Reference Pattern–Based Track-to-Track Association With Biased Data
A heuristic track-to-track association algorithm for sensors with azimuth biases and range biases is proposed based on a reference pattern (REP) of the target. For a given target, the structure revealed by its neighboring targets is named as its REP. To compensate for the relative azimuth bias between sensors, we modify the optimal subpattern assignment metric, which makes the derived association cost matrix insensitive to sensor biases. Simulation results demonstrate the superior performance of the proposed algorithm.
I. INTRODUCTION
In distributed multitarget-multisensor tracking systems, each sensor transmits its local tracks at defined times to the fusion center, which is responsible for associating and fusing the data. The diagram of a typical fusion system is shown in Fig. 1. It contains three core modules: track-to-track association (TTTA), track-level registration (i.e., sensor bias estimation and correction), and track-to-track fusion (TTTF). TTTA is the prerequisite for track-level registration and TTTF. Although data association and registration are usually addressed separately, they are actually coupled with each other [1]. On the one hand, biased data tend to cause misassociations [2]; on the other hand, misassociations disable the process of registration [3]. In practice, many interference factors complicate the TTTA problem. False and missed tracks diverge the targets reported by diverse sensors. Besides random errors, sensor biases lead to a large departure of the target position estimate from the true value. Since the position estimates of targets from each sensor are unreliable in the presence of sensor biases, traditional nearest neighbor (NN) and global nearest neighbor (GNN) algorithms degrade seriously in real-life applications. Some heuristic approaches, including the centroid matching and singleton matching algorithms, were developed in [4] to remove the relative sensor biases in TTTA. A viable approach was proposed in [5] to remove translational bias between two sets of tracks based on the fast Fourier transform (FFT) correlation technique. In recent years, more attention has been paid to the joint association and bias removal approach. Unlike the GNN problem, the global nearest pattern (GNP) problem presented by Levedahl accounts for sensor biases when constructing the association cost matrix [6]. As stated in [7], the GNP problem is a mixed-integer nonlinear programming (MINLP) problem that is very difficult to Manuscript received June 9, 2014; revised March 8, 2015; released for publication June 10, 2015. DOI. No. 10.1109/TAES.2015.140433. Refereeing of this contribution was handled by S. Mori. C 2016 IEEE 0018-9251/16/$26.00
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501
Fig. 1. Schematic diagram of distributed multitarget-multisensor tracking system.
solve, and a polynomial-time heuristic method, termed multistart local search (MSLS), is proposed. To achieve the global optimal solution, the MSLS algorithm utilizes multiple initial estimates of sensor biases to stimulate the process of alternative iteration between TTTA and registration. In [1], the expectation-maximization (EM) algorithm is incorporated with the Kalman filter (KF) to give the state and bias estimates simultaneously, while the TTTA problem is addressed at the E-step by a multidimensional assignment algorithm. In [8], a class of joint optimization algorithms was developed based on the branch and bound framework. However, these algorithms only consider the translational biases. In some applications, sensor biases occur in a non-Cartesian frame [9–11]. For example, sensors may have biases on their range and azimuth measurements. Under this scenario, traditional registration approaches, such as the standard least squares (LS) method, cannot provide stable estimates for biases [3], and the joint association and bias removal approaches cannot work equally well as they did with translational biases. Another way to solve the TTTA problem with the presence of sensor biases is to employ the reference topology (RET) feature to develop new association algorithms [12–14]. To avoid the abuse of the terminology, we correct the expression of “reference topology” into “reference pattern (REP)” hereinafter, because topology has its specific mathematical definition (i.e., a collection of open sets closed under unionization and finite intersection operations). Although the absolute position of a target is severely affected by sensor biases, the relative position relationship among targets is nearly insusceptible. Given a target, the structure revealed by its neighboring targets is named as the REP. In [12], a 0-1 matrix was used to describe the REP based on spatial discretization. When a target exists in a grid, the corresponding matrix element is set to be 1. Similarly, the element of the REP matrix was set according to the probability of the target existing scattered across grids in [13]. Both REP definitions given by [12] and [13] are descriptive, and the construction of the association cost matrix relies on ad hoc discretizing techniques. With dense grids, REP can exhibit more detail information, but the computational complexity of the derived TTTA algorithm explodes easily. In [14], a rigorous set definition of REP was proposed, and the 502
optimal subpattern assignment (OSPA) metric was utilized to measure the distance between two REPs. Nevertheless, as far as we know, the more complicated TTTA problem for sensors with azimuth biases and range biases is not solved. The main contribution of this paper is to extend [14] to the more challenging scenario where sensors have azimuth biases and range biases. The impact of azimuth bias and range bias on REP is analyzed, respectively. To make the association cost matrix insensitive to sensor biases, the relative azimuth bias is compensated in a robust way during the process of calculating the OSPA distance. One may argue that REP is a feature of a track scene that this work is trying to exploit, but it still belongs to the kinematic state of targets. Extensions for feature- or attribute-aided association have been made in [15–17]. In addition to the standard kinematic states of targets, either continuous or discrete, feature or attribute (e.g., size, cross section, target type, etc.) a posteriori probability distribution is employed to formulate the TTTA problem. The topic of feature- or attribute-aided association lies beyond the scope of this paper. II. PROBLEM DESCRIPTION
In this paper, we focus on the single-scan TTTA problem, which is a data association problem with two random point sets in a two-dimensional Euclidean space. A. Bias Model
Consider two sensors tracking multiple targets in the common surveillance space. The real target number is denoted by N. Both sensors have biases on their azimuth and range measurements. The bias vector for sensor s is represented as bs = [bsr , bsθ ] , s ∈ {A, B},
(1)
where bsr , bsθ represent the range and azimuth bias of sensor s, respectively. Sensor biases directly affect the position estimates of targets, while they have little impact on the velocity or acceleration estimates. Therefore, only the position estimates of targets are used in this paper. It is easy to take the velocity estimates of targets into consideration through augmenting the state dimension. In distributed multitarget-multisensor tracking systems, each local tracker ignores its sensor biases and obtains the local tracks. We assume that the data association problem at each local tracker has been well solved. The track set available at sensor s at a given time (without the time argument, for simplicity) in the system s ˆ is and Pis are the plane is denoted by (ˆxis , Pis )N i=1 , where x position-state estimate and error covariance matrix of the ith target, respectively, and Ns represents the number of tracks from sensor s. Generally, Ns may not be equal to N, as a result of false and missed tracks. It should be noted that in practice, many legacy sensors do not provide covariance [18].
