkÞ ; Z ðkÞ ¼ f ðX 0 Þ ∏ g Z j jX j ∏ f ðX i jX i  1 Þ j¼1

ð23Þ

i¼1

in which, f ðX 0 Þ is the prior probability density function of initial multiple target state vector. From conclusions in [5], segmenting vector X ¼ h iT T X α ; X Tβ to be estimated into two parts, the information matrix becomes " # J αα J αβ J¼ J βα J ββ

ð24Þ

Then, the error covariance matrix for estimation of X β is larger than the right-lower block of J  1 , that is h i1 1 Rβ Z J ββ J βα J αα J αβ ð25Þ h

Similarly, segmenting X ðkÞ into two parts, X ðkÞ ¼ iT . From Eq. (24), it comes out that

X Tðk  1Þ ; X Tk

" Ak   J X ðkÞ ¼ BTk

Bk Ck

#

o 2 n X E ΔX ððkk  11ÞÞ log pðkÞ 6 o ¼4 n X E ΔX kðk  1Þ log pðkÞ

n o3 E  ΔXX ðkk  1Þ log pðkÞ n o 7 5 E ΔXX kk log pðkÞ

ð26Þ So, with Eq. (25), information submatrix at time step k is J k ¼ J ðX k Þ ¼

here, J k and are N k nx  N k nx matrix, is Nk nx  N k þ 1 nx matrix, D21 is N k þ 1 nx  Nk nx matrix, D22 is k k N k þ 1 nx  Nk þ 1 nx matrix, J k þ 1 is Nk þ 1 nx  Nk þ 1 nx matrix. Especially, when Nk þ 1 aN k , Eq. (30) is still right in the view of calculation. Though similar recursive expressions of PCRLB have been derived in [5,,14], they only suit target tracking scene of single target and fixed number targets separately. Eq. (30) derived in this part suits scenes with targets of varying number, and performs excellently on evaluating optimal performance of complicated multiple target tracking problems. 3.2. Analysis on computation implementation of MT-PCRLB In Section 2 are proposed the Markov density function describing multiple target state set movement and measurement likelihood function describing the transition relations between multiple target state set and measurement set. To simplify calculation of Eq. (30), the motion density function and measurement likelihood function of set form are transformed to vector form, with concrete calculation formula unchanged, that is       f X k þ 1 jX k ¼ f X k þ 1 jX k ¼ eμðX k Þ f BðX k Þ X k þ 1 Uf ð∅jX k Þ U     X pS xk;i Uf xk þ 1;θ0 ðiÞ jxk;i     ∏i:θ0 ðiÞ 4 0  ð35Þ 1  pS xk;i UμðX k Þb xk þ 1;θ0 ðiÞ jX k 0 θ

        g Z k þ 1 jX k þ 1 ¼ g Z k þ 1 jX k þ 1 ¼ eλðX k þ 1 Þ g C ðX k þ 1 Þ Z k þ 1 U g ∅jX k þ 1 U

X θ

C k BTk Ak 1 Bk

ð27Þ

With Eq. (23), it comes out     pðk þ 1Þ ¼ pðkÞ g Z k þ 1 jX k þ 1 f X k þ 1 jX k

ð28Þ

With Eqs. (26) and (28), it comes out 2 3 Bk 0 Ak   6 T 11 12 7 Dk 5 J X ðk þ 1Þ ¼ 4 Bk C k þ Dk

ð29Þ

0

D21 k

D22 k

With Eq. (27), it could be achieved that " #  1" # h i Ak Bk 0 21 22 J k þ 1 ¼ Dk  0 Dk BTk C k þ D11 D12 k k h i1 21 11 T 1 D12 ¼ D22 k  Dk C k þ Dk Bk Ak Bk k h i1 21 11 ¼ D22 D12 k  Dk Dk þ J k k

ð30Þ

n  o  Xk  D11 k ¼ E ΔX k log f X k þ 1 X k

ð31Þ

D12 k

∏i:θðiÞ 4 0 

    pD xk þ 1;i U g zk þ 1;θðiÞ jxk þ 1;i       1  pD xk þ 1;i U λ X k þ 1 c zk þ 1;θðiÞ jX k þ 1

