- Email: [email protected]
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets
CJA 716 26 October 2016 Chinese Journal of Aeronautics, (2016), xxx(xx): xxx–xxx
No. of Pages 9
1
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
2
FULL LENGTH ARTICLE
6
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets
7
Yang Jinlong a,b,*, Yang Le a, Yuan Yunhao a, Ge Hongwei a
4
5
8 9
10
a b
School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Wuxi 214122, China
Received 31 December 2015; revised 22 September 2016; accepted 22 September 2016
11
13 14
KEYWORDS
15
Adaptive parameter estimation; Multiple target tracking; Multivariate Gaussian distribution; Particle filter; Probability hypothesis density
16 17 18 19 20 21 22
23
Abstract The probability hypothesis density (PHD) filter has been recognized as a promising technique for tracking an unknown number of targets. The performance of the PHD filter, however, is sensitive to the available knowledge on model parameters such as the measurement noise variance and those associated with the changes in the maneuvering target trajectories. If these parameters are unknown in advance, the tracking performance may degrade greatly. To address this aspect, this paper proposes to incorporate the adaptive parameter estimation (APE) method in the PHD filter so that the model parameters, which may be static and/or time-varying, can be estimated jointly with target states. The resulting APE-PHD algorithm is implemented using the particle filter (PF), which leads to the PF-APE-PHD filter. Simulations show that the newly proposed algorithm can correctly identify the unknown measurement noise variances, and it is capable of tracking multiple maneuvering targets with abrupt changing parameters in a more robust manner, compared to the multi-model approaches. Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author at: School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China. E-mail address: [email protected] (J. Yang). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
1. Introduction
24
Multiple target tracking (MTT) has gained wide attentions due to its theoretical and practical importance. Conventionally, the MTT problem was tackled from the perspective of data association. A number of tracking algorithms were developed in the literature on the basis of techniques including the joint probabilistic data association (JPDA),1 joint integrated probabilistic data association (JIPDA)2 and multiple hypothesis tracking (MHT).3 These methods are generally computationally intensive and some of them even have exponentially growing complexity as the target number increases. Reduced-
25
http://dx.doi.org/10.1016/j.cja.2016.09.010 1000-9361 Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
26 27 28 29 30 31 32 33 34
CJA 716 26 October 2016
No. of Pages 9
2 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
complexity techniques were proposed in Refs. 4–6. They are better for real-time applications at the cost of degraded estimation accuracy. Recently, the use of the random finite set (RFS) theory7–11 attracted great interests, because it provides an elegant formulation of the MTT problem. But the obtained multi-target Bayesian filter is intractable in most practical scenarios due to the inherent combinatorial nature of multi-target state densities and the need for evaluating set integrals over high dimensional spaces. To deal with the intractability, the probability hypothesis density (PHD) filter7 and the cardinalized PHD (CPHD) filter8 were developed using the first-order moment and cardinality distributions. Existing closed-form realizations of PHD filters include the particle filter PHD (PF-PHD),9,10 Gaussian mixture PHD (GM-PHD) filter11 and various modified versions.12–15 Different from the PHD and CPHD filters, the cardinality-balanced multi-target multi-Bernoulli (CBMeMBer) filter was proposed in Ref. 16 for MTT by directly propagating the approximate posterior density of the targets. These algorithms exhibit good performance only when the model parameters, such as the measurement noise variances, are known precisely. In the presence of unknown time-varying measurement noise variances, the variational Bayesian (VB) approximation method17–19 can be employed to recursively estimate the joint PHDs of the multi-target states and the measurement noise variance.20,21 However, these methods may suffer from performance degradation if targets manoeuver with unknown abruptly changing parameters. For maneuvering target tracking, the use of the jump Markov system (JMS) that switches among a set of candidate models in a Markovian fashion has proved to be effective.22,23 Pasha et al.24 introduced the linear JMS into PHD filters and derived a closed-form solution for the PHD recursion. Furthermore, the unscented transform (UT) and the linear fractional transformation (LFT) were combined with the closed-form solution for the nonlinear jump Markov multitarget models in Refs. 25, 26 In Ref. 27, a GM-PHD filter for jump Markov models was developed by employing the best-fitting Gaussian (BFG) approximation approach. These algorithms assume the Gaussianity of the PHD distribution, which may limit their application scope. The multiple-model particle PHD (MMP-PHD) filter, the MMP-CPHD filter and MMP-CBMeMBer filter are implemented by using the sequential Monte Carlo (SMC) method and their improved versions were presented in Refs. 28–30 Most of the MM-based filters track multiple maneuvering targets through the interaction of multiple models, which is realized via combining estimates from different models according to their respective model likelihoods. The difficulty of applying them in tracking targets with abruptly changing maneuvering parameters comes from the need to specify a prior set of candidate models. In other words, they may suffer from the curse of dimensionality: if we wish to account for multiple unknown parameters, the number of models needed would increase exponentially with the number of parameters. In this work, we incorporate the adaptive parameter estimation (APE) technique into the PHD filter for addressing the problem of multiple maneuvering target tracking, where both static and time varying unknown parameters, namely the measurement noise variance and the parameters associated with abrupt target maneuvers, are presented and need to be estimated. The inverse Gamma (IG) distribution is used to
J. Yang et al. approximate the posterior distribution of the measurement noise variances while the adaptive Liu and West (LW) filter is adopted to propagate the posterior marginal of the timevarying parameters as a mixture of multivariate Gaussian distributions.31–33 The obtained APE-PHD filter is realized using the particle filter (PF), which leads to the PF-APE-PHD algorithm for tracking multiple maneuvering targets in the presence of unknown model parameters. Simulation results show that the proposed algorithm exhibits better robustness and improved tracking performance over the MM-PHD and MM-CPHD algorithms. The remainder of this paper is organized as follows. Section 2 formulates the problem of tracking a target in the presence of unknown model parameters. It also briefly reviews the APE technique and the PHD filter. Section 3 develops the APE-PHD algorithm and presents the closed-form solution, the PF-APE-PHD algorithm. Simulation results are given in Section 4. Finally, conclusions are provided in Section 5.
97
2. Preliminary
115
2.1. Problem formulation
116
The state-space model for tracking a single target moving on a two-dimensional plane is given by
117
98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114
118
xkþ1 ¼ Fxk þ Gvk
ð1Þ
119 121
yk ¼ hðxk Þ þ wk
ð2Þ
122 124
where xk ¼ ½xk ; vxk ; yk ; vyk T denotes the target state at time k, ðxk ; yk Þ and ðvxk ; vyk Þ denote its position and velocity. F and G are the state transition matrix and the process noise gain matrix. yk is the measurement vector. vk and wk denote the process noise and the measurement noise. They are independent of each other and modeled as zero-mean Gaussian random processes with covariance Qk and Rk . In many practical applications, the state-space model in Eqs. (1) and (2) may contain unknown parameters. For example, if the target conducts a coordinated turn (CT),28 the state transition matrix would become 2 xT 3 0 1cos 1 sinxxT x 6 0 cos xT 0 sin xT 7 6 7 FðxÞ ¼ 6 ð3Þ 7 xT sin xT 4 0 1cos 5 1 x x 0 sin xT
0 cos xT
pðyk jxk ; Uk Þpðxk ; Uk jy1:k1 Þ pðyk jxk ; Uk Þpðxk ; Uk jy1:k1 Þdxk dUk
126 127 128 129 130 131 132 133 134 135 136
138
The turn rate x may be unknown and time-varying. Besides, the measurement noise covariance Rk may also be unknown. In these scenarios, we need to jointly estimate the posterior distribution of the target states and the unknown parameters from the measurements. Let Uk be a column vector that collects the static and timevarying parameters in the state-space model. The posterior probability density function (PDF) of the target state vector xk and Uk conditioned on the measurements up to time k is, according to Bayes’ rule, pðxk ; Uk jy1:k Þ ¼ RR
125
ð4Þ
where pðxk ; Uk jy1:k1 Þ is the predicted PDF given by
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
139 140 141 142 143 144 145 146 147 148 149 151 152 153
CJA 716 26 October 2016
155 156 157 158 159 160
161
162 163 164 165 166 167 168 169 170 171 172 174 175 176 177 178 180 181 182 183 184 185
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets 3 ZZ mik1 ¼ nik1 , which results in a posterior distribution overly pðxk ; Uk jy1:k1 Þ ¼ pðxk jxk1 ; Uk1 ÞpðUk jUk1 Þ dispersed in the sense that the covariance of the mixture is larger than Vk1 . The shrinkage factor introduced in Eq. (9) can pðxk1 ; Uk1 jy1:k1 Þdxk1 dUk1 ð5Þ force particles nik1 to move closer to their sample mean nk1 so Deriving exact recursive solutions for the posterior distributhat maintaining the same covariance Vk1 is achieved. tion pðxk ; Uk jy1:k Þ from Eqs. (4) and (5) is in general intractable In the case that the time instant k is a changepoint, and the and as a result, approximate solutions are usually resorted to. predicting distribution of the time-varying vector nk will be One such approach is the SMC method, also referred to as the reset to pn ðn0 Þ, its prior distribution. With the predicted PDF particle filter (PF).9,11,14 given in Eq. (8), the APE filter utilizes the PF to produce an approximation of the posterior distribution pðxk ; Uk jy1:k Þ in 2.2. Adaptive parameter estimation (APE) Eq. (4). Suppose at time k 1, the posterior distribution is In Refs. 31, 32, the Liu and West (LW) filter was proposed for the joint identification of static parameters and target states. In particular, the marginal posterior distribution of the unknown parameters is approximated and propagated using a mixture of multivariate Gaussian distributions. In Ref. 33, the particle learning technique was introduced into the LW filter. The obtained APE filter can handle both static and time-varying parameters. The development of the APE method starts with factorizing pðxk ; Uk jy1:k1 Þ into pðxk ; Uk jy1:k1 Þ ¼ pðxk jy1:k1 ; Uk ÞpðUk jy1:k1 Þ
pðUk jy1:k1 Þ ¼ pðhk ; nk jy1:k1 Þ ¼ pðhk jy1:k1 ; nk Þpðnk jy1:k1 Þ
190 191 192 193 194 195 196 197 198 199 201
205 206 207 208 209 210 211
8 N X > < xi
2 i k1 Nðnk jmk1 ;h Vk1 Þ
> :
with probability 1 b
i¼1
pn ðn0 Þ
ð8Þ
with probability b
where Nðnk jmik1 ; h2 Vk1 Þ is a Gaussian kernel with mean mik1 and covariance h2 Vk1 , and h 2 ð0; 1Þ denotes a scaling parameter that shrinks the kernel. xik1 is the weight of the ith com-
ponent. Here, b can model the temporal evolution of nk . It is defined as the probability that nk is subject to an abrupt change at time k, or equivalently speaking, time k is a changepoint.33 The time-varying vector nk is assumed to be piecewise constant between two neighboring changepoints. As defined in Eq. (8), if there is no abrupt change in nk , its predicted PDF follows a Gaussian mixture model of N components. The mean and covariance of each components are obtained by mik1 ¼ anik1 þ ð1 aÞnk1
202
204
ð7Þ
The predicted distribution pðhk jy1:k1 ; nk Þ of the static parameter vector hk is characterized using sufficient statistics sk , i.e., hk pðhjsk Þ.32 The predicted distribution of the time-varying parameter vector nk is approximated via
187
189
ð6Þ
Let Uk ¼ ðhk ; nk Þ, where hk and nk collect static and timevarying parameter vectors. The marginal predicting distribution of Uk can be expressed as
pðnk jy1:k1 Þ
188
No. of Pages 9
Vk1 ¼
N X
T xik1 ðnik1 nk1 Þðnik1 nk1 Þ
ð9Þ
N fxik1 ; hik1 ; nik1 gi¼1
represented by N particles with weights xik1 . At time k, each particle is given two weights33
i¼1
P i i where nk1 ¼ N i¼1 xk1 nk1 is the minimum mean square error (MMSE) estimate of nk1 at time k 1, and nik1 denotes the ith Gaussian component of the time-varying vector nk1 . pffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ 1 h2 is the shrinkage factor suggested in Ref. 34 to correct for the over-dispersion of the Gaussian mixture model Eq. (8). It is noted that standard kernel smoothing requires that kernel components be centered around the mean vectors
213 214 215 216 217 218 219 220 221 222 223 224 225
xik;1 / xik1 pðyk jlik ; hik1 ; nik Þ; nik Nðnk jmik1 ; h2 Vk1 Þ;
where
lik ¼ E½xik jxik1 ; hik1 ; nik
ð11Þ
227 228
xik;2 / xik1 pðyk jlik ; hik1 ; cik Þ;
where cik pn ðn0 Þ;
lik ¼ E½xik jxik1 ; hik1 ; cik
ð12Þ
which essentially leads to 2N particles. xik;1 and xik;2 correspond to the probability of the current measurement yk when there is no changepoint and when there is a changepoint, respectively. In the former case, the value of time-varying parameter vector nik is drawn from the Gaussian component Nðnk jmik1 ; h2 Vk1 Þ, while for the latter case, its value cik is produced using the prior distribution pn ðn0 Þ (see also Eq. (8)). A resampling is then performed on the basis of the weights ð1 bÞxik;1 and bxik;2 to select N particles out of 2N particles and propagate them to generate the approximation of the posterior pðxk ; Uk jy1:k Þ at time k. For more details on the APE filter for tracking a single maneuvering target, please refer to Ref. 33.