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Based on the first-order approximation, the position estimate of target i available at sensor s can be approximated as xˆ is ≈ xi + x˜ is + Bis bs ,
(2)
where xi is the true position of target i; x˜ is ∼ N (0, Pis ) is the position estimation error of target i at local tracker s; and Bis bs is the additional error induced by sensor biases. With the estimated range rˆsi and azimuth θˆsi of target i in the polar system of sensor s, one has ˆsi ) −ˆrsi sin(θˆsi ) cos( θ . (3) Bis = sin(θˆsi ) rˆsi cos(θˆsi ) B. The TTTA Problem
Most TTTA algorithms first construct a matrix of association costs between the two track sets and then determine the optimal solution by minimizing the overall cost [19]. The difference between these algorithms lies in the specific distance metric that is employed to construct the matrix of association costs. Let xˆ 0A and xˆ 0B represent the dummy tracks of sensor A and B, respectively. If a track in a list is not associated with any track from the other list, it will be assigned to the dummy track. Define the association hypothesis matrix H j A B between (ˆxiA )N xB )N i=0 and (ˆ j =0 specially as an (NA + 1) × (NB + 1) matrix with (i, j) element Hij for i = 1, . . ., NA + 1, and j = 1, . . ., NB + 1 such that j 1, if xˆ iA , xˆ B associated; Hij = (4) 0, otherwise. A The association cost matrix C between (ˆxiA )N i=0 and j NB (ˆxB )j =0 is defined as another (NA + 1) × (NB + 1) matrix with (i, j) element denoted by Cij , where Cij is the j association cost between xˆ iA and xˆ B , and its value depends on the chosen distance metric. Let H denote the set of all binary assignment matrices between the track sets. The standard TTTA problem determines the best association decision from all possibilities by minimizing the overall cost,
min
H∈H
NB NA
Cij Hij ,
(5)
i=0 j =0
subject to the constraints NA
Hij = 1,
j = 0,
(6)
Hij = 1,
i = 0,
(7)
i=0 NB j =0
Hij ∈ {0, 1}.
(8)
The constraints (6) and (7) guarantee that each track from a list is assigned to one and at most one track from the other list or to the dummy track. CORRESPONDENCE
Calculating the matrix of association costs based on the Mahalanobis positional distance metric, we have [20] ⎧ 2 ⎪ ⎨ dij + log(|Sij |) i = 0 and j = 0 i = 0 or j = 0 , Cij = G (9) ⎪ ⎩ +∞ i = 0 and j = 0 where
T
j j xˆ iA − xˆ B , dij2 = xˆ iA − xˆ B S−1 ij j
ij
ji
Sij = PiA + PB − PAB − PBA ,
(10) (11)
βT PDA PDB G = 2 log , 2πPNT A PNT B
(12)
PNT A = βT PDB (1 − PDA ) + βFB ,
(13)
PNT B = βT PDA (1 − PDB ) + βFA .