ð36Þ Theoretically speaking, after substitution of Eqs. (35) and 12 21 22 (36) into Eqs. (31)–(34) comes out D11 k , Dk , Dk and Dk . Substituting these matrice into Eq. (30), information matrix J k þ 1 at time step kþ1 could be recurred from information matrix J k at time step k. While, with analysis of Eqs. (35) and   (36), it is clear that in function f X k þ 1 jX k have been considered all possibilities for transition of multiple target state set at time step k into multiple target set at time step kþ 1, or all possible associations between estimated multiple target state   set and tracks, which means the function f X k þ 1 jX k itself is   an average of all possibilities. In function g Z k þ 1 jX k þ 1 have been considered all possibilities for transition of multiple target state set at time step kþ 1 to measurement set at time step kþ1, or all possible relations between multiple target mea  surement set and tracks. Similarly, the function g Z k þ 1 jX k þ 1 itself is also an average. Therefore, calculation of Eqs. (31)–(34) needs consideration of all possible associations analyzed above, which makes calculation of Eqs. (31)–(34) mathematically unpractical. Even it is practical, consideration of all possible

Y. Li et al. / Signal Processing 127 (2016) 156–167

associations is unnecessary, because at each time step only one possibility exists for associations between estimated multiple target states and tracks as well as association between multiple target measurement set and tracks. Some other means are needed to clarify such associations. Once such associations achieved, the weighted sum part in Eqs. (35) and (36) would be   deleted. Then, f X k þ 1 jX k becomes simple product of single   target state transition function, and g Z k þ 1 jX k þ 1 becomes simple product of single target measurement likelihood function. So, calculation of Eqs. (31)–(34) becomes practical. From above analysis, it is figured out that difficulties in calculation of Eqs. (31)–(34) and further Eq. (30) mainly lie in obtainment of associations between estimated multiple target state set and tracks as well as association between multiple target measurement set and tracks. Currently, such associations are obtained with traditional association algorithms such as nearest neighbor standard filter (NNSF), joint probabilistic data association filter (JPDAF) and multiple hypothesis tracking (MHT). However, because of low computational efficiency, algorithm's performance declines significantly as target number increases or filtering process lasts. Besides, the proposed MT-PCRLB is combined with the FISST based filtering method, whose prominent advantage is no need of traditional multiple target data association process. Therefore, in this research the labeled FISST based filtering method [19–24] is to be used for obtaining association relations between estimated multiple target state and track while estimating multiple target states, and further obtaining the association relations between multiple target measurement set and tracks. In Section 4, details on obtaining association relations is to be provided, taking the labeled PHD filter [19] as an example.

4. Improved labeled PHD filter for track extraction and data association From analysis above, obtainment of association relations between multiple target state set and tracks as well as association relations between multiple target measurement set and tracks is the prerequisite for computation of MTPCRLB in complicated multiple target tracking. Actually, estimation of association relations between multiple target state set and tracks is track extraction, association of multiple target measurement set and tracks is data association. Traditional data association based multiple target tracking algorithms tend to transfer multiple target tracking problem into several single target tracking problems, realizing estimation of target state by allocating measurement to its corresponding single target filter with data association algorithm, further realizing track extraction. Multiple target states could be estimated by FISST based multiple target tracking algorithms without data association, but not track extraction. The improved labeled particle PHD (IL-P-PHD) filter is proposed in [29] with adding label to the particle PHD filter, which performs track extraction well without data association. To improve performance of track extraction and data association, this section is to obtain track information