230 231 232 233 234 235 236 237 238 239 240 241 242 243
2.3. PHD filter
244
Under the RFS framework, we denote the multiple target state set and the measurement set at time k as Xk ¼ fxk;1 ; xk;2 ; . . . ; xk;Nk g and Yk ¼ fyk;1 ; yk;2 ; . . . ; yk;Mk g. Both Nk and Mk are random integers and they are the number of targets and measurements, respectively. Suppose Xk1 is the multiple target state set at time k 1, then Xk and Yk can be expressed as ! ! [ [ [ [ Xk ¼ Skjk1 ðxÞ Bkjk1 ðxÞ Ck ð13Þ
245
x2Xk1
Yk ¼ Kk
[
x2Xk1
[
! Hk ðxÞ
246 247 248 249 250 251
253 254
ð14Þ
x2Xk
ð10Þ
212
where Skjk1 ðxÞ is the RFS of targets surviving from time k 1 to k, Bkjk1 ðxÞ is the RFS of targets spawned from Xk1 and Ck is the RFS of targets that appear spontaneously at time k. Hk ðxÞ and Kk are the RFSs of measurements originating from the targets in Xk and the clutters. The optimal Bayesian recursions for propagating the multitarget posterior PDF are7 Z pkjk1 ðXk jY1:k1 Þ ¼ fkjk1 ðXk jXÞpk1 ðXjY1:k1 Þls ðdXÞ ð15Þ
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
256 257 258 259 260 261 262 263 264 266
CJA 716 26 October 2016
No. of Pages 9
4
J. Yang et al.
267
269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290
pkjk ðXk jY1:k Þ ¼ R
ð16Þ
where ls denotes the approximate state space Lebesgue measure, pkjk1 ðXk jY1:k1 Þ and pkjk ðXk jY1:k Þ are the predicted PDF and the posterior PDF, respectively. fkjk1 ðÞ is the state transition PDF and gk ðÞ is the measurement likelihood function. The PHD filter proposed by Mahler7 yields an approximation of the optimal Bayesian filter given in Eqs. (15) and (16) via propagating only the first-order moment of pkjk ðXk jY1:k Þ, i.e., the PHD. It is capable of tracking a variable number of targets and estimating both the number of targets and their states without utilizing data association techniques. The PHD is a multi-peak function in the state space. The number of peaks is often (but not necessarily) approximately equal to the number of targets, and the peak positions correspond to the expected values of target states, which can be extracted through the use of the expectation-maximum (EM) algorithm35,36 or clustering techniques.9,37 Let vkjk1 ðxÞ and vkjk ðxÞ denote the predicted and posterior intensity functions of pkjk ðXk jY1:k Þ. Their prediction and update equations are Z 0
vkjk1 ðxÞ ¼ 292
gk ðYk jXk Þpkjk1 ðXk jY1:k1 Þ gk ðYk jXÞpkjk1 ðXjY1:k1 Þls ðdXÞ
0
ðpS;kjk1 ðx Þfkjk1 ðxjx Þ 0
0
0
þ bkjk1 ðxjx ÞÞvk1jk1 ðx Þdðx Þ þ ck ðxÞ
ð17Þ
293
295
vkjk ðxÞ ¼ ½1 pD;k ðxÞvkjk1 ðxÞ X pD;k ðxÞgk ðyjxÞvkjk1 ðxÞ R þ j ðyÞ þ pD;k ðxÞgk ðyjxÞvkjk1 ðxÞdx y2Yk k
ð18Þ
When the latest measurements become available at time k, the joint posterior PHD becomes vkjk ðx; UÞ ¼ ð1 pD;k Þvkjk1 ðx; UÞ X p v ðx; UjyÞ RR D;k D;k þ j ðyÞ þ pD;k vD;k ðx0 ; U0 jyÞdðx0 ÞdðU0 Þ y2Yk k
vD;k ðx; UjyÞ ¼ gk ðyjx; UÞvkjk1 ðx; UÞ
ðiÞ ðiÞ ðiÞ ðiÞ L0 fx0 ; h0 ; n0 ; w0 gi¼1
are drawn from p0 ðX0 ; h0 ; n0 Þ,
304
We shall first generalize the PHD recursions in Eqs. (17) and (18) to take into account the presence of unknown model parameters. To simplify the presentation, it is assumed that the survival and the detection probabilities are independent of both the target state vector and the unknown parameter vector Uk . They will thus be denoted by pS;kjk1 and pD;k . Further drop the subscript k in Uk for notation simplicity and let vk1 ðx; UÞ be the joint posterior PHD at time k 1. According to Eq. (17), Eq. (18) and the Chapman-Kolmogorov equation, the predicted PHD ZZ vkjk1 ðx; UÞ can then be described as
305 306 307 308 309 310 311 312 313 314
vkjk1 ðx; UÞ ¼
ðpS;kjk1 fkjk1 ðx; Ujx0 ; U0 Þ
þ bkjk1 ðx; Ujx0 ; U0 ÞÞvk1 ðx0 ; U0 Þdðx0 ÞdðU0 Þ ZZ ðpS;kjk1 fkjk1 ðxjx0 ; U0 Þ þ ck ðx; UÞ ¼ 0
0
0
þ bkjk1 ðxjx ; U ÞÞpkjk1 ðUjU Þ 316
vk1 ðx0 ; U0 Þdðx0 ÞdðU0 Þ þ ck ðx; UÞ
ð19Þ
ðiÞ
and the weight is set to be w0 ¼ 1=N. For k P 1, the PFAPE-PHD recursions are as follows. 3.2.1. Prediction stage
329 330 331 332 333
ðiÞ ðiÞ Denote ¼ ðak1 ; bk1 Þ as the sufficient statistic for the staðiÞ tic parameter particle hk1 . For i ¼ 1; 2; . . . ; Lk1 , the parame-
ðiÞ pðhjsk1 Þ
336 337 338 339 340 341 342 343 344 345 346
347
ðiÞ sk1
ðiÞ hk1
3.1. APE-PHD recursions
328
335
ter particles can be obtained by
303
327
In this subsection, the PF is utilized to derive an approximation of the closed-form solution to the extended PHD recursions in Eqs. (19) and (20). The obtained PF-APE-PHD algorithm consists of two stages, namely the prediction and update stages. We assume that initially, there are N0 targets and N particles are produced for each target. The total number of particles is therefore L0 ¼ N N0 . Let X0 be the initial multiple target state set and p0 ðX0 ; h0 ; n0 Þ be the prior joint PDF. The initial
3. Tracking multiple maneuvering targets
300
326
334
302
299
ð21Þ
322 323 325
3.2. PF-APE-PHD algorithm
301
298
321
Note that in Eqs. (19)–(21), because U is unknown, the measurement likelihood gk ðyjx; UÞ and vkjk1 ðx; UÞ are hard to be obtained, this makes it difficult to calculate the analytic solution of the joint intensity function vD;k ðx; UjyÞ. However, its approximation solution can be obtained through the use of the APE technique combined with the PF. The proposed algorithm is therefore referred to as PF-APE-PHD algorithm, which will be presented in the following subsection.
particles
297
ð20Þ where
where bkjk1 ðxÞ and ck ðxÞ are the intensities of the RFSs of the spawned targets and spontaneous births. pS;kjk1 ðxÞ denotes the survival probability and pD;k ðxÞ is the detection probability. jk ðyÞ ¼ kk ck ðyÞ is the intensity of the clutter RFS, which is assumed to be Poisson distributed with mean rate kk , and ck ðyÞ is the distribution of the clutter.
296
317 318 319
348 349 350 351
ð22Þ
In this work, we consider the case that the static unknown parameter hk is the variance of the measurement noise. Its conjugate prior is approximated by an inverse-gamma (IG) distribution IGða; bÞ with parameters a and b, i.e., IGðh; a; bÞ ¼ R1 ba ha1 exp bh , where CðaÞ ¼ 0 ta1 exp ðtÞdt.18 The CðaÞ method for estimating a and b is similar to the method20 used to identify the unknown measurement noise covariance R. Therefore, the details are omitted here. To account for the possible abrupt changes in the timevarying parameters, we evaluate Eqs. (9) and (10) to obtain ðiÞ the estimate of the means mk1 and covariance Vk1 for the ðiÞ time-varying parameter particle nk1 . It is noted that P ðiÞ ðiÞ nk1 ¼ N i¼1 xk1 nk1 of Eqs. (9) and (10) in this algorithm is ðiÞ the Monte Carlo posterior mean of all nk1 which belong to the same target cluster. The particle clusters are formed in the stage of state extraction presented later in this subsection. We then generate 2Lk1 particles as in the APE filter. For ðiÞ ðiÞ this purpose, the proposal distributions qk ðjxk1 ; hk1 ;
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371
CJA 716 26 October 2016
No. of Pages 9
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets
5 427
372 373 374 375 376
ðiÞ
ðiÞ
ðiÞ
nk1 ; Yk Þ and pk ðjhk1 ; nk1 ; Yk Þ used to produce predicted particles as in Refs. 7, 14 are employed here. The first Lk1 particles are generated under the condition that the time-varying parameters do not change abruptly. They are obtained via k1ðiÞ
ðiÞ
ðiÞ
ðiÞ
378
xkj
qk ðxk jxk1 ; hk1 ; mk1 ; Yk Þ
379
Their weights are equal to
ð23Þ
Relabel the selected particles with indices i ¼ 1; 2; . . . ; Lk1 , ðiÞ xkjk1
ðli Þ xkjk1
ðiÞ wkjk1
ðli Þ wkjk1 .