(14)
Here, β T is the target density, β Fs is the false track density for sensor s, PDs is the probability that sensor s will have a track on a given target in the common field of view, and ij ji PAB and PBA are the cross-covariance matrices due to the common process noise. For convenience, it is usually presumed that estimation errors for each sensor reporting on a common target are uncorrelated [6, 7]. The parameter G is the so-called gate value, which is the cost in favor of assigning a track in a list to a dummy track. It is an open question to determine an optimal gate value. When sensors have translational biases, the joint problem for simultaneous TTTA and bias removal is cast as [6–8]: min u Ru +
H∈H,u∈U
NB NA
+ Cij (u)Hij ,
(15)
i=0 j =0
where u represents the relative translational bias between sensors, which is modeled as a Gaussian random vector having mean 0 and covariance R in a Cartesian coordinate frame; U is the feasible set of u; and Cij (u) is the association cost between tracks i and j based on the Mahalanobis distance, parameterized by a bias estimate u. For i = 0 and j = 0, we have xiA − xˆ B − u) + log(|Sij |). Cij (u) = (ˆxiA − xˆ B − u) S−1 ij (ˆ (16) j
j
Problem (15) is subject to the same constraints as problem (5). It is a MINLP problem, which is very hard to solve. A MSLS algorithm [7] pursues multiple local minima by explicitly using bias and assignment information in an iterative manner. To get the global optimized solution, the method has to maintain a number of initial bias estimates. However, the MSLS algorithm degrades (as shown in the performance studies in section V) with the occurrence of non-Cartesian biases, because it is difficult to get an accurate bias estimate under inaccurate association in the iteration process when biases are in a non-Cartesian frame [3]. 503
In this paper, we provide an approach on the basis of REP to solve the TTTA problem when sensors have range and azimuth biases. The mathematical form of the problem that is to be solved by the proposed approach is the same as problem (5), but the way to construct the matrix C of association costs is quite different from previous methods. The novelty of the proposed algorithm lies in the following two aspects: 1) The proposed algorithm is based on the REP of targets. On one hand, REP is insensitive to sensor biases. Owing to this good property, the proposed algorithm can effectively overcome the negative impact of sensor biases on TTTA. On the other hand, REP exploits the structural information among neighboring targets. Due to the additional information, the proposed algorithm is expected to achieve better performance. 2) To derive the cost between track i and j, we have to compare two REPs. In nature, for a given target, its REP is a set. Traditional cost functions, which aim to compare two points, are not applicable to compute the association cost between two REPs.
Fig. 2. Examples of REP definition (nt = 4). (a) REP of target A. (b) REP of target C.
III. DEFINITION AND PROPERTIES OF REP
Traditional TTTA algorithms using the absolute position information degrade seriously in practice because this information is severely affected by sensor biases. The REP feature of a target that makes use of the relative position information among targets provides a way to overcome this problem. A. Definition of REP
The rigorous set definition of REP has been given in [14]. The reference track set Ris of xˆ is can be defined using its nt nearest neighbors, i.e., t ˆ i,2 ˆ i,n Ris = {ˆxi,1 s }, s ,x s ,...,x
(17)
Fig. 3. Impact of range bias on element of REP.
where nt is called the reference number, and nt ≤ min{NA − 1, NB − 1}.
(18)
s bsr is applied to (ˆxis )N i=1 , the element of REP will be
bsr
Then the REP of xˆ is is defined as Tis
=
i,k {ti,k s |ts
=
xˆ i,k s
−
xˆ is , k
= 1, . . . , nt }.
i,k r ti,k s → ts [bs ].
(19)
For a simple example, there are five targets (A, B, C, D, E) in the surveillance space as shown in Fig. 2. Set the reference number nt as 4. The REP of target A is −→ −→ −→ −→ {AB, AC, AD, AE}, and the REP of target C is −→ −→ −→ −→ {CA, CB, CD, CE}. It can be seen that the REPs of different targets are quite different from each other, which helps to determine the true associations. It has been proved in [14] that the translational biases do not affect REP at all. Next, the impact of azimuth bias and range bias on REP will be analyzed theoretically.
As shown in Fig. 3, range bias lengthens or shortens the position estimates of targets radially. When range bias 504
Expressing the position estimates of targets in polar coordinates, we have i,k ti,k s = rˆs e
√ −1·θˆsi,k
− rˆsi e
√
−1·θˆsi
(21)
,
where rˆsi,k and θˆsi,k are the estimated range and azimuth of xˆ i,k s in the polar system of sensor s. Similarly, we have r rsi,k + bsr )e ti,k s [bs ] = (ˆ
= rˆsi,k e
√ −1·θˆsi,k
+ bsr (e
B. Impact of bsr on REP
(20)
√ −1·θˆsi,k
− rˆsi e
√ −1·θˆsi,k
r = ti,k s + bs (e
−e
√ −1·θˆsi,k
√
− (ˆrsi + bsr )e −1·θˆsi
√ −1·θˆsi
−e
√
√ −1·θˆsi
(22) (23)
)
−1·θˆsi
(24) ).
(25)
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Then we have θ TiA [bA ] = TiB [bBθ ]e
√
−1θ
,
(36)
where θ = bAθ − bBθ is the relative azimuth bias between sensor A and B. This shows that the REPs from different sensors at the same target can coincide approximately with each other, when one REP is rotated by θ in the proper direction. IV. THE PROPOSED ALGORITHM
Fig. 4. Impact of azimuth bias on REP.
Then, we have r i,k r ti,k s [bs ] − ts = bs (e
√ −1·θˆsi,k
−e
√ −1·θˆsi
).
(26)
It can be easily seen that r i,k r 0 ≤ ti,k s [bs ] − ts 2 ≤ 2bs .