161

with the IL-P-PHD filter, propose a new method for data association, breaking traditional algorithm's procedure of track extraction following data association. At time step kþ 1, multi-target tracking is to conduct filtering of measurement set Z k þ 1 from state set X k þ 1 , obtaining estimated multi-target state set X^ k þ 1 as well as track of each single target. As to traditional multi-target tracking algorithms, after association process, Z k þ 1 and X k goes through filtering process with single target filter independent of each other, leading to X^ k þ 1 . While in this part, estimated state X^ k þ 1 is achieved with the IL-P-PHD   filter, and calculation of g Z k þ 1 jX k þ 1 becomes simpler for data association of Z k þ 1 and X^ k þ 1 . In fact, association of Z k þ 1 and X^ k þ 1 is apparently easier than association of Z k þ 1 and X k , also of higher precision and simpler algorithm. In following part is given the algorithm for association of Z k þ 1 and X^ k þ 1 . For each x^ k þ 1;i A X^ k þ 1 , its predicted measurement is   z^ k þ 1;i ¼ g x^ k þ 1;i ð37Þ here, g ð:Þ is the sensor measurement function. As to each zk þ 1;j A Z k þ 1 and each x^ k þ 1;i A X^ k þ 1 , their association degree is n   T    1 o C j;i ¼ exp  1=2 zk þ 1;j  z^ k þ 1;i Rk þ 1 zk þ 1;j  z^ k þ 1;i ð38Þ here, Rk þ 1 is the covariance of sensor measurement noise. Larger C j;i means higher association degree, and the asso  ciation degree matrix is C ¼ C j;i M N^ . k k Then, matrix C is to be processed. For measurement corresponding to each x^ k þ 1;i , find
 max C j;i ð39Þ C p;q ¼ 1 r j rM k ^k 1 r jr N here, C p;q means association degree between zk þ 1;p and x^ k þ 1;q is the highest and the two could be associated. Then, elements of pth row and qth column in matrix are all set to be 0, coming a new matrix C. Repeating Eq. (39), another ^k association of measurement and state comes out. With N iterations of such process, association between multitarget measurement set and estimated target state set is completed. The data association algorithm here is essentially different from data association algorithm applied in traditional data association based tracking algorithms (such as JPDAF). As to traditional tracking algorithms, multi-target state set is estimated after data association between measurement set and obtained tracks. When target number is large or varying, or target distance is too small relative to sensor's resolution, association would become unreliable and complicated, which further affects estimation of target number and target states. While, the data association algorithm proposed in this part is to be performed after estimation of target states. As the multitarget state estimation and target number estimation is completed with the labeled FISST based filters which need no data association and perform well in multi-target

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tracking, the data association algorithm is to be of higher precision and easier for computation realization. Remark 1. 1) As sensors are generally of high detection probability and false measurements from clutter usually exist, measurement number is generally large than target number, that is M Z N^ k . 2) With association process proposed above, associating several targets with one measurement and associating one target with several measurements could be avoided, and algorithm performs well when targets are close and tracks cross.

5. Computation implementation of MT-PCRLB With the results from track extraction and data association in Section 4, mapping θ and θ0 in Eqs. (35) and (36) become unique     f X k þ 1 jX k ¼ eμðX k Þ f BðX k Þ X k þ 1 U f ð∅jX k Þ U     pS xk;i Uf xk þ 1;θ0 ðiÞ jxk;i     ð40Þ ∏ 1  pS xk;i UμðX k Þb xk þ 1;θ0 ðiÞ jX k i 



    g C ðX k þ 1 Þ Z k þ 1 U g ∅jX k þ 1 U g Z k þ 1 jX k þ 1 ¼ e     pD xk þ 1;i U g zk þ 1;θðiÞ jxk þ 1;i       ∏ 1  pD xk þ 1;i Uλ X k þ 1 c zk þ 1;θðiÞ jX k þ 1 i λðX k þ 1 Þ

ð41Þ

Eqs. (40) and (41) are further simplified into 

N k þ 1;sur    Nk þ 1;spo   f X k þ 1 jX k ¼ αk þ 1 U ∏ f sur xk þ 1;i jxk;i ∏ f spo xk þ 1;j jX k U i¼1

N k þ 1;spa

∏ f spa

m¼1



j¼1

 Nk þ 1;dea   xk þ 1;m j∅ ∏ f dea ∅jxk;n

ð42Þ

n¼1

Mk þ 1;det     g Z k þ 1 jX k þ 1 ¼ βk þ 1 U ∏ g det zk þ 1;i jxk þ 1;i i¼1

M k þ 1;f _d

∏

j¼1

  g f _d zk þ 1;j jX k þ 1 U

 ∏ g f _i zk þ 1;m j∅

M k þ 1;f _i

 Mk þ 1;mis

m¼1

∏

n¼1

  g mis ∅jxk þ 1;n

  detection respectively, g mis ∅jxk þ 1;n target miss-detection probability. M k þ 1;det , Mk þ 1;f _d , M k þ 1;f _i and M k þ 1;mis represent target measurement number, state relative false detection number, state irrelative false detection number and miss-detection target number at time step k þ1 respectively. M k þ 1;det þM k þ 1;f _d þ Mk þ 1;f _i ¼ Mk þ 1

ð46Þ

M k þ 1;det þM k þ 1;mis ¼ N k þ 1

ð47Þ

In fact, as real value of target state is unknown, only estimated target state could be obtained with filtering.     Therefore, calculation of f X k þ 1 jX k and g Z k þ 1 jX k þ 1 is



to be approximated with f X^ k þ 1 jX^ k and g Z k þ 1 jX^ k þ 1 .