¼ with ¼ Sample Jk new-born particles with indices i ¼ Lk1 þ 1; Lk1 þ 2; . . . ; Lk1 þ Jk from the ðiÞ ðiÞ proposal distribution pk ðxk jh0 ; n0 ; Yk Þ via7,14 k1ðiÞ
xkj
ðiÞ
ðiÞ
382
ðiÞ
i ¼ Lk1 þ 1;
ðiÞ
384 385 386
ði;jÞ
389 390 391 392 393 394 395 396 398
ð24Þ ðiÞ
ðiÞ
ðiÞ
where i ¼ 1; 2; . . . ; Lk1 and /ðxkjk1 ; xk1 ; hk1 ; mk1 Þ is the transitional PDF. At time k, each particle is given another weight proportional to the predictive likelihood corresponding ðiÞ to no changepoint parameter mk1 , i.e.,
387
x1
ðiÞ ðiÞ ðiÞ / p Yk jxkjk1 ; hk1 ; mk1
ð25Þ
The remaining Lk1 particles are produced under the assumption that abrupt changes occurred. As in the APE technique, the values of the time-varying parameters are now drawn from ðiÞ their prior distributions, i.e., ck1 pn ðn0 Þ. The predicted particles and their weights are obtained via, for i ¼ Lk1 þ 1; Lk1 þ 2; . . . ; 2Lk1 , k1ðiÞ xkj
ðiL Þ ðiL Þ ðiL Þ qk ðxk jxk1 k1 ; hk1 k1 ; ck1 k1 ; Yk Þ
ð26Þ
399 k1ðiÞ wkj
401 402 403 404 405 407
¼
ðiL Þ ðiL Þ ðiL Þ ðiL Þ /ðxkjk1k1 ; xk1 k1 ; hk1 k1 ; ck1 k1 Þ
ðiÞ wk1
Lk1 þ 2; . . . ; Lk1 þ Jk
ð29Þ
At time k, each particle is also given another weight proportional to the predictive likelihood corresponding to changeðiÞ point parameter ck1 , i.e., ðiÞ ðiL Þ ðiL Þ ðiL Þ ð28Þ x2 / p Yk jxkjk1k1 ; hk1 k1 ; ck1 k1
411
(1) For i ¼ 1; 2; . . . ; Lk1 , select indices li with probability
412 413 414 415 416 417 418 419 420 421 422 423 424 425 426
ðli Þ
ðli Þ
ð1 bÞx1 from ½1; 2; . . . ; Lk1 and bx2 from ½Lk1 þ 1; . . . ; 2Lk1 , where b is the probability that an abrupt change occurred and it is assumed to be known (see also Section 2.2). (2) If li 2 f1; 2; :::; Lk1 g, then update the time-varying i
ðiÞ
ðl Þ
parameter particles using nk Nðjmk1 ; h2 Vk1 Þ, where V k1 is given in Eq. (10). Set the composite parameter ðiÞ
ðiÞ
ðiÞ T
ðli Þ
ðiÞ
particle as Uk ¼ ½hk1 ; nk with hk1 ¼ hk1 and the sufficient statistics for the static parameters as ðli Þ sk1 .
ðiÞ sk1 i
¼ (3) If l 2 fLk1 þ 1; Lk1 þ 2; . . . ; 2Lk1 g, then set the timeðiÞ
i
ðl Þ
varying parameter particles to be nk ¼ ck . The composite parameter particle and the sufficient statistics ðiÞ
remain to be denoted by Uk ¼ ðiÞ hk1
¼
ðli Þ hk1
and
ðiÞ sk1
¼
ðli Þ sk1 .
ðiÞ
ðiÞ
wkjk1
ðiÞ
ðiÞ
1 ck ðxk ; h0 ; n0 Þ ; ¼ Jk p ðxðiÞ hðiÞ ; nðiÞ ; Y Þ k k 0 0 k
i ¼ Lk1 þ 1;
Lk1 þ 2; . . . ; Lk1 þ Jk
ð30Þ
3.2.2. Update stage
h ðiÞ ^ðiÞ w k ¼ ð1 PD;k ðxk ÞÞ
# ðiÞ ðiÞ ðiÞ ðiÞ pD;k ðxk Þgk ðyjxk ;hk1 ;nk1 Þ k1ðiÞ ð31Þ wkj P ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ y2Yk kk ck ðyÞ þ y2Yk pD;k ðxk Þgk ðyjxk ;hk1 ;nk1 Þwkjk1
þ
ðiÞ ðiÞ T ½hk1 ; nk
437
438
After receiving the measurement at time k, the Lk1 þ Jk particle weights can be updated by
439 440 441
X
3.2.3. Computation of the total mass
443
444 445 446
Nk ¼
Lk1 þJk X
^ðiÞ w k
ð32Þ 448
3.2.4. Resampling
410
409
434
ð27Þ
We select Lk1 out of the 2Lk1 obtained particles. Denote their indices as li 2 f1; 2; . . . ; 2Lk1 g, where i ¼ 1; 2; . . . ; Lk1 , the selection process is as follows.
408
431 432
ðiÞ
i¼1
ðiLk1 Þ ðiLk1 Þ ðiL Þ ðiL Þ xk1 ; hk1 k1 ; ck1 k1 ; Yk Þ
qk ðxk
430
435
ðiÞ
/ðxkjk1 ; xk1 ; hk1 ; mk1 Þ ðiÞ wk1 ¼ ðiÞ ðiÞ ðiÞ ðiÞ qk ðxk xk1 ; hk1 ; mk1 ; Yk Þ ðiÞ
383
429
pk ðxk jh0 ; n0 ; Yk Þ;
380 ðiÞ wkjk1
428
with
ðiÞ
ðiÞ
449 ðiÞ
ðiÞ
Lk1 þJk
^k =Nk gi¼1 Resample fxk ; hk ; nk ; w
through the weights to
Lk ðiÞ ðiÞ ðiÞ ðiÞ fxk ; hk ; nk ; wk =Nk gi¼1 .
450
obtain a new particle set Each particle is assigned the same weight Nk =Lk after resampling, where Lk ¼ Lk1 þ Jk .
451
3.2.5. Extraction of target states
454
Target states can be obtained by clustering the particles and
452 453
455
^
Nk ^ k ¼ f^ the cluster centers are the estimated states X xk;j gj¼1 , where ^ Nk ¼ roundðNk Þ is the estimate of the target number, and roundðÞ denotes the rounding operator.
456 457 458
4. Simulations
459
In order to illustrate the performance of the proposed PFAPE-PHD algorithm, a two-dimensional tracking scenario is simulated. The benchmark techniques are the MMP-PHD,28 MMP-CPHD and MMP-CBMeMBer filters.29 In the considered scenario, the measurements are obtained at four stationary sensors located at ð0; 0Þ m, ð0; 1 104 Þ m, ð1 104 ; 0Þ m, and ð1 104 ; 1 104 Þ m. At time k, each sensor outputs the measured bearing of the received signal, which is given by yk ySi þ wk ySk i ¼ tan1 ð33Þ xk xSi
460
where ðxSi ; ySi Þ denotes the location of the ith sensor, i ¼ 1; 2; 3; 4. wk is the zero-mean Gaussian noise with variance r2w ¼ 1 104 rad2 .
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
461 462 463 464 465 466 467 468 470 471 472 473
CJA 716 26 October 2016
No. of Pages 9
6 474 475 476 477 478 479 480 481 482 483 484 485
487 488 489 490 491 492
J. Yang et al.
There are three maneuvering targets. Targets 1 and 2 remain active throughout the whole simulation process and their initial positions are at ð3 103 ; 5 103 Þ m and ð1:4 104 ; 8 103 Þ m, as in Ref. 28. Target 3 is a spontaneous birth at 10th min with initial position ð2 103 ; 10:5 103 Þ m and disappears at 50th min. The true tracks of the three targets are depicted in Fig. 1. When realizing the MMP-PHD and MMP-CPHD algorithms, we set that they both consist of a constant velocity (CV) model and two CT models. The transition probability matrix is assumed to be 2 3 T T 1 sT1 2s1 2s1 6 T 7 T 7 1 sT2 ½hab ¼ 6 ð34Þ 2s2 5 4 2s2 T T T 1 s2 2s2 2s2 where the sampling interval is T ¼ 1 min, and the sojourn durations are s1 ¼ 200 min and s2 ¼ 100 min. The initial model probabilities for the three models are all equal to 1=3. The state evolution for CV and CT models is 2 3 1 T 0 0 60 1 0 0 7 6 7 i xik ¼ 6 ð35Þ 7x þ vik 4 0 0 1 T 5 k1 0
494
0
0
1
495 497 498 499 500 501 502 503
xik
¼
FðxÞxik1
507 508 509
0
0
T =2 2
T
where r2v ¼ 1 104 m2 s3 . For the three algorithms in consideration, we model the birth process using a Poisson RFS with intensity ðiÞ
511
ð36Þ
where xik ¼ ½xik ; x_ ik ; zik ; z_ik is the state vector of the ith target. FðxÞ is the state transition matrix of the CT model (see Eq. (3) for its definition). We set the turn rates of the two CT models to be x ¼ 9 =min, and vik is a zero-mean white Gaussian process noise with covariance 2 3 3 0 0 T =3 T2 =2 6 T2 =2 T 0 0 7 6 7 2 Q¼6 ð37Þ 7rv 3 2 4 0 0 T =3 T =2 5
505 506
þ
vik
Ck ðxÞ ¼
3 X
ðiÞ
ðiÞ
0:2Nðx; mC ; PC Þ;
i ¼ 1; 2; 3
i¼1
Fig. 1
ð38Þ
ð1Þ
ð2Þ
where mC ¼ ð3 103 m; 0 m=s; 5 103 m; 0 m=sÞ, mC ¼ ð3Þ ð1:4 104 m; 0 m=s; 8 103 m; 0 m=sÞ, mC ¼ ð2 103 m; ð1Þ ð2Þ ð3Þ 3 0 m=s; 10:5 10 m; 0 m=sÞ, and PC ¼ PC ¼ PC ¼ diag ð400; 1; 400; 1Þ. The clutter is modeled as a Poisson RFS with the mean rate r = 10 over the observation space. The probabilities of the target survival and detection are pS;k ¼ 0:99 and pD;k ¼ 0:98, respectively. The initial parameters of the inverse Gamma distribution are set as a ¼ b ¼ 1.17,20 At each time, a maximum 1500 and minimum 300 particles perhypothesized track are imposed so that the number of particles representing each hypothesized track is proportional to its existence probability after resampling in the update step. To verify the effectiveness of the proposed algorithm, simulations are performed on a Lenovo T430 desktop with Intel (R) Core(TM) CPU i5-3210 M, 2.50 GHz and 8 GB RAM. Two performance metrics are used. One is the statistics of the target number estimate. The other is the optimal subpattern assignment (OSPA)38 distance defined as !!1=p m p X 1 ðcÞ ðcÞ p min d xi ; ypðiÞ þ c ðn mÞ dp ðX; YÞ ¼ n p2Pn i¼1 ð39Þ where X ¼ fx1 ; x2 ; ; xm g and Y ¼ fy1 ; y2 ; ; yn g are arbitrary finite subsets, 1 6 p < 1, c > 0, m; n 2 No ¼ ðcÞ f0; 1; 2; . . .g. If m > n, dðcÞ p ðX; YÞ ¼ dp ðY; XÞ. In the simulation, the parameters of OSPA distance are set to be p ¼ 2 and c ¼ 1000. Three simulation experiments are performed and the results shown are obtained from Monte Carlo simulations of 200 ensemble runs. The first experiment is to evaluate the performance for multiple abruptly maneuvering target tracking, where only the maneuvering parameters (e.g., turn rates) are unknown. The second experiment is to compare the performance of the proposed algorithm in the presence of unknown maneuvering parameters as well as unknown measurement noise variances. The last experiment is conducted using different measurement noise variances to evaluate the robustness of the proposed algorithm.
512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530
532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
4.1. Multiple abruptly maneuvering target tracking
549
In this experiment, the standard deviation of the measurement noise is set to be r ¼ 0:01 rad and it is assumed known for the considered PF-APE-PHD, MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms. The turn rate x is considered as an unknown and time-varying parameter for the proposed PF-APE-PHD algorithm. The MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms use one CV and two CT models of Eqs. (35) and (36) as the target motion models. Although in practice, the true turn rates are unavailable for the IMM-based filters, we realize the CT models with the real turn rates x ¼ 9 =min and x ¼ 9 =min so that the MMPbased methods would have the ‘optimal’ performance. Simulation results for this experiment are shown in Figs. 2–4. Fig. 2 shows the average target number estimates obtained by the PF-APE-PHD, MMP-PHD, MMP-CPHD and MMPCBMeMBer filters. It can be seen that the proposed PF-APEPHD algorithm can even provide more accurate target number estimates than the benchmark techniques. The reason is that
550
True target tracks.