(27)
Here, bsr is usually a little larger than or similar to the random noises. Hence, the impact of range bias on the REP is negligible. C. Impact of bsθ on REP
j
As shown in Fig. 4, when the azimuth bias bsθ is s applied to (ˆxis )N i=1 , we can have bsθ
i,k θ ti,k s → ts [bs ],
(28) (29)
θ Expressing ti,k s [bs ] in polar coordinates, we get √
= (ˆrsi,k e =
−1·(θˆsi,k +bsθ )
√
−1·θˆsi,k
− rˆsi e
− rˆsi e
√
−1·(θˆsi +bsθ )
√ −1·θˆsi
)e
√ −1·bsθ
(30) (31) (32)
√ θ Tis e −1·bs .
(33)
That is, azimuth bias makes REP rotate by bsθ . As to the target xi , we denote its ideal REP by Ti . Considering the impact of bsθ , while ignoring the influence of other interferences, we have θ bA
θ Ti → TiA [bA ] = Ti e bBθ
√
Ti → TiB [bBθ ] = Ti e CORRESPONDENCE
hkl ≤ 1,
l = 1, . . . , nt ,
(39)
hkl ≤ 1,
k = 1, . . . , nt ;
(40)
k=1
s Assuming that there are no false tracks in (ˆxis )N i=1 , we can have
≈
where j 1) h, an assignment matrix between TiA and TB , is an nt × nt matrix with (k, l) element hkl for k = 1, . . ., nt , and l = 1, . . ., nt , such that j,l 1, if ti,k A , tB paired, hkl = (38) 0, otherwise, nt
√ −1·bsθ ti,k . s e
Tis [bsθ ]
dp,c (TiA , TB ) 1 min cp (2nt − 2qh ) = 2nt − qh h∈HT ,θ∈Q nt nt √ i,k j,l 1/p [d(tA , tB · e −1θ )]p hkl , (37) + k=1 l=1
bsθ
Tis → Tis [bsθ ].
θ i,k ti,k s [bs ] = rˆs e
To construct the association cost matrix on the basis of REP, we have to compare two REPs. Due to the set nature of REP, conventional cost functions that measure the distance between two points are not applicable [19]. As discussed in [14], the OSPA metric is suitable to compare two REPs. In deriving the association cost matrix, only the cost under H0 (i.e., the two tracks that belong to the same target) can be computed reasonably, since the likelihood function under H1 (i.e. the two tracks that do not belong to the same target) is not available. When measuring the distance between two REPs under H0 , the impact of θ should be removed in order to make the derived association cost insensitive to sensor biases. The modified j OSPA-distance between TiA and TB with relative azimuth bias compensation can be defined as
θ −1·bA
√ −1·bBθ
, .
(34) (35)
nt l=1
HT is the set of all binary assignment matrices satisfying (38–40); and h is the first optimization variable in (37); 2) θ, the relative azimuth bias between sensor A and B, is the second optimization variable in (37); where Q represents the possible range of θ, Q ⊂ ; 3) qh , the number of pairs between the sets, is
qh =
nt nt
hkl ;
(41)
k=1 l=1
505
4) 2nt – 2qh is the number of unpaired elements in the two sets, and 2nt – qh is the total number of pairs and unpaired elements in the two sets; j,l i,k j,l 5) d(ti,k A , tB ) is the base distance between tA and t B , where the Mahalanobis distance is often defined as the base distance, and when the state estimation error covariance matrix is not available, the Euclidean distance can be utilized; 6) 1 ≤ p < ∞ is the OSPA metric order parameter, which is usually set as 2, leading to smooth distance curves, and is applied extensively in many other metrics; and 7) c determines the penalty for an unpaired track. It compares the cost of pairing two points of REPs in favor of a null assignment. The determination of parameters c obeys the guidelines of the original OSPA metric [21]. Then the matrix C of association costs based on REP can be set as ⎧ ⎨ dp,c (TiA , TjB ) i = 0 and j = 0 Cij = c (42) i = 0 or j = 0 ⎩ +∞ i = 0 and j = 0 Equation (37) is a complex MINLP problem, which is difficult to solve. Fortunately, given a bias θ, we can efficiently compute an optimal assignment h* with the Munkres algorithm [22]. In addition, given an assignment h, an optimal bias estimate θ * can be obtained by solving min
θ∈Q
nt nt
j,l
d(ti,k A , tB · e
√
−1·θ p
) hkl ,
which is a simple optimization problem in one variable. However, h often contains false assignments. Here, we present a simple approach to obtain a robust bias estimate on the basis of the median operator as ∗
(44)
where med is a function to take the median value of a sequence of real numbers. If h is a null matrix, there is no need to calculate θ * anymore, and the distance between two REPs can be set as c. In summary, the OSPA distance between Tai and Tbj with the relative azimuth bias compensation can be obtained by going through Algorithm 1. In Algorithm 1, the OSPA distances are computed twice. Let c1 and c2 represent the penalty for an unpaired track before and after the relative azimuth bias compensation, respectively. When θ is compensated for two REPs that belong to the same target, the base distance between two associated reference tracks gets small. Hence, the parameter c2 should be smaller than c1 , due to the bias compensation. It appears that the process of constructing the association cost matrix on the basis of REP and the OSPA metric has high computational complexity, since each entry in the association cost matrix should be calculated 506
Association cost between two REPs.