From analysis on track extraction in Section 4, f X^ k þ 1 jX^ k is of the same form as Eq. (42), that is

  f X k þ 1 jX k  f X^ k þ 1 jX^ k ¼ α0k þ 1 U

  Nk þ 1;spo ∏ f spo x^ k þ 1;j jX^ k U ∏ f sur x^ k þ 1;i jx^ k;i ^

^ N k þ 1;sur i¼1

j¼1

^ N k þ 1;spa

 N^ k þ 1;dea

 ∏ f spa x^ k þ 1;m j∅

m¼1

^ N kþ1   ¼ β0k þ 1 U ∏ g det zk þ 1;i jx^ k þ 1;i

ð44Þ

N k þ 1;sur þ Nk þ 1;dea ¼ Nk

ð45Þ

In Eq. (43), βk þ 1 is a constant, g det ð Uj U Þ represents the   measurement likelihood function, g f _d zk þ 1;j jX k þ 1 and   g f _i zk þ 1;m j∅ represent the probability density function of state relative false detection and state irrelative false

ð48Þ

ð49Þ

i¼1

The difference between Eqs. (49) and (43) shows that the IL-P-PHD filter is capable of limiting false alarm and miss-detection. Therefore,

  log f X k þ 1 jX k  log f X^ k þ 1 jX^ k ^ N kX þ 1;sur

  log f sur x^ k þ 1;i jx^ k;i

i¼1

ð43Þ

N k þ 1;sur þ Nk þ 1;spo þ Nk þ 1;spa ¼ N k þ 1

  f dea ∅jx^ k;n



As to g Z k þ 1 jX^ k þ 1 , when false alarm rate and missdetection rate are high, difference between X^ k þ 1 and X k þ 1 might exist. From analysis on data association in Section 4, expression of g Z k þ 1 jX^ k þ 1 is different from that of Eq. (43), that is

  g Z k þ 1 jX k þ 1  g Z k þ 1 jX^ k þ 1

¼ ck þ 1 þ

In Eq. (42), αk þ 1 is a constant, f sur ð Uj U Þ is the transition   density function of surviving target state, f spo xk þ 1;j jX k   and f spa xk þ 1;m j∅ representing the probability density function of target spontaneous and target spawn respec  tively, f dea ∅jxk;n representing the probability density function of target death. Nk þ 1;sur , Nk þ 1;spo , Nk þ 1;spa and N k þ 1;dea represent surviving target number, spawn target number, spontaneous target number and dead target number at time step k þ1 respectively, and satisfy

∏

n¼1

^ N k þ 1;spo

þ

X



log f spo x^ k þ 1;j jX^ k

j¼1 ^ N k þ 1;spa

þ

X

^

þ 1;dea   NkX   log f spa x^ k þ 1;m j∅ þ log f dea ∅jx^ k;n

m¼1

n¼1

ð50Þ

  log g Z k þ 1 jX k þ 1  log g Z k þ 1 jX^ k þ 1 ¼ dk þ 1 þ

^ N kþ1 X

  log g det zk þ 1;i jx^ k þ 1;i

ð51Þ

i¼1 12 21 22 As to calculation of D11 k , Dk , Dk and Dk respectively corresponding to Eqs. (31)–(34), derivation is in ascending order of target label, expectation Ef U g in Eqs. (31)–(33) corresponds to X k þ 1 and X k , Ef U g in Eq. (34) corresponds to Z k þ 1 , X k þ 1 and X k . In multi-target tracking problem, probability functions of spawn target, spontaneous target