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567
CJA 716 26 October 2016
No. of Pages 9
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607
608 609
610 611
the proposed algorithm can effectively estimate jointly the unknown model parameter x which can be well matched with the motion model of each target. While for the MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms, the tracking accuracy is affected by the model interference due to the interaction of multiple models, an inevitable phenomenon of IMM-based techniques, which renders their performance under ‘optimal’ parameter settings still inferior to the proposed technique. Moreover, it is noticed that the MMP-CPHD and MMP-CBMeMBer algorithms have better performance in terms of more precise target number estimates than the MMP-PHD algorithm. The reason is that the MMPCBMeMBer method propagates the parameterized approximation of the posterior cardinality distribution, and the MMP-CPHD method jointly propagates the cardinality distribution and the intensity function, whereas the MMP-PHD method propagates the cardinality mean only with a single Poisson parameter. Fig. 3 compares the OSPA distances of the four simulated algorithms, and it is clear that the proposed algorithm again outperforms the MMP-PHD, MMP-CPHD and MMPCBMeMBer algorithms. This is also due to the fact that the proposed method can adapt to the temporal evolution of the target maneuvering parameters. It is worth noting that when the third target disappears at 50th min, the OSPA distance of the MMP-CPHD algorithm increases suddenly, which indicates that the ‘spooky action’ problem steps in, i.e., it is beneficial when missed detection occurs in MMP-PHD but is harmful when targets really disappear. Fig. 4 shows the average run time of the four algorithms in consideration. It can be seen that the average run time of the proposed PF-APE-PHD algorithm is slightly larger than that of the MMP-PHD algorithm. The reason is that with the APE technique, the proposed algorithm generates twice more particles with different parameter predictions in the predicted step and has an additional particle selection step. It is noted that the complexity of the MMP-CBMeMBer is slightly lower than that of the MMP-PHD algorithm, because this method allows reliable and inexpensive extraction of state estimates without particle clustering. 4.2. Multiple abruptly maneuvering target tracking with unknown measurement noise variance In this experiment, the true standard deviation of the measurement noise is fixed at r ¼ 0:01 rad, but it is unknown for the
Fig. 2
Target number estimates.
Fig. 3
7
OSPA distance statistics.
Fig. 4
Average run time.
proposed algorithm. We then apply the PF-APE-PHD algorithm to identifying it together with the time-varying turn rate x and the target states. For comparison purposes, we also simulate the PF-APE-PHD filters with other assumed values of the measurement noise variance (i.e., r = 0.005, 0.01, 0.015, 0.03, 0.06, 0.1). The simulation results are summarized in Figs. 5 and 6. It is clear that when the measurement noise variance is estimated jointly with the target states, the performance of the PFAPE-PHD algorithm is very close to that when the measurement noise variance is accurately known in advance (r ¼ 0:01 rad). This indicates that the proposed PF-APEPHD algorithm can achieve accurate joint parameter and target state estimations. On the other hand, if PF-APE-PHD simply operates with an incorrect setting of the measurement noise variance, it would suffer from significant performance degradation, mainly due to model mismatch.
612
4.3. Performance with different measurement noise variance settings
629
In this experiment, we realize two versions of the PF-APEPHD filter, one filter with unknown measurement noise variance and another filter with true measurement noise variance. The simulation results with different measurement noise standard deviations (i.e., r = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07) are shown in Fig. 7. As can be seen the estimation accuracy of the PF-APE-PHD algorithm with unknown r is close to that of the PF-APE-PHD algorithm with the true value of r known in advance. It is shown that the proposed algorithm has a good performance for multiple target tracking with
631
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628
630
632 633 634 635 636 637 638 639 640
CJA 716 26 October 2016
No. of Pages 9
8
J. Yang et al.
Fig. 5
Target number estimates with different measurement noise standard deviations.
Fig. 6
OSPA distance statistics with different measurement noise standard deviations.
Fig. 7 Average OSPA distance statistics with different measurement noise standard deviations.
642
unknown measurement noise parameters and the time-varying abruptly changing maneuver parameters.
643
5. Conclusions
644
In this paper, we developed a new MTT algorithm, the PFAPE-PHD filter, to handle the presence of unknown model parameters including e.g., the measurement noise variance that
641
645 646
is static and the parameters in accordance with the target maneuvers that may be time-varying and subject to abrupt changes. The development started with extending the PHD filter to take into account the unknown parameters and the APE technique was incorporated to achieve online parameter estimation. The SMC approach was utilized to derive the approximate closed-form solution. Simulations showed that the newly proposed PF-APE-PHD filter can offer higher tracking accuracy in the case of multiple maneuvering targets over the existing MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms. It is also applicable to the case with unknown measurement noise parameters for multiple maneuvering target tracking. In future works, we shall consider introducing the APE technique into the spline PHD filter,39 the CPHD filter8 and the CBMeMBer filter40,41 to obtain good algorithms for tracking multiple targets with unknown abrupt changing parameters.
647
Acknowledgments
665
This paper is supported by the National Natural Science Foundation of China (Nos. 61305017, 61304264) and the Natural Science Foundation of Jiangsu Province (No. BK20130154).
666
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664
667 668 669
CJA 716 26 October 2016
No. of Pages 9
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets 670
671 672 673
674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732
Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cja.2016. 09.010. References 1. Fortmann T, Bar-Shalom Y, Scheffe M. Sonar tracking of multiple targets using joint probabilistic data association. IEEE J Ocean Eng 1983;8(3):173–84. 2. Musicki D, Evans R. Joint integrated probabilistic data association: JIPD.. IEEE Trans Aerosp Electron Syst 2004;40(3):1093–9. 3. Blackman S. Multiple hypothesis tracking for multiple target tracking. IEEE Aerosp Electron Syst Mag 2004;19(1):5–18. 4. Roecker J, Pillis G. Suboptimal joint probabilistic data association. IEEE Trans Aerosp Electron Syst 1993;29(2):504–10. 5. Roecker J. A class of near optimal JPDA algorithms. IEEE Trans Aerosp Electron Syst 1994;30(2):504–10. 6. Purank S, Tugnait JK. Tracking of multiple maneuvering targets using multiscan JPDA and IMM filter. IEEE Trans Aerosp Electron Syst 2007;43(1):23–34. 7. Mahler R. Multitarget Bayes filtering via first-order multitarget moments. IEEE Trans Aerosp Electron Syst 2003;29(4):1152–78. 8. Vo BT, Vo BN, Cantoni A. Analytic implementations of the cardinalized probability hypothesis density filter. IEEE Trans Signal Process 2007;55(7):3553–67. 9. Vo BN, Singh S, Doucet A. Sequential Monte Carlo methods for multi-target filtering with random finite sets. IEEE Trans Aerosp Electron Syst 2005;41(4):1224–45. 10. Whiteley N, Singh S, Godsil S. Auxiliary particle implementation of probability hypothesis density filter. IEEE Trans Aerosp Electron Syst 2010;46(3):1437–54. 11. Vo BN, Ma WK. The Gaussian mixture probability hypothesis density filter. IEEE Trans Signal Process 2006;54(11):4091–104. 12. Clark D, Vo BT, Vo BN. Gaussian particle implementations of probability hypothesis density filtersProceedings of IEEE aerospace conference, 2007 March 3-10, Big Sky, MT, USA. Piscataway, NJ: IEEE Press; 2007. p. 1–11. 13. Braca P, Marano S, Matta V. Asymptotic efficiency of the PHD in multitarget/multisensor estimation. IEEE J Sel Top Signal Process 2013;7(3):553–64. 14. Yang JL, Ji HB. A novel track maintenance algorithm for PHD/ CPHD filter. Signal Process 2012;92(10):2371–80. 15. Beard M, Vo BT, Vo BN, Arulampalam S. A partially uniform target birth model for Gaussian mixture PHD/CPHD filtering. IEEE Trans Aerosp Electron Syst 2013;49(4):2835–44. 16. Vo BT, Vo BN, Cantoni A. The Cardinality balanced multi-target multi-Bernoulli filter and its implementations. IEEE Trans Signal Process 2009;57(2):409–23. 17. Sa¨rkka¨ S, Nummenmaa A. Recursive noise adaptive Kalman filtering by variational Bayesian approximations. IEEE Trans Autom Control 2009;54(3):596–600. 18. Li WL, Jia YM. State estimation for jump Markov linear systems by variational Bayesian approximation. IET Control Theory Appl 2012;6(2):319–26. 19. Gao XB, Chen JG, Tao DC. Multi-sensor centralized fusion without measurement noise covariance by variational Bayesian approximation. IEEE Trans Aerosp Electron Systs 2011;47 (1):718–27. 20. Yang JL, Ge HW. Adaptive probability hypothesis density filter based on variational Bayesian approximation for multi-target tracking. IET Radar, Sonar Navigat 2013;7(9):959–67. 21. Zhang GH, Lian F, Han CZ, Han SY. An improved PHD filter based on variational Bayesian method for multi-target trackingProceedings of the 17th international conference information
22.
23.
24.
25.
26.
27.
28.
29.
30. 31.
32. 33.
34. 35.
36.
37. 38.
39.
40.
41.
9
fusion, 2014 July 7–10, Salamanca, Spain. Piscataway, NJ: IEEE Press; 2014. p. 1–6. Hernandez ML, Ristic B, Farina A. Performance measure for Markovian switching systems using best-fitting Gaussian distributions. IEEE Trans Aerosp Electron Syst 2008;44(2):724–47. Li XR, Jilikov VP. Survey of maneuvering target tracking. Part V: multiple-model methods. IEEE Trans Aerosp Electron Syst 2005;41(4):1255–321. Pasha A, Vo BN, Tuan HD. Closed-form PHD filtering for linear jump Markov modelsProceedings of the int. conf. on information fusion; 2016 July 10–13, Florence, Italy. IEEE; 2006. p. 1–8. Pasha SA, Vo BN, Tuan HD. A, Gaussian mixture PHD filter for jump Markov system models. IEEE Trans Aerosp Electron Syst 2009;45(3):919–36. Pasha SA, Tuan HD, Apkarian P. The LFT based PHD filter for nonlinear jump Markov models in multi-target trackingProceedings of the IEEE conference on decision and control, 2009 December 15– 18, Shanghai, China. Piscataway, NJ: IEEE Press; 2009. p. 5478–83. Li WL, Jia YM. Gaussian mixture PHD filter for jump Markov models based on best-fitting Gaussian approximation. Signal Process 2011;91(4):1036–42. Punithakumar K, Kirubarajan T, Sinha A. Multiple-model probability hypothesis density filter for tracking maneuvering targets. IEEE Trans Aerosp Electron Syst 2008;44(1):87–98. Yang JL, Ji HB, Ge HW. Multi-model particle cardinalitybalanced multi-target multi-Bernoulli algorithm for multiple manoeuvring target tracking. IET Radar, Sonar Navigat 2013;7 (2):101–12. Ouyang C, Ji HB, Guo ZQ. Extensions of the SMC-PHD filters for jump Markov systems. Signal Process 2012;92(6):1422–30. Liu J, West M. Combined parameter and state estimation in simulation-based filtering. In: Doucet A, Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer; 2001. p. 197–223. Carvalho C, Johannes M, Lopes H, Polson N. Particle learning and smoothing. Stat Sci 2010;25(1):88–106. Nemeth C, Fearnhead P, Mihaylova L. Sequential Monte Carlo methods for state and parameter estimation in abruptly changing environments. IEEE Trans Signal Process 2014;62(5):1245–55. West M. Approximating posterior distributions by mixture. J Roy Stat Soc 1993;55(2):409–22. Tobias M, Lanterman AD. Probability hypothesis density-based multi-target tracking with bistatic range and Doppler observation. IET Radar, Sonar Navigat 2005;152(3):195–205. Lian F, Han CZ, Liu WF. Estimating Unknown clutter intensity for PHD filter. IEEE Trans Aerosp Electron Syst 2010;46 (4):2066–78. Xu BL, Xu HG, Zhu JH. Ant clustering PHD filter for multipletarget tracking. Appl Soft Comput 2011;11:1074–86. Schuhmacher D, Vo BT, Vo BN. A consistent metric for performance evaluation of multi-object filters. IEEE Trans Signal Process 2008;56(8):3447–57. Sithiravel R, Chen X, Tharmarasa R. The spline probability hypothesis density filter. IEEE Trans Signal Process 2013;61 (24):6188–203. Vo BT, Vo BN, Cantoni A. The cardinality balanced multi-target multi-Bernoulli filter and its implementations. IEEE Trans Signal Process 2009;57(2):409–23. Beard MA, Vo BT, Vo BN. Bayesian multi-target tracking with merged measurements using labelled random finite sets. IEEE Trans Signal Process 2015;63(6):1433–47.
Yang Jinlong is an associate professor in Jiangnan University. He received his M.S. degree in circuit and system from Northwest Normal University, China in 2009, and his Ph.D. degree in Pattern Recognition and Intelligent System from Xidian University, China, in 2012. His research interests include target tracking, information fusion and signal processing.