1) Set the initial estimate of θ to be 0. Then obtain h* by solving problem (37) using the Munkres algorithm. 2) Given h* , get θ * by calculating (44). 3) Substitute θ * into problem (37), compute it again, and get the j association cost dp,c (TiA , TB ).
Algorithm 2
The proposed algorithm.
1) Initialize the threshold parameter g, the penalty parameters c1 and c2 , and the reference number nt . 2) According to the definition of REP (see (19)), construct the REPs for for each track in each track list. 3) For any i = 1,. . ., NA and j = 1,. . ., NB , if (45) is satisfied, then j directly set dp,c (TiA , TB ) = ∞; otherwise, perform Algorithm 1 to get the matrix of association costs. 4) Run a linear assignment algorithm on the derived association cost matrix to determine the best association in terms of the overall cost.
online by performing Algorithm 1. As stated in [14], two techniques contribute to reducing the computation burden. First, the size of problem (37) is restricted by parameter nt . Ignoring the bias compensation procedure, the computation complexity of constructing the cost matrix is O(NA NB n3t ), since for each of the NA NB pairs of tracks, we have to solve a very small linear assignment problem, which has O(n3t ) complexity. Second, a gating technique is utilized. If the distance between two tracks satisfies j
ˆxiA − xˆ B 2 > g,
(43)
k=1 l=1
j,l θ = med{θˆAi,k − θˆB |hkl = 1, k = 1, . . . , nt ; l = 1, . . . , nt },
Algorithm 1
(45)
where g is a predetermined threshold parameter, then we can directly set j
dp,c (TiA , TB ) = ∞,
(46)
without performing Algorithm 1. To identify the optimal track associations that cause the minimal global cost, a linear assignment algorithm can be run on the derived association cost matrix. A sketch of the proposed algorithm used to find the single best TTTA decision with biased data is outlined in Algorithm 2. Similar to MSLS, the proposed algorithm could easily be adapted to provide the K-best TTTAs. It should be noted that in the proposed algorithm, the covariance matrix associated with each sensor track, as well as the a priori statistics of the sensor biases, is not used. There are two reasons. The first one is that the range and azimuth biases are assumed to be unknown constants [10]. The second is that deriving the covariance matrix between two REPs is difficult. When covariance matrices are available, it is easy to adjust our algorithm to make use of the covariance information. The Mahalanobis distance can be defined as the base distance in (37), while the values of parameters g and c should be adjusted accordingly. V. SIMULATIONS
In this section, we test the proposed single-scan TTTA algorithm (REP for short) using simulations.
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The simulation scenario is set as follows. Two sensors track N targets in the common two-dimensional space. It is assumed that the sensors have been synchronized, and they transmit their local tracks to the fusion center at defined times. Sensors A and B are located at (−50, 0) km and (50, 0) km, respectively. Random errors of all local tracks are identically subject to Gaussian distribution with mean zero and covariance σ 2 I. The standard deviation σ of the random errors is set as 100 m. Since calculating the cross-covariance as a result of common process noises in real-time applications is not practical, the approximation with a single correlation coefficient ρ for all the state components is taken into account [20]. The parameter ρ is set as 0.4. It is further presumed that the problem of measurement-to-track association has been well solved at each sensor, but the local tracker may report extraneous tracks or lose some targets. For both sensors, the number of false tracks satisfies a Poisson distribution with expected value lF , and the rate of missed tracks is denoted by Pm . Targets are randomly distributed in a 20 km × 20 km region the center of which is at (0, 80) km. The performance criterion is the probability of correct association, which is defined as PCA =
1 M
M
QAB¯ (m) + QAB ¯ (m) + QAB (m) , TAB¯ (m) + TAB ¯ (m) + TAB (m) m=1
(47)
where M is the total number of Monte Carlo runs. In the mth Monte Carlo run, TAB (m) is the true number of targets reported by both sensors, TAB¯ (m) is the true number of targets reported by sensor A but missed by sensor B, and TAB ¯ (m) is the true number of targets reported by sensor B but missed by sensor A. Correspondingly, QAB (m) is the number of correctly associated track pairs based on the TTTA output in simulations, QAB¯ (m) is the number of correctly unpaired tracks of sensor A, and QAB ¯ (m) is the number of correctly unpaired tracks of sensor B. The PCA , defined by (47), accounts for not only the correctly paired tracks but also the correctly unpaired tracks. It is suitable to compare the performance of TTTA algorithms in the presence of false and missed tracks. Algorithms used for comparison include the GNN algorithm, which is ignorant of sensor biases [20] and the MSLS algorithm [7]. Unlike the original MSLS algorithm, which only deals with the translational biases, the MSLS algorithm used here has been extended to deal with the scenario where sensors have azimuth biases and range biases. It carries out association and registration in an alternate iteration manner. The registration module employs least squares to estimate the absolute biases of both sensors instead of the relative translational biases, while the TTTA module adopts the GNN algorithm. Thirty initial estimates of sensor biases are utilized to stimulate the process of alternate iteration. For each starting point, the maximal iteration number is set to be 6. The initial estimates of biases are generated randomly according to the prior knowledge about the bias range. For both sensors, the lower and upper bounds of azimuth biases are set as CORRESPONDENCE
Fig. 5. Performance comparison among MSLS, GNN, and proposed REP algorithms in terms of probability of correct association when target number changes with fixed target region.