Y. Li et al. / Signal Processing 127 (2016) 156–167

and target death are usually set as constant, commonly used as target spawn probability, spontaneous probability and detection probability, which contributes to calculation 12 21 of D11 and D22 k , Dk , Dk k . Based on Eqs. (50) and (51), 11 21 22 expressions for Dk , D12 k , Dk and Dk are derived as  o n ^ Xk ^ D11 k  E  ΔX k log f X k þ 1 X k  n  oi x^  ð52Þ ¼ E  Δx^ k;j log f X^ k þ 1 X^ k ^ ^ N k N k

k;i

in which  o n x^  E Δx^ k;j log f X^ k þ 1 X^ k 8k;i    < Ef  Δx^ k;i log f ^ ^ ; sur x k þ 1;i x k;i x^ k;i ¼ : 0n n ; x

x

i¼j others

 o n  Xk þ 1 D12 log f X^ k þ 1 X^ k k  E  ΔX k  n  oi x^  ¼ E  Δx^ k þ 1;j log f X^ k þ 1 X^ k ^

ð54Þ

N k N^ k þ 1

k;i

ð53Þ

in which n

o x^ E Δx^ k þ 1;j log f X^ k þ 1 jX^ k k;i

8 < Ef Δx^ k þ 1;i log f ðx^ k þ 1;i jx^ k;i Þg; i ¼ j and target i surviving sur x^ k;i ¼ : 0nx nx ; others

ð55Þ h iT 12 D21 k ¼ Dk

ð56Þ

 o n ^ Xk þ 1 ^ D22 k ¼ E  ΔX k þ 1 log f X k þ 1 X k  n

o  X þE ΔX kk þþ 11 log g Z k þ 1 X^ k þ 1  n  oi x^  ¼ E  Δx^ k þ 1;j log f X^ k þ 1 X^ k k þ 1;i N^ k þ 1 N^ k þ 1  n 

oi x^ k þ 1;j ^ log g Z k þ 1 X k þ 1 þ E Δ x^ k þ 1;i

N^ k þ 1 N^ k þ 1

x

x

ð57Þ

target i surviving

ð58Þ

x

x

i¼j others

ð59Þ

As to given multi-target tracking problem, once expressions of the state transition density function f sur ðxk jxk Þ and the measurement likelihood function   12 21 g det zk þ 1 jxk þ 1 for single target are fixed, D11 k , Dk , Dk and 22 Dk could be computed out with Eqs. (52)–(59). Then, with J k , the information matrix J k þ 1 at time step k þ1 comes out, for which the inverse matrix is the MT-PCRLB at time step kþ1. Remark 2.

xk þ 1 ¼ F k þ 1 xk þvk

ð60Þ

  zk þ 1 ¼ hk þ 1 xk þ 1 þ wk þ 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2ffi 3 xk þ 1  xs;k þ 1 þ yk þ 1 ys;k þ 1   6 7

hk þ 1 xk þ 1 ¼ 4 5 y  ys;k þ 1 arctan xkk þþ 11  xs;k þ1 T

is the motion state of target, here, xk ¼ xk ; x_ k ; yk ; y_ k   xs;k þ 1 ; ys;k þ 1 radar sensor position at time step kþ1, F k þ 1 A Rnx nx state transition matrix at time step kþ1,   process noise vk  N vk ; 0nv 1 ; Q k , measurement noise   wk þ 1  N wk þ 1 ; 0nw 1 ; Rk þ 1 . Then, it comes   T 1  log f sur xk þ 1 jxk ¼ c1 þ xk þ 1  F k þ 1 xk Q k 1 2    xk þ 1  F k þ 1 xk ð62Þ

others

 n

o x^  E Δx^ k þ 1;j log g Z k þ 1 X^ k þ 1 k þ 1;i 8    < Ef  Δx^ k þ 1;i log g ^ ; det zk þ 1;i x k þ 1;i x^ k þ 1;i ¼ : 0n n ;

As to radar target tracking problem, a typical complicated multi-target tracking problem, calculation formula for Eqs. (53), (55), (58) and (59) are derived in following part. Assuming target moves in 2D plane and radar measures target distance and azimuth, the target motion equation and sensor measurement equation are

ð61Þ

k þ 1;i

and

1) The MT-PCRLB is specific for labeled FISST based multitarget tracking algorithms, and suits complicated multitarget tracking scenes which include target death, target birth, uncertain detection, uncertain measurement source and uncertain data association. It is to scale the lower bound of multi-target tracking algorithm for complicated scenes, and its performance metric is of rigid theoretical explanation and physical connotation, which could further improve precision and stability of multi-target tracking. 2) As many model parameters, target motion parameters, sensor parameters and track scene parameters are involved in computation of MT-PCRLB, these parameters differ in different multi-target tracking problems and should be fixed in certain situation. Precision of such parameters directly determines precision of computing MT-PCRLB.



