Joint Target Tracking and Classification via ... - Semantic Scholar

Book Title Book Editors IOS Press, 2003

1

Joint Target Tracking and Classification via Sequential Monte Carlo Filtering 1 Donka Angelova a,2 , and Lyudmila Mihaylova b Bulgarian Academy of Sciences, 25A Acad. G. Bonchev St, 1113 Sofia, Bulgaria b Department of Electrical and Electronic Engineering, University of Bristol, UK

a

Abstract. A sequential Monte Carlo algorithm is suggested for joint maneuvering target tracking and classification, based on kinematic measurements. A mixture Kalman filter is designed for two-class identification of air targets: commercial and military aircraft. Speed and acceleration constraints are imposed on the target behaviour models in order to improve the classification process. The class is modeled as an independent random variable, which can take values over the discrete class space with an equal probability. As a result, the multiple-model structure in the class space, required for reliable classification, is achieved. The performance of the proposed algorithm is evaluated by simulation over typical target scenarios. Keywords. Joint tracking and classification, sequential Monte Carlo methods, mixture Kalman filtering

1. Introduction Monte Carlo techniques have been extensively applied recently in the field of multisensor data processing [1]. In particular, a number of simulation-based algorithms have been published in the specialized literature, devoted to the important problem of joint tracking and classification (JTC). Sequential Monte Carlo (SMC) methods are very suitable for classification purposes: the highly non-linear relationships between state and class (feature) measurements and non-Gaussian noise processes can be easily processed by the particle filtering technique. Moreover, flight envelope constraints, especially useful for this task, can be incorporated into the filtering procedure in a natural and consistent way. A robust particle filter for JTC is suggested by Gordon, Maskell and Kirubarajan in [2]: a bank of independent filters, covering the state and feature space are run in parallel with each filter matched to a different target class. An example of a successful application of this approach to littoral tracking is proposed in [3]. Interesting JTC algorithms are reported in [4,5]. Based on this concept, a multiple model particle filter and a mixture Kalman filter (MKF) for maneuvering target tracking and classification are developed in [6]. The classification task is carried out by processing kinematic measurements only, primarily in the air surveillance context. The features of the proposed algorithms include the following: for each target class a separate filter is designed; class filters operate in parallel, cov1 partially supported by the UK MOD Data and Information Fusion DT Center and Bulgarian Foundation for Scientific Investigations under grants I-1202/02 and I-1205/02. 2 Corresponding author, E-mail: [email protected]

2

/ Joint Target Tracking and Classification

ering the class space; each class-dependent filter represents a switching multiple model filtering procedure, covering the kinematic state space of the maneuvering target. This kind of multiple model configuration provides precise and reliable tracking and correct class identification. However, the good results have been achieved at the cost of a rather complex algorithm. The incorporation of the filter into multiple target tracking systems will be difficult. In the present paper, an improved MKF is proposed for tracking and identification of air targets in two classes: commercial and military aircraft. The parallel work of class filters are simulated by utilizing a random class variable. The independent sets of particles for each class are replaced by a randomly generated at each time step two class-dependent sets of particles. The class variable is modeled as an independent random variable, taking values over a finite discrete class space with an equal probability. The proposed filtering algorithm has a relatively simple structure and exhibits the same performance as the MKF, proposed in [6]. The elimination of the unlikely filters after classification decision can be realized in a easy way. Section 2 summarizes the Bayesian formulation of the JTC problem and a mixture Kalman filtering algorithm. Section 3 deals with the implementation of the MKF for JTC. Simulation results are given in Section 5. Section 6 contains concluding remarks. 2. Bayesian formulation of JTC. Mixture Kalman Filtering Consider the following model of a discrete-time jump Markov linear system xk = F (λk ) xk−1 + G (λk ) uk (λk ) + B (λk ) wk ,

(1)

z k = H (λk ) xk + D (λk ) v k ,

(2)