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801
No. of Pages 9
1
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
2
FULL LENGTH ARTICLE
6
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets
7
Yang Jinlong a,b,*, Yang Le a, Yuan Yunhao a, Ge Hongwei a
4
5
8 9
10
a b
School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Wuxi 214122, China
Received 31 December 2015; revised 22 September 2016; accepted 22 September 2016
11
13 14
KEYWORDS
15
Adaptive parameter estimation; Multiple target tracking; Multivariate Gaussian distribution; Particle filter; Probability hypothesis density
16 17 18 19 20 21 22
23
Abstract The probability hypothesis density (PHD) filter has been recognized as a promising technique for tracking an unknown number of targets. The performance of the PHD filter, however, is sensitive to the available knowledge on model parameters such as the measurement noise variance and those associated with the changes in the maneuvering target trajectories. If these parameters are unknown in advance, the tracking performance may degrade greatly. To address this aspect, this paper proposes to incorporate the adaptive parameter estimation (APE) method in the PHD filter so that the model parameters, which may be static and/or time-varying, can be estimated jointly with target states. The resulting APE-PHD algorithm is implemented using the particle filter (PF), which leads to the PF-APE-PHD filter. Simulations show that the newly proposed algorithm can correctly identify the unknown measurement noise variances, and it is capable of tracking multiple maneuvering targets with abrupt changing parameters in a more robust manner, compared to the multi-model approaches. Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author at: School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China. E-mail address: [email protected] (J. Yang). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
1. Introduction
24
Multiple target tracking (MTT) has gained wide attentions due to its theoretical and practical importance. Conventionally, the MTT problem was tackled from the perspective of data association. A number of tracking algorithms were developed in the literature on the basis of techniques including the joint probabilistic data association (JPDA),1 joint integrated probabilistic data association (JIPDA)2 and multiple hypothesis tracking (MHT).3 These methods are generally computationally intensive and some of them even have exponentially growing complexity as the target number increases. Reduced-
25
http://dx.doi.org/10.1016/j.cja.2016.09.010 1000-9361 Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
26 27 28 29 30 31 32 33 34
CJA 716 26 October 2016
No. of Pages 9
2 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
complexity techniques were proposed in Refs. 4–6. They are better for real-time applications at the cost of degraded estimation accuracy. Recently, the use of the random finite set (RFS) theory7–11 attracted great interests, because it provides an elegant formulation of the MTT problem. But the obtained multi-target Bayesian filter is intractable in most practical scenarios due to the inherent combinatorial nature of multi-target state densities and the need for evaluating set integrals over high dimensional spaces. To deal with the intractability, the probability hypothesis density (PHD) filter7 and the cardinalized PHD (CPHD) filter8 were developed using the first-order moment and cardinality distributions. Existing closed-form realizations of PHD filters include the particle filter PHD (PF-PHD),9,10 Gaussian mixture PHD (GM-PHD) filter11 and various modified versions.12–15 Different from the PHD and CPHD filters, the cardinality-balanced multi-target multi-Bernoulli (CBMeMBer) filter was proposed in Ref. 16 for MTT by directly propagating the approximate posterior density of the targets. These algorithms exhibit good performance only when the model parameters, such as the measurement noise variances, are known precisely. In the presence of unknown time-varying measurement noise variances, the variational Bayesian (VB) approximation method17–19 can be employed to recursively estimate the joint PHDs of the multi-target states and the measurement noise variance.20,21 However, these methods may suffer from performance degradation if targets manoeuver with unknown abruptly changing parameters. For maneuvering target tracking, the use of the jump Markov system (JMS) that switches among a set of candidate models in a Markovian fashion has proved to be effective.22,23 Pasha et al.24 introduced the linear JMS into PHD filters and derived a closed-form solution for the PHD recursion. Furthermore, the unscented transform (UT) and the linear fractional transformation (LFT) were combined with the closed-form solution for the nonlinear jump Markov multitarget models in Refs. 25, 26 In Ref. 27, a GM-PHD filter for jump Markov models was developed by employing the best-fitting Gaussian (BFG) approximation approach. These algorithms assume the Gaussianity of the PHD distribution, which may limit their application scope. The multiple-model particle PHD (MMP-PHD) filter, the MMP-CPHD filter and MMP-CBMeMBer filter are implemented by using the sequential Monte Carlo (SMC) method and their improved versions were presented in Refs. 28–30 Most of the MM-based filters track multiple maneuvering targets through the interaction of multiple models, which is realized via combining estimates from different models according to their respective model likelihoods. The difficulty of applying them in tracking targets with abruptly changing maneuvering parameters comes from the need to specify a prior set of candidate models. In other words, they may suffer from the curse of dimensionality: if we wish to account for multiple unknown parameters, the number of models needed would increase exponentially with the number of parameters. In this work, we incorporate the adaptive parameter estimation (APE) technique into the PHD filter for addressing the problem of multiple maneuvering target tracking, where both static and time varying unknown parameters, namely the measurement noise variance and the parameters associated with abrupt target maneuvers, are presented and need to be estimated. The inverse Gamma (IG) distribution is used to
J. Yang et al. approximate the posterior distribution of the measurement noise variances while the adaptive Liu and West (LW) filter is adopted to propagate the posterior marginal of the timevarying parameters as a mixture of multivariate Gaussian distributions.31–33 The obtained APE-PHD filter is realized using the particle filter (PF), which leads to the PF-APE-PHD algorithm for tracking multiple maneuvering targets in the presence of unknown model parameters. Simulation results show that the proposed algorithm exhibits better robustness and improved tracking performance over the MM-PHD and MM-CPHD algorithms. The remainder of this paper is organized as follows. Section 2 formulates the problem of tracking a target in the presence of unknown model parameters. It also briefly reviews the APE technique and the PHD filter. Section 3 develops the APE-PHD algorithm and presents the closed-form solution, the PF-APE-PHD algorithm. Simulation results are given in Section 4. Finally, conclusions are provided in Section 5.
97
2. Preliminary
115
2.1. Problem formulation
116
The state-space model for tracking a single target moving on a two-dimensional plane is given by
117
98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114
118
xkþ1 ¼ Fxk þ Gvk
ð1Þ
119 121
yk ¼ hðxk Þ þ wk
ð2Þ
122 124
where xk ¼ ½xk ; vxk ; yk ; vyk T denotes the target state at time k, ðxk ; yk Þ and ðvxk ; vyk Þ denote its position and velocity. F and G are the state transition matrix and the process noise gain matrix. yk is the measurement vector. vk and wk denote the process noise and the measurement noise. They are independent of each other and modeled as zero-mean Gaussian random processes with covariance Qk and Rk . In many practical applications, the state-space model in Eqs. (1) and (2) may contain unknown parameters. For example, if the target conducts a coordinated turn (CT),28 the state transition matrix would become 2 xT 3 0 1cos 1 sinxxT x 6 0 cos xT 0 sin xT 7 6 7 FðxÞ ¼ 6 ð3Þ 7 xT sin xT 4 0 1cos 5 1 x x 0 sin xT
0 cos xT
pðyk jxk ; Uk Þpðxk ; Uk jy1:k1 Þ pðyk jxk ; Uk Þpðxk ; Uk jy1:k1 Þdxk dUk
126 127 128 129 130 131 132 133 134 135 136
138
The turn rate x may be unknown and time-varying. Besides, the measurement noise covariance Rk may also be unknown. In these scenarios, we need to jointly estimate the posterior distribution of the target states and the unknown parameters from the measurements. Let Uk be a column vector that collects the static and timevarying parameters in the state-space model. The posterior probability density function (PDF) of the target state vector xk and Uk conditioned on the measurements up to time k is, according to Bayes’ rule, pðxk ; Uk jy1:k Þ ¼ RR
125
ð4Þ
where pðxk ; Uk jy1:k1 Þ is the predicted PDF given by
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
139 140 141 142 143 144 145 146 147 148 149 151 152 153
CJA 716 26 October 2016
155 156 157 158 159 160
161
162 163 164 165 166 167 168 169 170 171 172 174 175 176 177 178 180 181 182 183 184 185
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets 3 ZZ mik1 ¼ nik1 , which results in a posterior distribution overly pðxk ; Uk jy1:k1 Þ ¼ pðxk jxk1 ; Uk1 ÞpðUk jUk1 Þ dispersed in the sense that the covariance of the mixture is larger than Vk1 . The shrinkage factor introduced in Eq. (9) can pðxk1 ; Uk1 jy1:k1 Þdxk1 dUk1 ð5Þ force particles nik1 to move closer to their sample mean nk1 so Deriving exact recursive solutions for the posterior distributhat maintaining the same covariance Vk1 is achieved. tion pðxk ; Uk jy1:k Þ from Eqs. (4) and (5) is in general intractable In the case that the time instant k is a changepoint, and the and as a result, approximate solutions are usually resorted to. predicting distribution of the time-varying vector nk will be One such approach is the SMC method, also referred to as the reset to pn ðn0 Þ, its prior distribution. With the predicted PDF particle filter (PF).9,11,14 given in Eq. (8), the APE filter utilizes the PF to produce an approximation of the posterior distribution pðxk ; Uk jy1:k Þ in 2.2. Adaptive parameter estimation (APE) Eq. (4). Suppose at time k 1, the posterior distribution is In Refs. 31, 32, the Liu and West (LW) filter was proposed for the joint identification of static parameters and target states. In particular, the marginal posterior distribution of the unknown parameters is approximated and propagated using a mixture of multivariate Gaussian distributions. In Ref. 33, the particle learning technique was introduced into the LW filter. The obtained APE filter can handle both static and time-varying parameters. The development of the APE method starts with factorizing pðxk ; Uk jy1:k1 Þ into pðxk ; Uk jy1:k1 Þ ¼ pðxk jy1:k1 ; Uk ÞpðUk jy1:k1 Þ
pðUk jy1:k1 Þ ¼ pðhk ; nk jy1:k1 Þ ¼ pðhk jy1:k1 ; nk Þpðnk jy1:k1 Þ
190 191 192 193 194 195 196 197 198 199 201
205 206 207 208 209 210 211
8 N X > < xi
2 i k1 Nðnk jmk1 ;h Vk1 Þ
> :
with probability 1 b
i¼1
pn ðn0 Þ
ð8Þ
with probability b
where Nðnk jmik1 ; h2 Vk1 Þ is a Gaussian kernel with mean mik1 and covariance h2 Vk1 , and h 2 ð0; 1Þ denotes a scaling parameter that shrinks the kernel. xik1 is the weight of the ith com-
ponent. Here, b can model the temporal evolution of nk . It is defined as the probability that nk is subject to an abrupt change at time k, or equivalently speaking, time k is a changepoint.33 The time-varying vector nk is assumed to be piecewise constant between two neighboring changepoints. As defined in Eq. (8), if there is no abrupt change in nk , its predicted PDF follows a Gaussian mixture model of N components. The mean and covariance of each components are obtained by mik1 ¼ anik1 þ ð1 aÞnk1
202
204
ð7Þ
The predicted distribution pðhk jy1:k1 ; nk Þ of the static parameter vector hk is characterized using sufficient statistics sk , i.e., hk pðhjsk Þ.32 The predicted distribution of the time-varying parameter vector nk is approximated via
187
189
ð6Þ
Let Uk ¼ ðhk ; nk Þ, where hk and nk collect static and timevarying parameter vectors. The marginal predicting distribution of Uk can be expressed as
pðnk jy1:k1 Þ
188
No. of Pages 9
Vk1 ¼
N X
T xik1 ðnik1 nk1 Þðnik1 nk1 Þ
ð9Þ
N fxik1 ; hik1 ; nik1 gi¼1
represented by N particles with weights xik1 . At time k, each particle is given two weights33
i¼1
P i i where nk1 ¼ N i¼1 xk1 nk1 is the minimum mean square error (MMSE) estimate of nk1 at time k 1, and nik1 denotes the ith Gaussian component of the time-varying vector nk1 . pffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ 1 h2 is the shrinkage factor suggested in Ref. 34 to correct for the over-dispersion of the Gaussian mixture model Eq. (8). It is noted that standard kernel smoothing requires that kernel components be centered around the mean vectors
213 214 215 216 217 218 219 220 221 222 223 224 225
xik;1 / xik1 pðyk jlik ; hik1 ; nik Þ; nik Nðnk jmik1 ; h2 Vk1 Þ;
where
lik ¼ E½xik jxik1 ; hik1 ; nik
ð11Þ
227 228
xik;2 / xik1 pðyk jlik ; hik1 ; cik Þ;
where cik pn ðn0 Þ;
lik ¼ E½xik jxik1 ; hik1 ; cik
ð12Þ
which essentially leads to 2N particles. xik;1 and xik;2 correspond to the probability of the current measurement yk when there is no changepoint and when there is a changepoint, respectively. In the former case, the value of time-varying parameter vector nik is drawn from the Gaussian component Nðnk jmik1 ; h2 Vk1 Þ, while for the latter case, its value cik is produced using the prior distribution pn ðn0 Þ (see also Eq. (8)). A resampling is then performed on the basis of the weights ð1 bÞxik;1 and bxik;2 to select N particles out of 2N particles and propagate them to generate the approximation of the posterior pðxk ; Uk jy1:k Þ at time k. For more details on the APE filter for tracking a single maneuvering target, please refer to Ref. 33.