in which  o n x^  E Δx^ k þ 1;j log f X^ k þ 1 X^ k

8 < Ef Δx^ k þ 1;i log f ðx^ k þ 1;i jx^ k;i Þg; i ¼ j sur x^ k þ 1;i ¼ : 0n n ;

163

   T 1  log g det zk þ 1 jxk þ 1 ¼ c2 þ zk þ 1  hk þ 1 xk þ 1 2    Rkþ11 zk þ 1  hk þ 1 xk þ 1

ð63Þ

nh
   T i E  Δxxkk log f sur xk þ 1 jxk ¼ E ∇xk F k þ 1 xk

h  T iT Q k 1 ∇xk F k þ 1 xk

ð64Þ

nh
   T io  1 x Qk E  Δxkk þ 1 log f sur xk þ 1 jxk ¼ E ∇xk F k þ 1 xk ð65Þ (

 x E  Δxkk þþ 11 log f sur xk þ 1 jxk

) 

( )   x þ E  Δxkk þþ 11 log g det zk þ 1 jxk þ 1 nh   T i ¼ Q k 1 þ E ∇xk þ 1 hk þ 1 xk þ 1

164

Y. Li et al. / Signal Processing 127 (2016) 156–167

h   T iT Rkþ11 ∇xk þ 1 hk þ 1 xk þ 1

ð66Þ

in which  T  T ∇xk F k þ 1 xk ¼ F k þ 1

ð67Þ

2

xk þ 1  xs;k þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 ðxk þ 1  xs;k þ 1 Þ2 þ ðyk þ 1  ys;k þ 1 Þ2 6   T 6 0 ∇xk þ 1 hk þ 1 xk þ 1 ¼6 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yk þ 1  ys;k þ 1 ffi 6 4 ðxk þ 1  xs;k þ 1 Þ2 þ ðyk þ 1  ys;k þ 1 Þ2 0

3  ðyk þ 1  ys;k þ 1 Þ 2 2 ðxk þ 1  xs;k þ 1 Þ þ ðyk þ 1  ys;k þ 1 Þ 7 7 7 0 7 7 xk þ 1  xs;k þ 1 2 2 7 ðxk þ 1  xs;k þ 1 Þ þ ðyk þ 1  ys;k þ 1 Þ 5 0

ð68Þ So far, computation of MT-PCRLB for specific multitarget tracking problem is completed.

As p ¼ 2, OSPA distance of Eq. (69) becomes

OSPA2c X k ; X^ k ¼

1 2 ^ c Nk  Nk ^ Nk Nk X    2 min c; d xk;i ; x^ k;πðiÞ

þ min π A ∏N^

k

!!12

i¼1

ð70Þ ^ k ¼ N k and track association is known and correct, If N   d xk;i ; x^ k;π ðiÞ ¼ ‖xk;i  x^ k;πðiÞ ‖2 , c is large enough for ‖xk;i  x^ k;i ‖2 o c, then Eq. (70) is further simplified as 0 112 ^ N

k  X 2 1 ^ ‖xk;i  x^ k;i ‖2 A ð71Þ OSPA X k ; X k ¼ @ ^k N i¼1

6. Simulation experiments and result analysis In this section is constructed a typical complicated multi-target tracking scene in which target birth, target death and track cross occur, and measurement data from target and clutter are obtained with active radar sensor. The inner coherence between the OPSA distance [3] and the proposed MT-PCRLB is analyzed, leading to the derivation of metric Mea(MT-PCRLB) for MT-PCRLB. Besides, the IL-P-PHD filter's performance in multi-target state estimation and track extraction and the proposed data association algorithm's performance are tested. Based on that, MT-PCRLB is validated as the lower bound of multitarget tracking algorithm's performance in coping with complicated multi-target tracking problem. Performance indicators applied include estimated target number per step, OSPA distance and Mea(MT-PCRLB). The OSPA distance is used to scale multi-target tracking performance of the IL-P-PHD filter, order and cut-off of OSPA are set p ¼2 and c¼80 respectively, Mea(MT-PCRLB) is used to scale lower bound of multi-target tracking algorithm's performance in coping with complicated multi-target tracking problem. The result is an average from 100 independently repeated experiments. Experiments use measurement data from simulations, and are coded with Matlab and run on a PC of 2.7 GHz and 2 GB RAM. 6.1. Indicator based on MT-PCRLB for multi-target tracking performance evaluation In [3] is given the OSPA distance between multi-target
N sate set X k ¼ xk;i i ¼k 1 and estimated target set X^ k ¼
N^ k x^ k;i i ¼ 1 at time step k, that is