−3◦ and 3◦ , respectively; the lower and upper bounds of range biases are set as −2 km and 2 km, respectively. All results are based on 500 Monte Carlo runs. A. The Influence of nt on the Performance of REP
We examine the influence of the reference number nt on the performance of the proposed REP algorithm under different target densities. The target spatial density varies with the change of the target number N, while the target region remains unchanged. The N varies from 10 to 40 in increments of 5. Other scenario parameters are: bA = [1 km, 1◦ ] , bB = [−1 km, −1◦ ] , lF = 1, and Pm = 4%. In the REP algorithm, the Euclidean distance is employed as the base distance metric. The reference number nt varies from 1 to 8 in increments of 1. The threshold value g, which is the maximal possible distance between two targets when random noises and the prior bounds of biases are taken into consideration, is set to be 9.3 km. The penalty parameter c1 for an unpaired track before bias compensation is determined based on the following experience formula: √ c1 = 2 · lT · θpossi + 5 · σ, where lT is the length of target region, and θ possi is the possible range value of θ. Considering that the lower and upper bounds of azimuth biases are set as −3◦ and 3◦ , respectively, θ is 6◦ π/180◦ . In the present simulation, the length of the target region lT is 20 km, so the value of c1 is 3.46 km. The penalty c2 after compensating for the relative azimuth bias is set as half of c1 . Fig. 5 demonstrates the impact of nt on the performance of the proposed REP algorithm in terms of the probability of correction association. Compared with the MSLS and GNN algorithms, the proposed REP algorithm, with nt no less than 5, has an obvious advantage when the target spatial density is not too high. This superiority of REP shrinks as the target spatial density increases. The reason lies in that, when target 507
Fig. 6. Performance comparison among MSLS, GNN, and proposed REP algorithms in terms of run time when target number changes with fixed target region.
spatial density is high, the impact of random noises on REP becomes remarkable, and the random errors may mess up the structural information revealed by REP. Another simulation experiment will be presented in section VE to further study the impact of target density on the proposed REP algorithm. From Fig. 5, we can also see that the increase of nt can promote the probability of correct association of the REP algorithm, but when nt is larger than 5, no obvious growth can be achieved by unceasingly increasing nt . Fig. 6 shows the comparison of run time among the REP, MSLS, and GNN algorithms. These results were obtained using Matlab on a computer with Intel(R) Core(TM) i7 central processing unit 860 @2.8 GHz and 3.24 GB random-access memory. The run time of the MSLS algorithm increases quite rapidly when N grows. When N or nt increase, the run time of the proposed REP algorithm grows slowly. The computation complexity of the proposed REP algorithm is a little higher than the GNN algorithm. With comprehensive consideration for the association performance and computation complexity, the proposed REP algorithm is more acceptable in practice than the other two algorithms. B. Sensitivity to the Threshold Parameter g
We examine the sensitivity of the proposed REP algorithm to the threshold parameter g. The simulation scenario is the same as section VA, while the target number is set as 15. The reference number nt of the REP algorithm is set to be 5. The parameter c1 is set to be 3.46 km, while c2 is set as half of c1 . The threshold parameter g varies from 1 km to 15 km in increments of 1 km. Fig. 7 demonstrates the probability of correct association of the proposed REP algorithm as a function of the threshold parameter g. This threshold parameter aids the TTTA process in two aspects. On one hand, it can help decrease the computation complexity of the proposed 508
Fig. 7. Impact of threshold parameter g on performance of proposed REP algorithm (nt = 5) in terms of PCA .
algorithm; on the other hand, a proper g can help improve the probability of correct association by definitely pointing out the impossible associations. In this point of view, this parameter can introduce some prior knowledge about the truth of TTTA. From Fig. 7, it can be concluded that the proposed REP algorithm is not sensitive to g, as long as g is not too small. Under the scene with sensor biases, a small g (e.g., less than 3 in Fig. 7) can remarkably increase the risk that two tracks, even originating from the same target, do not associated with each other. How to determine an optimal g remains an open question. In practice, various factors (including sensor biases, the distance from target region to sensors, the standard deviation of random errors, etc.) should be taken into consideration to select a proper g. C. Sensitivity to c
We examine the sensitivity of the proposed REP algorithm to parameter c. The parameter c determines the weighting of how the metric penalizes cardinality errors as opposed to localization errors [21]. In other words, it is used to penalize the outlier that is deemed “unassignable” because of the false tracks and (or) missed detections. Parameters c1 and c2 are the penalties for an unpaired track in Algorithm 1 before and after the relative azimuth bias compensation, respectively. After the relative azimuth bias is corrected, the localization errors between real tracks, which originate from the same target, become small. Thus, c2 should be smaller than c1 to better identify the “unassignable” objects. According to our experience, the parameter c2 can be set as half of c1 . In the simulations, c1 varies from 3σ to 60σ in increments of σ . The value of σ is the standard deviation of random errors of the tracks, and it is set to be 0.1 km. The simulation scenario parameters are set as follows. The target number is set as 15. The target region and bias parameters are the same as section VA. The reference number nt of the REP algorithm is set to be 5. The
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Fig. 8. Impact of c1 on performance of proposed REP algorithm (nt = 5) in terms of PCA , with Pm = 4%, and lF = 1.