230 231 232 233 234 235 236 237 238 239 240 241 242 243
2.3. PHD filter
244
Under the RFS framework, we denote the multiple target state set and the measurement set at time k as Xk ¼ fxk;1 ; xk;2 ; . . . ; xk;Nk g and Yk ¼ fyk;1 ; yk;2 ; . . . ; yk;Mk g. Both Nk and Mk are random integers and they are the number of targets and measurements, respectively. Suppose Xk1 is the multiple target state set at time k 1, then Xk and Yk can be expressed as ! ! [ [ [ [ Xk ¼ Skjk1 ðxÞ Bkjk1 ðxÞ Ck ð13Þ
245
x2Xk1
Yk ¼ Kk
[
x2Xk1
[
! Hk ðxÞ
246 247 248 249 250 251
253 254
ð14Þ
x2Xk
ð10Þ
212
where Skjk1 ðxÞ is the RFS of targets surviving from time k 1 to k, Bkjk1 ðxÞ is the RFS of targets spawned from Xk1 and Ck is the RFS of targets that appear spontaneously at time k. Hk ðxÞ and Kk are the RFSs of measurements originating from the targets in Xk and the clutters. The optimal Bayesian recursions for propagating the multitarget posterior PDF are7 Z pkjk1 ðXk jY1:k1 Þ ¼ fkjk1 ðXk jXÞpk1 ðXjY1:k1 Þls ðdXÞ ð15Þ
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
256 257 258 259 260 261 262 263 264 266
CJA 716 26 October 2016
No. of Pages 9
4
J. Yang et al.
267
269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290
pkjk ðXk jY1:k Þ ¼ R
ð16Þ
where ls denotes the approximate state space Lebesgue measure, pkjk1 ðXk jY1:k1 Þ and pkjk ðXk jY1:k Þ are the predicted PDF and the posterior PDF, respectively. fkjk1 ðÞ is the state transition PDF and gk ðÞ is the measurement likelihood function. The PHD filter proposed by Mahler7 yields an approximation of the optimal Bayesian filter given in Eqs. (15) and (16) via propagating only the first-order moment of pkjk ðXk jY1:k Þ, i.e., the PHD. It is capable of tracking a variable number of targets and estimating both the number of targets and their states without utilizing data association techniques. The PHD is a multi-peak function in the state space. The number of peaks is often (but not necessarily) approximately equal to the number of targets, and the peak positions correspond to the expected values of target states, which can be extracted through the use of the expectation-maximum (EM) algorithm35,36 or clustering techniques.9,37 Let vkjk1 ðxÞ and vkjk ðxÞ denote the predicted and posterior intensity functions of pkjk ðXk jY1:k Þ. Their prediction and update equations are Z 0
vkjk1 ðxÞ ¼ 292
gk ðYk jXk Þpkjk1 ðXk jY1:k1 Þ gk ðYk jXÞpkjk1 ðXjY1:k1 Þls ðdXÞ
0
ðpS;kjk1 ðx Þfkjk1 ðxjx Þ 0
0
0
þ bkjk1 ðxjx ÞÞvk1jk1 ðx Þdðx Þ þ ck ðxÞ
ð17Þ
293
295
vkjk ðxÞ ¼ ½1 pD;k ðxÞvkjk1 ðxÞ X pD;k ðxÞgk ðyjxÞvkjk1 ðxÞ R þ j ðyÞ þ pD;k ðxÞgk ðyjxÞvkjk1 ðxÞdx y2Yk k
ð18Þ
When the latest measurements become available at time k, the joint posterior PHD becomes vkjk ðx; UÞ ¼ ð1 pD;k Þvkjk1 ðx; UÞ X p v ðx; UjyÞ RR D;k D;k þ j ðyÞ þ pD;k vD;k ðx0 ; U0 jyÞdðx0 ÞdðU0 Þ y2Yk k
vD;k ðx; UjyÞ ¼ gk ðyjx; UÞvkjk1 ðx; UÞ
ðiÞ ðiÞ ðiÞ ðiÞ L0 fx0 ; h0 ; n0 ; w0 gi¼1
are drawn from p0 ðX0 ; h0 ; n0 Þ,
304
We shall first generalize the PHD recursions in Eqs. (17) and (18) to take into account the presence of unknown model parameters. To simplify the presentation, it is assumed that the survival and the detection probabilities are independent of both the target state vector and the unknown parameter vector Uk . They will thus be denoted by pS;kjk1 and pD;k . Further drop the subscript k in Uk for notation simplicity and let vk1 ðx; UÞ be the joint posterior PHD at time k 1. According to Eq. (17), Eq. (18) and the Chapman-Kolmogorov equation, the predicted PHD ZZ vkjk1 ðx; UÞ can then be described as
305 306 307 308 309 310 311 312 313 314
vkjk1 ðx; UÞ ¼
ðpS;kjk1 fkjk1 ðx; Ujx0 ; U0 Þ
þ bkjk1 ðx; Ujx0 ; U0 ÞÞvk1 ðx0 ; U0 Þdðx0 ÞdðU0 Þ ZZ ðpS;kjk1 fkjk1 ðxjx0 ; U0 Þ þ ck ðx; UÞ ¼ 0
0
0
þ bkjk1 ðxjx ; U ÞÞpkjk1 ðUjU Þ 316
vk1 ðx0 ; U0 Þdðx0 ÞdðU0 Þ þ ck ðx; UÞ
ð19Þ
ðiÞ
and the weight is set to be w0 ¼ 1=N. For k P 1, the PFAPE-PHD recursions are as follows. 3.2.1. Prediction stage
329 330 331 332 333
ðiÞ ðiÞ Denote ¼ ðak1 ; bk1 Þ as the sufficient statistic for the staðiÞ tic parameter particle hk1 . For i ¼ 1; 2; . . . ; Lk1 , the parame-
ðiÞ pðhjsk1 Þ
336 337 338 339 340 341 342 343 344 345 346
347
ðiÞ sk1
ðiÞ hk1
3.1. APE-PHD recursions
328
335
ter particles can be obtained by
303
327
In this subsection, the PF is utilized to derive an approximation of the closed-form solution to the extended PHD recursions in Eqs. (19) and (20). The obtained PF-APE-PHD algorithm consists of two stages, namely the prediction and update stages. We assume that initially, there are N0 targets and N particles are produced for each target. The total number of particles is therefore L0 ¼ N N0 . Let X0 be the initial multiple target state set and p0 ðX0 ; h0 ; n0 Þ be the prior joint PDF. The initial
3. Tracking multiple maneuvering targets
300
326
334
302
299
ð21Þ
322 323 325
3.2. PF-APE-PHD algorithm
301
298
321
Note that in Eqs. (19)–(21), because U is unknown, the measurement likelihood gk ðyjx; UÞ and vkjk1 ðx; UÞ are hard to be obtained, this makes it difficult to calculate the analytic solution of the joint intensity function vD;k ðx; UjyÞ. However, its approximation solution can be obtained through the use of the APE technique combined with the PF. The proposed algorithm is therefore referred to as PF-APE-PHD algorithm, which will be presented in the following subsection.
particles
297
ð20Þ where
where bkjk1 ðxÞ and ck ðxÞ are the intensities of the RFSs of the spawned targets and spontaneous births. pS;kjk1 ðxÞ denotes the survival probability and pD;k ðxÞ is the detection probability. jk ðyÞ ¼ kk ck ðyÞ is the intensity of the clutter RFS, which is assumed to be Poisson distributed with mean rate kk , and ck ðyÞ is the distribution of the clutter.
296
317 318 319
348 349 350 351
ð22Þ
In this work, we consider the case that the static unknown parameter hk is the variance of the measurement noise. Its conjugate prior is approximated by an inverse-gamma (IG) distribution IGða; bÞ with parameters a and b, i.e., IGðh; a; bÞ ¼ R1 ba ha1 exp bh , where CðaÞ ¼ 0 ta1 exp ðtÞdt.18 The CðaÞ method for estimating a and b is similar to the method20 used to identify the unknown measurement noise covariance R. Therefore, the details are omitted here. To account for the possible abrupt changes in the timevarying parameters, we evaluate Eqs. (9) and (10) to obtain ðiÞ the estimate of the means mk1 and covariance Vk1 for the ðiÞ time-varying parameter particle nk1 . It is noted that P ðiÞ ðiÞ nk1 ¼ N i¼1 xk1 nk1 of Eqs. (9) and (10) in this algorithm is ðiÞ the Monte Carlo posterior mean of all nk1 which belong to the same target cluster. The particle clusters are formed in the stage of state extraction presented later in this subsection. We then generate 2Lk1 particles as in the APE filter. For ðiÞ ðiÞ this purpose, the proposal distributions qk ðjxk1 ; hk1 ;
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371
CJA 716 26 October 2016
No. of Pages 9
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets
5 427
372 373 374 375 376
ðiÞ
ðiÞ
ðiÞ
nk1 ; Yk Þ and pk ðjhk1 ; nk1 ; Yk Þ used to produce predicted particles as in Refs. 7, 14 are employed here. The first Lk1 particles are generated under the condition that the time-varying parameters do not change abruptly. They are obtained via k1ðiÞ
ðiÞ
ðiÞ
ðiÞ
378
xkj
qk ðxk jxk1 ; hk1 ; mk1 ; Yk Þ
379
Their weights are equal to
ð23Þ
Relabel the selected particles with indices i ¼ 1; 2; . . . ; Lk1 , ðiÞ xkjk1
ðli Þ xkjk1
ðiÞ wkjk1
ðli Þ wkjk1 .
¼ with ¼ Sample Jk new-born particles with indices i ¼ Lk1 þ 1; Lk1 þ 2; . . . ; Lk1 þ Jk from the ðiÞ ðiÞ proposal distribution pk ðxk jh0 ; n0 ; Yk Þ via7,14 k1ðiÞ
xkj
ðiÞ
ðiÞ
382
ðiÞ
i ¼ Lk1 þ 1;
ðiÞ
384 385 386
ði;jÞ
389 390 391 392 393 394 395 396 398
ð24Þ ðiÞ
ðiÞ
ðiÞ
where i ¼ 1; 2; . . . ; Lk1 and /ðxkjk1 ; xk1 ; hk1 ; mk1 Þ is the transitional PDF. At time k, each particle is given another weight proportional to the predictive likelihood corresponding ðiÞ to no changepoint parameter mk1 , i.e.,
387
x1
ðiÞ ðiÞ ðiÞ / p Yk jxkjk1 ; hk1 ; mk1
ð25Þ
The remaining Lk1 particles are produced under the assumption that abrupt changes occurred. As in the APE technique, the values of the time-varying parameters are now drawn from ðiÞ their prior distributions, i.e., ck1 pn ðn0 Þ. The predicted particles and their weights are obtained via, for i ¼ Lk1 þ 1; Lk1 þ 2; . . . ; 2Lk1 , k1ðiÞ xkj
ðiL Þ ðiL Þ ðiL Þ qk ðxk jxk1 k1 ; hk1 k1 ; ck1 k1 ; Yk Þ
ð26Þ
399 k1ðiÞ wkj
401 402 403 404 405 407
¼
ðiL Þ ðiL Þ ðiL Þ ðiL Þ /ðxkjk1k1 ; xk1 k1 ; hk1 k1 ; ck1 k1 Þ
ðiÞ wk1
Lk1 þ 2; . . . ; Lk1 þ Jk
ð29Þ
At time k, each particle is also given another weight proportional to the predictive likelihood corresponding to changeðiÞ point parameter ck1 , i.e., ðiÞ ðiL Þ ðiL Þ ðiL Þ ð28Þ x2 / p Yk jxkjk1k1 ; hk1 k1 ; ck1 k1
411
(1) For i ¼ 1; 2; . . . ; Lk1 , select indices li with probability
412 413 414 415 416 417 418 419 420 421 422 423 424 425 426
ðli Þ
ðli Þ
ð1 bÞx1 from ½1; 2; . . . ; Lk1 and bx2 from ½Lk1 þ 1; . . . ; 2Lk1 , where b is the probability that an abrupt change occurred and it is assumed to be known (see also Section 2.2). (2) If li 2 f1; 2; :::; Lk1 g, then update the time-varying i
ðiÞ
ðl Þ
parameter particles using nk Nðjmk1 ; h2 Vk1 Þ, where V k1 is given in Eq. (10). Set the composite parameter ðiÞ
ðiÞ
ðiÞ T
ðli Þ
ðiÞ
particle as Uk ¼ ½hk1 ; nk with hk1 ¼ hk1 and the sufficient statistics for the static parameters as ðli Þ sk1 .