OSPApc X k ; X^ k ¼



1 ^ k N k cp N ^k N Nk X    p min c; d xk;i ; x^ k;π ðiÞ

þ min π A ∏N^

k

!!1p

i¼1

ð69Þ here, dð U; U Þ represents the distance between two vectors, π association relations between estimated target state set and existing tracks, ∏N^ k set for all possible association relations, p the order, c cut-off value.

From Eq. (71), the OSPA distance is in essence the average of multi-target error distance at time step k. The MT-PCRLB J k 1 at time step k represents the lower bound of multi-target tracking algorithm's performance 



T E X^ k  X k X^ k X k ð72Þ ZJ k 1 Based on Eq. (72), it comes that 

T



X^ k  X k E X^ k  X k Ztr J k 1

ð73Þ

In Eq.

(73), tr ð U Þ means calculation of matrix trace. tr J k 1 is a performance metric from MT-PCRLB, scaling the lower bound of the mean-square error of estimated multi-target

states at time step k. For consistency with Eq. (71), tr J k 1 is further improved, deriving an indicator for lower bound of average performance in estimating multitarget states, that is !12 1  1

MeaðMT  PCRLBÞ ¼ tr J k ð74Þ ^k N It could be expected, MeaðMT  PCRLBÞ is certainly the lower bound of OSPA distance, which is to be validated in following simulation experiment part. 6.2. Constructing simulation scene The surveillance area for active radar is ½  50; 50 m ½50; 50 m, in which target moves. The motion model for single target is Eq. (60), with Q k ¼ diagð½1; 0:1; 1; 0:1Þ for     1 T 1 T process noise, F k ¼ diag ; , time interval 0 1 0 1 is T ¼ 1 s. The measurement equation for active radar sensor is Eq. (61) with Rk ¼ diagð½1; 0:005Þ for measurement noise. The position of sensor in experiment is set as ð40; 40Þ m, its measurement space is about [ π, π] rad  [0, 100] m. Clutter is uniformly distributed in surveillance area, and clutter measurement is uniformly distributed in measurement space. The number of clutter measurements from once sensor measuring is subject to Poisson distribution of r ¼ 3, the clutter PHD is κ ¼ r=ð2π U 100Þ= ðrad mÞ. In experiments, target surviving probability is 0.95, target birth probability 0.9, radar sensor detection probability 0.9. Number of particle for single target is

Y. Li et al. / Signal Processing 127 (2016) 156–167

1000, new particles are distributed in a 5 m  5 m net with one particle in one grid. Sampling density for new target is subject to Nðx; x; Q Þ, x ¼ ½x; 0; y; 0T , Q ¼ diagð½1; 0: 1; 1; 0:1Þ, ðx; yÞ is location of grid center. A typical complicated multi-target scene with target birth, target death and track across is simulated in experiment. Experiment lasts for 40 s, on 1 s, target 1 start moving from original point ð0; 0Þ m at ð2:1; 1:6Þ m=s and continues for 20 s, target 2 appears on 5 s and disappears on 35 s with initial state ½40; 2:7; 35; 2:0T , target 3 appears on 25 s and disappears on 40 s with initial state ½45; 5:0; 35; 5:0T , tracks of target 2 and target 3 cross on about 29 s and 30 s.

6.3. Experiment results and analysis The IL-P-PHD filter deals with the measurement data set obtained by the active radar, and performs multi-target state estimation and multi-target track extraction jointly. In Fig. 1 is shown results of multi-target state estimation and track extraction, tracks of different targets are labeled differently by the IL-P-PHD filter, ‘O’ track of target 1, ‘’ track of target 2, ‘ þ ’ track of target 3. It is apparent that the IL-P-PHD filter is capable of initiating new track and terminating old track immediately after target