k = 1, 2, . . . ,

where xk ∈ Rnx is the base (continuous) state vector, z k ∈ Rnz specifies the measurement vector, uk ∈ Rnu represents a known control input and k is a discrete time. The input noise process wk and the measurement noise v k are independent identically distributed Gaussian processes having characteristics wk ∼ N (0, Q) and v k ∼ N (0, R), respectively. The modal (discrete) state λk , characterizing the different system modes (regimes), can take values over a finite set S, i.e. λk ∈ S , {1, 2, . . . , s}. We assume that the mode λk is evolving according to a first-order Markov chain with transition probabilities πij , P r {λk = j | λk−1 = i} , (i, j ∈ S) and initial probability distribution Ps P0 (i) , P r {λ0 = i} for i ∈ S, such that P0 (i) ≥ 0, and i=1 P0 (i) = 1. Next we suppose that the target belongs to one of M classes c ∈ C where C , {1, 2, . . . , M } represents the set of the target classes. Generally, the number of the discrete states c s = s(c), £ the ¤ initial probability distribution P0 (i) and the transition probability matrix c π(c) = πij , i, j ∈ S(c) are different for each target class. Denote with ω k , {z k , y k } the set of kinematic z koand class (feature) y k measuren k k ments obtained at time instant k. Then Ω = Z , Y k specifies the cumulative set of kinematic and feature measurements, available up to time k. The goal of the joint tracking and classification task is to estimate simultaneously the³ base state xk , the modal state λk and the posterior classification probabilities ´ P c | Ωk , c ∈ C based on the observation set Ωk . The problem can be stated in the Bayesian framework of ³estimating the ´posterior joint state-mode-class probability density function (pdf) p xk , λk , c | Ωk , which


3

can be computed recursively from the joint pdf at the previous time step in two stages – prediction ´and measurement update [1]. The predicted state-mode-class pdf ³ p xk , λk , c | Ωk−1 at time k is given by the equation ³ ´ p xk , λk , c | Ωk−1 = X λk−1 ∈S(c)

Z xk−1 ∈Rnx

(3)

³ ´ ³ ´ p xk , λk | xk−1 , λk−1 , c, Ωk−1 × p xk−1 , λk−1 , c | Ωk−1 dxk−1 ,

´ ³ where the state prediction pdf p xk , λk | xk−1 , λk−1 , c, Ωk−1 is obtained from the state transition equation (1). The form of the measurement likelihood function p (ω k | xk , λk , c) is usually known. When the measurement ω k arrives, the update step can be completed ³ ´ ³ ´ p (ω k | xk , λk , c) p xk , λk , c | Ωk−1 ³ ´ p xk , λk , c | Ωk = (4) p ω k | Ωk−1 The recursion (3)-(4) begins with the prior density P {x0 , λ0 , c}, assumed known, where x0 ∈ Rnx , c ∈ C, λ0 ∈ S(c). The target classification probability is calculated by ³ P c|Ω

k

´

¡ ¢ ¡ ¢ p ω k | c, Ωk−1 P c | Ωk−1 = PM ¡ ¢ ¡ ¢, k−1 P ci | Ωk−1 i=1 p ω k | ci , Ω

(5)

P with an initial prior target classification probability P0 (c), c∈C P0 (c) = 1. The state ˆ ck for each class c estimate x ³ ´ X Z c ˆk = x xk p xk , λk , c | Ωk dxk , c ∈ C (6) λk ∈S(c)

xk ∈Rnx

³ ´ P ˆ ck P c | Ωk . ˆ k = c∈C x takes part in the calculation of the combined state estimate x It is obvious that the estimates, needed for each class, can be calculated independently from the other classes. Therefore, the JTC task can be accomplished by the simultaneous work of M independent filters. SMC methods provide a number of useful suboptimal algorithms to approximate the optimal JTC solution, given by (3)-(6). In the general case of nonlinear state and measurement equations, particle filters represent the above complicated probability disn oN (j) (j) (j) for tributions by a set of N discrete, weighted (by W ) samples λk , xk , Wk j=1

each class c, and utilize importance sampling and weighted resampling to complete the filtering task [1]. However, for a particular case of a system model (1)-(2), more efficient mixture Kalman filtering algorithm is proposed in [7,8]. The dynamic system model (1)-(2) under consideration, belongs to the class of conditional dynamic linear models (CDLM). In the terminology of the CDL models, the modal state variable λk is denoted as an indicator variable. For a given trajectory of the indicator λk , k = 1, 2, . . . , the system is both linear and Gaussian and can be estimated by the Kalman filter. The MKF exploits the conditional Gaussian property and utilizes a marginalization operation to improve the efficiency of the conventional SMC procedure. The samples are generated only in the indicator space and the target state distribution is approximated by a mixture of Gaussian distributions. Let Λk = {λ0 , λ1 , λ2 , . . . , λk } be the set of indicator variables up to time instant k. By

4

/ Joint Target Tracking and Classification (j)