ðiÞ sk1 i
¼ (3) If l 2 fLk1 þ 1; Lk1 þ 2; . . . ; 2Lk1 g, then set the timeðiÞ
i
ðl Þ
varying parameter particles to be nk ¼ ck . The composite parameter particle and the sufficient statistics ðiÞ
remain to be denoted by Uk ¼ ðiÞ hk1
¼
ðli Þ hk1
and
ðiÞ sk1
¼
ðli Þ sk1 .
ðiÞ
ðiÞ
wkjk1
ðiÞ
ðiÞ
1 ck ðxk ; h0 ; n0 Þ ; ¼ Jk p ðxðiÞ hðiÞ ; nðiÞ ; Y Þ k k 0 0 k
i ¼ Lk1 þ 1;
Lk1 þ 2; . . . ; Lk1 þ Jk
ð30Þ
3.2.2. Update stage
h ðiÞ ^ðiÞ w k ¼ ð1 PD;k ðxk ÞÞ
# ðiÞ ðiÞ ðiÞ ðiÞ pD;k ðxk Þgk ðyjxk ;hk1 ;nk1 Þ k1ðiÞ ð31Þ wkj P ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ y2Yk kk ck ðyÞ þ y2Yk pD;k ðxk Þgk ðyjxk ;hk1 ;nk1 Þwkjk1
þ
ðiÞ ðiÞ T ½hk1 ; nk
437
438
After receiving the measurement at time k, the Lk1 þ Jk particle weights can be updated by
439 440 441
X
3.2.3. Computation of the total mass
443
444 445 446
Nk ¼
Lk1 þJk X
^ðiÞ w k
ð32Þ 448
3.2.4. Resampling
410
409
434
ð27Þ
We select Lk1 out of the 2Lk1 obtained particles. Denote their indices as li 2 f1; 2; . . . ; 2Lk1 g, where i ¼ 1; 2; . . . ; Lk1 , the selection process is as follows.
408
431 432
ðiÞ
i¼1
ðiLk1 Þ ðiLk1 Þ ðiL Þ ðiL Þ xk1 ; hk1 k1 ; ck1 k1 ; Yk Þ
qk ðxk
430
435
ðiÞ
/ðxkjk1 ; xk1 ; hk1 ; mk1 Þ ðiÞ wk1 ¼ ðiÞ ðiÞ ðiÞ ðiÞ qk ðxk xk1 ; hk1 ; mk1 ; Yk Þ ðiÞ
383
429
pk ðxk jh0 ; n0 ; Yk Þ;
380 ðiÞ wkjk1
428
with
ðiÞ
ðiÞ
449 ðiÞ
ðiÞ
Lk1 þJk
^k =Nk gi¼1 Resample fxk ; hk ; nk ; w
through the weights to
Lk ðiÞ ðiÞ ðiÞ ðiÞ fxk ; hk ; nk ; wk =Nk gi¼1 .
450
obtain a new particle set Each particle is assigned the same weight Nk =Lk after resampling, where Lk ¼ Lk1 þ Jk .
451
3.2.5. Extraction of target states
454
Target states can be obtained by clustering the particles and
452 453
455
^
Nk ^ k ¼ f^ the cluster centers are the estimated states X xk;j gj¼1 , where ^ Nk ¼ roundðNk Þ is the estimate of the target number, and roundðÞ denotes the rounding operator.
456 457 458
4. Simulations
459
In order to illustrate the performance of the proposed PFAPE-PHD algorithm, a two-dimensional tracking scenario is simulated. The benchmark techniques are the MMP-PHD,28 MMP-CPHD and MMP-CBMeMBer filters.29 In the considered scenario, the measurements are obtained at four stationary sensors located at ð0; 0Þ m, ð0; 1 104 Þ m, ð1 104 ; 0Þ m, and ð1 104 ; 1 104 Þ m. At time k, each sensor outputs the measured bearing of the received signal, which is given by yk ySi þ wk ySk i ¼ tan1 ð33Þ xk xSi
460
where ðxSi ; ySi Þ denotes the location of the ith sensor, i ¼ 1; 2; 3; 4. wk is the zero-mean Gaussian noise with variance r2w ¼ 1 104 rad2 .
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
461 462 463 464 465 466 467 468 470 471 472 473
CJA 716 26 October 2016
No. of Pages 9
6 474 475 476 477 478 479 480 481 482 483 484 485
487 488 489 490 491 492
J. Yang et al.
There are three maneuvering targets. Targets 1 and 2 remain active throughout the whole simulation process and their initial positions are at ð3 103 ; 5 103 Þ m and ð1:4 104 ; 8 103 Þ m, as in Ref. 28. Target 3 is a spontaneous birth at 10th min with initial position ð2 103 ; 10:5 103 Þ m and disappears at 50th min. The true tracks of the three targets are depicted in Fig. 1. When realizing the MMP-PHD and MMP-CPHD algorithms, we set that they both consist of a constant velocity (CV) model and two CT models. The transition probability matrix is assumed to be 2 3 T T 1 sT1 2s1 2s1 6 T 7 T 7 1 sT2 ½hab ¼ 6 ð34Þ 2s2 5 4 2s2 T T T 1 s2 2s2 2s2 where the sampling interval is T ¼ 1 min, and the sojourn durations are s1 ¼ 200 min and s2 ¼ 100 min. The initial model probabilities for the three models are all equal to 1=3. The state evolution for CV and CT models is 2 3 1 T 0 0 60 1 0 0 7 6 7 i xik ¼ 6 ð35Þ 7x þ vik 4 0 0 1 T 5 k1 0
494
0
0
1
495 497 498 499 500 501 502 503
xik
¼
FðxÞxik1
507 508 509
0
0
T =2 2
T
where r2v ¼ 1 104 m2 s3 . For the three algorithms in consideration, we model the birth process using a Poisson RFS with intensity ðiÞ
511
ð36Þ
where xik ¼ ½xik ; x_ ik ; zik ; z_ik is the state vector of the ith target. FðxÞ is the state transition matrix of the CT model (see Eq. (3) for its definition). We set the turn rates of the two CT models to be x ¼ 9 =min, and vik is a zero-mean white Gaussian process noise with covariance 2 3 3 0 0 T =3 T2 =2 6 T2 =2 T 0 0 7 6 7 2 Q¼6 ð37Þ 7rv 3 2 4 0 0 T =3 T =2 5
505 506
þ
vik
Ck ðxÞ ¼
3 X
ðiÞ
ðiÞ
0:2Nðx; mC ; PC Þ;
i ¼ 1; 2; 3
i¼1
Fig. 1
ð38Þ
ð1Þ
ð2Þ
where mC ¼ ð3 103 m; 0 m=s; 5 103 m; 0 m=sÞ, mC ¼ ð3Þ ð1:4 104 m; 0 m=s; 8 103 m; 0 m=sÞ, mC ¼ ð2 103 m; ð1Þ ð2Þ ð3Þ 3 0 m=s; 10:5 10 m; 0 m=sÞ, and PC ¼ PC ¼ PC ¼ diag ð400; 1; 400; 1Þ. The clutter is modeled as a Poisson RFS with the mean rate r = 10 over the observation space. The probabilities of the target survival and detection are pS;k ¼ 0:99 and pD;k ¼ 0:98, respectively. The initial parameters of the inverse Gamma distribution are set as a ¼ b ¼ 1.17,20 At each time, a maximum 1500 and minimum 300 particles perhypothesized track are imposed so that the number of particles representing each hypothesized track is proportional to its existence probability after resampling in the update step. To verify the effectiveness of the proposed algorithm, simulations are performed on a Lenovo T430 desktop with Intel (R) Core(TM) CPU i5-3210 M, 2.50 GHz and 8 GB RAM. Two performance metrics are used. One is the statistics of the target number estimate. The other is the optimal subpattern assignment (OSPA)38 distance defined as !!1=p m p X 1 ðcÞ ðcÞ p min d xi ; ypðiÞ þ c ðn mÞ dp ðX; YÞ ¼ n p2Pn i¼1 ð39Þ where X ¼ fx1 ; x2 ; ; xm g and Y ¼ fy1 ; y2 ; ; yn g are arbitrary finite subsets, 1 6 p < 1, c > 0, m; n 2 No ¼ ðcÞ f0; 1; 2; . . .g. If m > n, dðcÞ p ðX; YÞ ¼ dp ðY; XÞ. In the simulation, the parameters of OSPA distance are set to be p ¼ 2 and c ¼ 1000. Three simulation experiments are performed and the results shown are obtained from Monte Carlo simulations of 200 ensemble runs. The first experiment is to evaluate the performance for multiple abruptly maneuvering target tracking, where only the maneuvering parameters (e.g., turn rates) are unknown. The second experiment is to compare the performance of the proposed algorithm in the presence of unknown maneuvering parameters as well as unknown measurement noise variances. The last experiment is conducted using different measurement noise variances to evaluate the robustness of the proposed algorithm.
512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530
532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
4.1. Multiple abruptly maneuvering target tracking
549
In this experiment, the standard deviation of the measurement noise is set to be r ¼ 0:01 rad and it is assumed known for the considered PF-APE-PHD, MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms. The turn rate x is considered as an unknown and time-varying parameter for the proposed PF-APE-PHD algorithm. The MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms use one CV and two CT models of Eqs. (35) and (36) as the target motion models. Although in practice, the true turn rates are unavailable for the IMM-based filters, we realize the CT models with the real turn rates x ¼ 9 =min and x ¼ 9 =min so that the MMPbased methods would have the ‘optimal’ performance. Simulation results for this experiment are shown in Figs. 2–4. Fig. 2 shows the average target number estimates obtained by the PF-APE-PHD, MMP-PHD, MMP-CPHD and MMPCBMeMBer filters. It can be seen that the proposed PF-APEPHD algorithm can even provide more accurate target number estimates than the benchmark techniques. The reason is that
550
True target tracks.