Fig. 9. Impact of c1 on performance of proposed REP algorithm (nt = 5) in terms of PCA , with Pm = 20%, and lF = 1.
threshold parameter g is fixed to be 9.3 km. Three scenarios are considered in the simulations. When Pm is 4% and lF is 1, the scenario corresponds to few missed tracks and few false tracks. Under this scenario, Fig. 8 demonstrates that the proposed REP algorithm is not sensitive to parameter c1 , and it is not difficult for the REP algorithm to choose a feasible c1 with few missed tracks and few false tracks in practice. When Pm is 20% and lF is 1, the scenario corresponds to few false tracks but more missed tracks. Fig. 9 demonstrates that the proposed REP algorithm becomes sensitive to parameter c1 when the probability of missed detections increases. As the probability of missed detections increases, a small gate value should be employed. When Pm is 4% and lF is 4, the scenario corresponds to few missed tracks but more false tracks. Fig. 10 demonstrates that the proposed REP algorithm becomes sensitive to parameter c1 when the number of false tracks increases. As the number of false tracks increases, a small gate value should be chosen. Comparing Fig. 9 with Fig. 10, we can see that the proposed REP algorithm is more CORRESPONDENCE
Fig. 10. Impact of c1 on performance of proposed REP algorithm (nt = 5) in terms of PCA , with Pm = 4%, and lF = 4.
Fig. 11. Performance comparison among MSLS, GNN, and proposed REP algorithms (nt = 5) in terms of probability of correct association when azimuth biases change.
sensitive to missed tracks than to false tracks. That is because missed detections make the REPs lose part of the structure revealed by the neighboring targets, while false tracks make the structure more complicated. D. The Influence of Various Interference Factors on REP
We test the influence of sensor biases, false tracks, and missed tracks on the association performance of the proposed REP algorithm, respectively. In the following simulations, the target number N is fixed to be 15. The reference number nt is set to be 5. The threshold parameter g is set as 9.3 km. The value for c1 is set as 3.46 km, while the value for c2 is set as half of c1 . Set the scenario parameters as: bAθ = −bBθ , r bA = −bBr = 1 km, lF = 1, and Pm = 4%. The bAθ varies from 0◦ to 3◦ in increments of 0.6◦ . Fig. 11 shows the impact of azimuth biases on the probability of correct association of the TTTA algorithms. The GNN algorithm degrades seriously when large azimuth biases occur. Both the REP and MSLS algorithms are not sensitive to the 509
Fig. 12. Performance comparison among MSLS, GNN, and proposed REP (nt = 5) algorithms in terms of probability of correct association with changes of range biases.
Fig. 14. Performance comparison among MSLS, GNN, and proposed REP (nt = 5) algorithms in terms of probability of correct association with changes of Pm . TABLE I Cases of lF and Pm Case index
lF Pm
Fig. 13. Performance comparison among MSLS, GNN, and proposed REP (nt = 5) algorithms in terms of probability of correct association with changes of lF .