x/m

50

-50

appearing or disappearing and stably keeping track unaffected when tracks across. In Fig. 2 is shown the comparison of estimated target number between the IL-P-PHD filter and the P-PHD filter, target number estimated by IL-P-PHD filter is correct all the time, but target number estimated by P-PHD filter is incorrect when tracks across. When target tracks across, target 2 and target 3 are so close that only one measurement could be obtained by sensor, so the P-PHD filter estimated only one target state. However, because of introducing track label into state vector, the IL-P-PHD filter will not produce confusion when target particles are similar on location dimension and track labels are different. Based on the precise track extracted with the IL-P-PHD filter, the association relation between measurement set at current step and tracks at current step is obtained with the data association algorithm proposed in Section 4. In Fig. 3 are shown the real distance and azimuth of targets relative to sensor, target measurements and clutter measurements obtained by sensor as well as data association results. It is apparent that data association has not been affected by clutter measurements and tracks crossing and data association result is almost correct. Therefore, it is fully proved

true position label 1 label 2 label 3

0

0

5

10

15

20

25

30

35

40

45

50

time/s

y/m

50 true position label 1 label 2 label 3

0

-50

0

5

10

15

20

25

30

35

40

45

50

time/s

Fig. 1. Illustration of estimated target position and track extraction.

165

Fig. 3. Illustration of data association result.

Fig. 2. Comparison of estimated target number.

166

Y. Li et al. / Signal Processing 127 (2016) 156–167

that the track extraction based data association algorithm proposed in this research is effective and reliable. With precise track extraction and data association result, the MT-PCRLB could be calculated. In Fig. 4 is shown the comparison between OSPA and Mea(MT-PCRLB) along time. It is apparent that Mea(MT-PCRLB) is smaller than OSPA all the time, proving that Mea(MT-PCRLB) proposed in this research is the lower bound of multi-target tracking algorithm's performance. In Figs. 5–7 are shown the Mea(MT-PCRLB) along with time on different Q k 、Rk 、r separately. It is discovered that Mea(MT-PCRLB) is significantly affected by process noise covariance matrix Q k and measurement covariance matrix Rk , stronger process noise and measurement noise causing larger Mea(MT-PCRLB). Also, Mea(MT-PCRLB) is nearly unaffected by clutter strength r. With further analysis, the reason underlying could be discovered. The motion model and measurement model are mathematical description of multi-target tracking problem, of which the precision is scaled by process noise covariance matrix Q k and measurement noise covariance matrix Rk respectively, and Mea(MT-PCRLB) is the performance lower bound of multi-target tracking algorithms (such as IL-P-PHD filter) based on such two models. Therefore, MT-PCRLB is to be affected by Q k and Rk , which means with different Q k and Rk algorithms for one complicated multi-target tracking problem might be of different performance lower bounds. As clutter r is an external factor, it will not affect MT-PCRLB for complicated multi-target tracking problem, in spite of its influence on tracking algorithm's performance, or it is not the natural characteristic of complicated multi-target tracking problem. For example, when clutter is too strong in a complicated multi-target tracking problem, though performance of some multi-target tracking algorithms (such as JPDAF) might decline, the IL-P-PHD filter still performs well.

Fig. 4. Comparison between OSPA and Mea(MT-PCRLB).

Fig. 5. Comparison of Mea (MT-PCRLB) on different Q k .

6.5

Mea(MT-PCRLB) with different

6 5.5 5 4.5 4 3.5

7. Conclusion

3 2.5 2 1.5

0

5

10

15

20

25

30

35

time/s

Fig. 6. Comparison of Mea(MT-PCRLB) on different Rk .

40

In this paper is derived multi-target PCRLB (MT-PCRLB), the performance lower bounds of multi-target tracking algorithms in dealing with complicated multi-target tracking scenes. This lower bound could be used jointly with labeled FISST based filters. The multi-target tracks and association relations between measurement set and tracks obtained from labeled FISST based filters contribute to the recursive calculation of MT-PCRLB. Simulation experiment proves that MT-PCRLB is the performance lower bound of tracking algorithm in dealing with complicated multi-target tracking problem, the IL-P-PHD filter performs well on jointly estimating target states and extracting tracks, the new data association algorithm precisely acquires association relations between measurement set and tracks.

Acknowledgments

Fig. 7. Comparison of Mea(MT-PCRLB) on different r.

The anonymous reviewers are thanked for their valuable comments. This work was supported by the National Natural Science Foundation of China under Grant No.

Y. Li et al. / Signal Processing 127 (2016) 156–167

61372159 and supported by the Hunan Provincial Innovation Foundation for Postgraduate.

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