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567
CJA 716 26 October 2016
No. of Pages 9
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607
608 609
610 611
the proposed algorithm can effectively estimate jointly the unknown model parameter x which can be well matched with the motion model of each target. While for the MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms, the tracking accuracy is affected by the model interference due to the interaction of multiple models, an inevitable phenomenon of IMM-based techniques, which renders their performance under ‘optimal’ parameter settings still inferior to the proposed technique. Moreover, it is noticed that the MMP-CPHD and MMP-CBMeMBer algorithms have better performance in terms of more precise target number estimates than the MMP-PHD algorithm. The reason is that the MMPCBMeMBer method propagates the parameterized approximation of the posterior cardinality distribution, and the MMP-CPHD method jointly propagates the cardinality distribution and the intensity function, whereas the MMP-PHD method propagates the cardinality mean only with a single Poisson parameter. Fig. 3 compares the OSPA distances of the four simulated algorithms, and it is clear that the proposed algorithm again outperforms the MMP-PHD, MMP-CPHD and MMPCBMeMBer algorithms. This is also due to the fact that the proposed method can adapt to the temporal evolution of the target maneuvering parameters. It is worth noting that when the third target disappears at 50th min, the OSPA distance of the MMP-CPHD algorithm increases suddenly, which indicates that the ‘spooky action’ problem steps in, i.e., it is beneficial when missed detection occurs in MMP-PHD but is harmful when targets really disappear. Fig. 4 shows the average run time of the four algorithms in consideration. It can be seen that the average run time of the proposed PF-APE-PHD algorithm is slightly larger than that of the MMP-PHD algorithm. The reason is that with the APE technique, the proposed algorithm generates twice more particles with different parameter predictions in the predicted step and has an additional particle selection step. It is noted that the complexity of the MMP-CBMeMBer is slightly lower than that of the MMP-PHD algorithm, because this method allows reliable and inexpensive extraction of state estimates without particle clustering. 4.2. Multiple abruptly maneuvering target tracking with unknown measurement noise variance In this experiment, the true standard deviation of the measurement noise is fixed at r ¼ 0:01 rad, but it is unknown for the
Fig. 2
Target number estimates.
Fig. 3
7
OSPA distance statistics.
Fig. 4
Average run time.
proposed algorithm. We then apply the PF-APE-PHD algorithm to identifying it together with the time-varying turn rate x and the target states. For comparison purposes, we also simulate the PF-APE-PHD filters with other assumed values of the measurement noise variance (i.e., r = 0.005, 0.01, 0.015, 0.03, 0.06, 0.1). The simulation results are summarized in Figs. 5 and 6. It is clear that when the measurement noise variance is estimated jointly with the target states, the performance of the PFAPE-PHD algorithm is very close to that when the measurement noise variance is accurately known in advance (r ¼ 0:01 rad). This indicates that the proposed PF-APEPHD algorithm can achieve accurate joint parameter and target state estimations. On the other hand, if PF-APE-PHD simply operates with an incorrect setting of the measurement noise variance, it would suffer from significant performance degradation, mainly due to model mismatch.
612
4.3. Performance with different measurement noise variance settings
629
In this experiment, we realize two versions of the PF-APEPHD filter, one filter with unknown measurement noise variance and another filter with true measurement noise variance. The simulation results with different measurement noise standard deviations (i.e., r = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07) are shown in Fig. 7. As can be seen the estimation accuracy of the PF-APE-PHD algorithm with unknown r is close to that of the PF-APE-PHD algorithm with the true value of r known in advance. It is shown that the proposed algorithm has a good performance for multiple target tracking with
631
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628
630
632 633 634 635 636 637 638 639 640
CJA 716 26 October 2016
No. of Pages 9
8
J. Yang et al.
Fig. 5
Target number estimates with different measurement noise standard deviations.
Fig. 6
OSPA distance statistics with different measurement noise standard deviations.
Fig. 7 Average OSPA distance statistics with different measurement noise standard deviations.
642
unknown measurement noise parameters and the time-varying abruptly changing maneuver parameters.
643
5. Conclusions
644
In this paper, we developed a new MTT algorithm, the PFAPE-PHD filter, to handle the presence of unknown model parameters including e.g., the measurement noise variance that
641
645 646
is static and the parameters in accordance with the target maneuvers that may be time-varying and subject to abrupt changes. The development started with extending the PHD filter to take into account the unknown parameters and the APE technique was incorporated to achieve online parameter estimation. The SMC approach was utilized to derive the approximate closed-form solution. Simulations showed that the newly proposed PF-APE-PHD filter can offer higher tracking accuracy in the case of multiple maneuvering targets over the existing MMP-PHD, MMP-CPHD and MMP-CBMeMBer algorithms. It is also applicable to the case with unknown measurement noise parameters for multiple maneuvering target tracking. In future works, we shall consider introducing the APE technique into the spline PHD filter,39 the CPHD filter8 and the CBMeMBer filter40,41 to obtain good algorithms for tracking multiple targets with unknown abrupt changing parameters.
647
Acknowledgments
665
This paper is supported by the National Natural Science Foundation of China (Nos. 61305017, 61304264) and the Natural Science Foundation of Jiangsu Province (No. BK20130154).
666
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664
667 668 669
CJA 716 26 October 2016
No. of Pages 9
Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets 670
671 672 673
674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732
Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cja.2016. 09.010. References 1. Fortmann T, Bar-Shalom Y, Scheffe M. Sonar tracking of multiple targets using joint probabilistic data association. IEEE J Ocean Eng 1983;8(3):173–84. 2. Musicki D, Evans R. Joint integrated probabilistic data association: JIPD.. IEEE Trans Aerosp Electron Syst 2004;40(3):1093–9. 3. Blackman S. Multiple hypothesis tracking for multiple target tracking. IEEE Aerosp Electron Syst Mag 2004;19(1):5–18. 4. Roecker J, Pillis G. Suboptimal joint probabilistic data association. IEEE Trans Aerosp Electron Syst 1993;29(2):504–10. 5. Roecker J. A class of near optimal JPDA algorithms. IEEE Trans Aerosp Electron Syst 1994;30(2):504–10. 6. Purank S, Tugnait JK. Tracking of multiple maneuvering targets using multiscan JPDA and IMM filter. IEEE Trans Aerosp Electron Syst 2007;43(1):23–34. 7. Mahler R. Multitarget Bayes filtering via first-order multitarget moments. IEEE Trans Aerosp Electron Syst 2003;29(4):1152–78. 8. Vo BT, Vo BN, Cantoni A. Analytic implementations of the cardinalized probability hypothesis density filter. IEEE Trans Signal Process 2007;55(7):3553–67. 9. Vo BN, Singh S, Doucet A. Sequential Monte Carlo methods for multi-target filtering with random finite sets. IEEE Trans Aerosp Electron Syst 2005;41(4):1224–45. 10. Whiteley N, Singh S, Godsil S. Auxiliary particle implementation of probability hypothesis density filter. IEEE Trans Aerosp Electron Syst 2010;46(3):1437–54. 11. Vo BN, Ma WK. The Gaussian mixture probability hypothesis density filter. IEEE Trans Signal Process 2006;54(11):4091–104. 12. Clark D, Vo BT, Vo BN. Gaussian particle implementations of probability hypothesis density filtersProceedings of IEEE aerospace conference, 2007 March 3-10, Big Sky, MT, USA. Piscataway, NJ: IEEE Press; 2007. p. 1–11. 13. Braca P, Marano S, Matta V. Asymptotic efficiency of the PHD in multitarget/multisensor estimation. IEEE J Sel Top Signal Process 2013;7(3):553–64. 14. Yang JL, Ji HB. A novel track maintenance algorithm for PHD/ CPHD filter. Signal Process 2012;92(10):2371–80. 15. Beard M, Vo BT, Vo BN, Arulampalam S. A partially uniform target birth model for Gaussian mixture PHD/CPHD filtering. IEEE Trans Aerosp Electron Syst 2013;49(4):2835–44. 16. Vo BT, Vo BN, Cantoni A. The Cardinality balanced multi-target multi-Bernoulli filter and its implementations. IEEE Trans Signal Process 2009;57(2):409–23. 17. Sa¨rkka¨ S, Nummenmaa A. Recursive noise adaptive Kalman filtering by variational Bayesian approximations. IEEE Trans Autom Control 2009;54(3):596–600. 18. Li WL, Jia YM. State estimation for jump Markov linear systems by variational Bayesian approximation. IET Control Theory Appl 2012;6(2):319–26. 19. Gao XB, Chen JG, Tao DC. Multi-sensor centralized fusion without measurement noise covariance by variational Bayesian approximation. IEEE Trans Aerosp Electron Systs 2011;47 (1):718–27. 20. Yang JL, Ge HW. Adaptive probability hypothesis density filter based on variational Bayesian approximation for multi-target tracking. IET Radar, Sonar Navigat 2013;7(9):959–67. 21. Zhang GH, Lian F, Han CZ, Han SY. An improved PHD filter based on variational Bayesian method for multi-target trackingProceedings of the 17th international conference information
22.
23.
24.
25.
26.
27.
28.
29.
30. 31.
32. 33.
34. 35.
36.
37. 38.
39.
40.
41.
9
fusion, 2014 July 7–10, Salamanca, Spain. Piscataway, NJ: IEEE Press; 2014. p. 1–6. Hernandez ML, Ristic B, Farina A. Performance measure for Markovian switching systems using best-fitting Gaussian distributions. IEEE Trans Aerosp Electron Syst 2008;44(2):724–47. Li XR, Jilikov VP. Survey of maneuvering target tracking. Part V: multiple-model methods. IEEE Trans Aerosp Electron Syst 2005;41(4):1255–321. Pasha A, Vo BN, Tuan HD. Closed-form PHD filtering for linear jump Markov modelsProceedings of the int. conf. on information fusion; 2016 July 10–13, Florence, Italy. IEEE; 2006. p. 1–8. Pasha SA, Vo BN, Tuan HD. A, Gaussian mixture PHD filter for jump Markov system models. IEEE Trans Aerosp Electron Syst 2009;45(3):919–36. Pasha SA, Tuan HD, Apkarian P. The LFT based PHD filter for nonlinear jump Markov models in multi-target trackingProceedings of the IEEE conference on decision and control, 2009 December 15– 18, Shanghai, China. Piscataway, NJ: IEEE Press; 2009. p. 5478–83. Li WL, Jia YM. Gaussian mixture PHD filter for jump Markov models based on best-fitting Gaussian approximation. Signal Process 2011;91(4):1036–42. Punithakumar K, Kirubarajan T, Sinha A. Multiple-model probability hypothesis density filter for tracking maneuvering targets. IEEE Trans Aerosp Electron Syst 2008;44(1):87–98. Yang JL, Ji HB, Ge HW. Multi-model particle cardinalitybalanced multi-target multi-Bernoulli algorithm for multiple manoeuvring target tracking. IET Radar, Sonar Navigat 2013;7 (2):101–12. Ouyang C, Ji HB, Guo ZQ. Extensions of the SMC-PHD filters for jump Markov systems. Signal Process 2012;92(6):1422–30. Liu J, West M. Combined parameter and state estimation in simulation-based filtering. In: Doucet A, Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer; 2001. p. 197–223. Carvalho C, Johannes M, Lopes H, Polson N. Particle learning and smoothing. Stat Sci 2010;25(1):88–106. Nemeth C, Fearnhead P, Mihaylova L. Sequential Monte Carlo methods for state and parameter estimation in abruptly changing environments. IEEE Trans Signal Process 2014;62(5):1245–55. West M. Approximating posterior distributions by mixture. J Roy Stat Soc 1993;55(2):409–22. Tobias M, Lanterman AD. Probability hypothesis density-based multi-target tracking with bistatic range and Doppler observation. IET Radar, Sonar Navigat 2005;152(3):195–205. Lian F, Han CZ, Liu WF. Estimating Unknown clutter intensity for PHD filter. IEEE Trans Aerosp Electron Syst 2010;46 (4):2066–78. Xu BL, Xu HG, Zhu JH. Ant clustering PHD filter for multipletarget tracking. Appl Soft Comput 2011;11:1074–86. Schuhmacher D, Vo BT, Vo BN. A consistent metric for performance evaluation of multi-object filters. IEEE Trans Signal Process 2008;56(8):3447–57. Sithiravel R, Chen X, Tharmarasa R. The spline probability hypothesis density filter. IEEE Trans Signal Process 2013;61 (24):6188–203. Vo BT, Vo BN, Cantoni A. The cardinality balanced multi-target multi-Bernoulli filter and its implementations. IEEE Trans Signal Process 2009;57(2):409–23. Beard MA, Vo BT, Vo BN. Bayesian multi-target tracking with merged measurements using labelled random finite sets. IEEE Trans Signal Process 2015;63(6):1433–47.
Yang Jinlong is an associate professor in Jiangnan University. He received his M.S. degree in circuit and system from Northwest Normal University, China in 2009, and his Ph.D. degree in Pattern Recognition and Intelligent System from Xidian University, China, in 2012. His research interests include target tracking, information fusion and signal processing.
Please cite this article in press as: Yang J et al. Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.010
733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801