azimuth biases. Moreover, the former is obviously better than the latter. Set the scenario parameters as: bAθ = −bBθ = 1◦ , r bA = −bBr , lF = 1, and Pm = 4%. The bAr varies from 0 to 2 km in increments of 0.4 km. Fig. 12 demonstrates the impact of range biases on the probability of correct association of the TTTA algorithms. The GNN algorithm degrades when the range biases get larger. Compared with Fig. 11, we can see that the GNN algorithm is less sensitive to range biases than to azimuth biases. That is because the azimuth bias of a sensor has greater influence, which expands as the target range increases, on the target position estimate. The REP and MSLS algorithms are both insensitive to range biases, and the former is better than the latter. Set the scenario parameters as: bAθ = −bBθ = 1◦ , r bA = −bBr = 1 km, and Pm = 4%. The lF varies from 0 to 5 in increments of 1. Fig. 13 demonstrates the impact of false tracks on the probability of correct association of the TTTA algorithms. Although all three of the TTTA 510
0 0 0
1 1 5%
2 2 10%
3 3 15%
4 4 20%
5 5 25%
algorithms degrade with the increase of lF , the proposed REP algorithm can deal with false tracks better than the other two algorithms. Set the scenario parameters as: bAθ = −bBθ = 1◦ , r bA = −bBr = 1 km, and lF = 1. The Pm varies from 0 to 25% in increments of 5%. Fig. 14 shows the advantage of the proposed REP algorithm when dealing with missed tracks. Comparing Fig. 14 with Fig. 13, we can see that the proposed REP algorithm is more robust to false tracks than to missed tracks. That is because the missed tracks result in the loss of part of the structure revealed by REP, while false tracks only add interference information to REP by complicating the existing structure of REP. Finally, we test the performance of the TTTA algorithms under six cases of lF and Pm as shown in Table I. Fig. 15 demonstrates that the proposed REP algorithm is superior to the GNN and MSLS algorithms when the rates of missed tracks and false tracks increase simultaneously. E. The Influence of Target Density on REP
We test the performance of the proposed REP algorithm as target density varies. The minimum Mahalanobis positional distance (MMD), a unitless metric sometimes referred to as the minimum nearest neighbor distance, is employed as a proxy for the target density [23]. The reference number nt of the REP algorithm is set to be 5. The threshold parameter g is set as 9.3 km. Parameter c1 is set as 3.46 km, while c2 is set as half of c1 . The target number is fixed to be 15, while the length of the target region is set to be 0.5, 1, 2, 4, 5, 10, 15, 20, 30, 40, 50, 60 km. Random errors of all local tracks are
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Fig. 15. Performance comparison among MSLS, GNN, and proposed REP (nt = 5) algorithms in terms of probability of correct association when lF and Pm change simultaneously.
A medium-density scene corresponds to an average MMD in the rough interval [1.4, 13.2] (see Fig. 16). With a medium-density scene, the proposed REP algorithm has remarkable advantage over the MSLS and GNN algorithms. The reason lies in two aspects. First, when targets are not too dense, the relative position relationship is hardly influenced by random errors. Thus, the REP can exploit clear structural information reflected by neighboring targets. Second, the REP is insensitive to sensor biases. A low-density scene corresponds to an average MMD in the rough interval [15, ∞) (see Fig. 16). With a low-density scene, the probability of correct association of the three algorithms gets close. That is because when targets are sparse enough, the negative impact of sensor biases on the TTTA problem becomes insignificant. Under this circumstance, the TTTA problem becomes simple, and even the traditional GNN algorithm can yield high probability of correct association. VI. CONCLUSIONS AND FUTURE WORK
Fig. 16. Performance comparison among MSLS, GNN, and proposed REP (nt = 5) algorithms in terms of the probability of correct association when target density changes.
identically subject to Gaussian distribution with mean zero and covariance σ 2 I, where σ is set to be 0.1 km. The other parameters about the simulation scenario are the same as section VA. Without loss of generality, we calculate the MMD based on the track set from sensor A. Then, the MMD used in the simulation can be defined as j j MMD = min (ˆxiA − xˆ A ) (2σ 2 I)−1 (ˆxiA − xˆ A ). i=j,i,j ∈{1,...,NA }
(48) For a given target region, we perform 500 Monte Carlo runs, on the basis of which the average MMD is obtained. Fig. 16 demonstrates the probability of correct association of the REP, MSLS, and GNN algorithms as a function of the average MMD. When the average MMD is small (e.g., less than 1 in Fig. 16), the targets are quite dense, and the proposed REP algorithm is inferior to the MSLS and GNN algorithms. That is because with a high-density scene, the structural information reflected by neighboring targets tends to be messed up by the random errors. CORRESPONDENCE
It is very challenging to determine track-to-track associations in the presence of random noises, sensor biases, false tracks, and missed tracks. On the basis of the reference pattern of targets, a heuristic track-to-track association algorithm is proposed for sensors that have azimuth biases and range biases. In order to make the association cost matrix insensitive to sensor biases, we modify the optimal subpattern assignment metric by compensating for the relative azimuth bias in a simple but robust way. Simulation results demonstrate that our proposed algorithm can achieve better association performance with acceptable computation complexity than that of the traditional algorithms. Future work can be made in the following directions: 1) The major limitation of the work is that the performance of the proposed algorithm relies on careful selection of parameters g and c. However, it is still an open question to determine the optimal values for these parameters. 2) In this paper, we only make use of the position-only state of targets. Naturally, we would like to extend our results by utilizing velocity information of targets; in addition, other attributes or features can also be introduced by adjusting the global cost function [24]. Theoretically, more information will improve the probability of correct association. 3) Extending the REP idea to three or more sensors is another direction for the future work. The likelihood function based on the REPs of targets for the track-to-track association problem from multiple sources should be derived. WEI TIAN Data Analysis Center Unit 91715, PLA Guangzhou, 510450, China E-mail: (
[email protected]) 511
YUE WANG Tsinghua University Department of Electronic Engineering Qinghuayuan Street Beijing, 100084, China XIONGJIE DU National Computer Network Emergency Response Technical Team/Coordination Center of China (CNCERT/CC) Box 06, No. A3 Yumin Road, Chaoyang District Beijing 100029, China XIUMING SHAN JIAN YANG Tsinghua University Department of Electronic Engineering Qinghuayuan Street Beijing, 100084, China
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