(j)

recursively generating a set of properly weighted random samples {(Λk , Wk )}N j=1 to represent the pdf p(Λk |Ωk ) (for class c), the MKF approximates the state pdf p(xk |Ωk ) by a random mixture of Gaussian distributions [7] N X

(j)

(j)

(j)

Wk N (µk , Σk ),

(7)

j=1 (j)

(j)

(j)

(j)

where µk = µk (Λk ) and Σk = Σk (Λk ) are obtained by a KF, designed with the (j) (j) (j) system model (1)-(2), corresponding to class c. We denote by KFk = {µk , Σk } the sufficient statistics that characterize the posterior mean and covariance matrix of the state xk , conditional on the set of accumulated observations Ωk and the indicator trajectory (j) (j) (j) (j) Λk . Then based on the set of samples {(Λk−1 , KFk−1 , Wk−1 )}N j=1 at the previous (j)

(j)

(j)

time (k − 1), the MKF produces a respective set of samples {(Λk , KFk , Wk )}N j=1 at the current time k. The correctness of the ³procedure is proven in [8]. Using the likeli´ hood function of class c ∈ C at time k – p ω k | c, Ωk−1 , the class probabilities are calculated according to (5). 3. MKF-based algorithm for JTC Target model. In the two-dimensional target dynamics given by (1), the state vector x = (x, x, ˙ y, y) ˙ 0 contains target positions and velocities in the horizontal (Oxy) Cartesian coordinate frame. The control input vector u = (ax , ay )0 includes target accelerations along x and y coordinates. The process noise w = (wx , wy )0 models perturbations in the accelerations. The transition matrices F and G are [9] µ ¶ ³ 2 ´0 1T F = diag (F 1 , F 1 ) , G = diag (g 1 , g 1 ) , for F 1 = , g 1 = T2 , T , 01 where T is the sampling interval and B = G. The target is assumed to belong to one of two classes (M = 2), representing either a lower speed commercial aircraft with limited maneuvering capability (c1 ) or a highly maneuvering military aircraft (c2 ). The flight envelope informationp comprises speed and acceleration constrains, characterizing each class. The speed v = x˙ 2 + y˙ 2 of each class is limited respectively to the interval: {c1 : v ∈ (100, 300)} [m/s] and {c2 : v ∈ (150, 650)} [m/s]. The control inputs are restricted to the following sets of accelerations: {c1 : u ∈ (0, +2g, −2g)} [m/s2 ] and {c2 : u ∈ (0, +5g, −5g)} [m/s2 ], where g = 9.81 [m/s2 ] is the acceleration due to gravity. The acceleration process uk is a Markov chain with five states (modes) s(c1 ) = s(c2 ) = 5. The following sets of modes (ax , ay ) are selected in the implementation: {(0, 0), (0, A), (0, −A), (−A, 0), (0, A)}, where A = 2g stands for class c1 target and A = 5g refers to the class c2 . The initial probabilities of the Markov chain are selected equal for the two classes: P0 (1) = 0.6, P0 (i) = 0.1, i = 2, . . . , 5. The transition matrix π(c) is assumed of the same form for both types of targets: pij = 0.7 for i = j; p1j = 0.075 for j = 2, . . . , 5; pi1 = 0.15 for i = 2, . . . , 5; pij = 0.05 for j 6= i, i, j = 2, . . . , 5. 2 2 The process noise is Gaussian, w ∼ N (0, Q), Q = diag(σw , σw ), having different standard deviations for each mode and class:


5

© ª 1 j = 7.5; σw = 14.0 [m/s2 ], j = 2, . . . , 5ª and © c1 : σw 1 j c2 : σw = 7.5, σw = 17.5 [m/s2 ], j = 2, . . . , 5 Measurement model. The measurement vector z = (D, β)0 consists of the distance D to the target and bearing β, measured by the radar. The covariance matrix of the mea2 surement error vector v ∼ N (0, R) has the form R = diag(σD , σβ2 ). For the purposes of the MKF design, a measurement conversion is performed from polar (D , β) to Cartesian (x , y) coordinates: z = (D sin(β), D cos(β))0 . Thus, the measurement equation becomes linear with a nz × nx measurement matrix H, where all elements are zeros, except for H11 = H23 = 1. The components of the corresponding measurement noise covariance matrix R can be found in [9]. The following sensor parameters are selected in the simulations: σD = 120.0 [m] σβ = 0.20 [deg]. The sampling interval is T = 5 s. Speed constraints. Acceleration constraints are imposed on the filter operation by the use of a control input in the target model. The speed constraints are enforced through the speed likelihood functions. They are constructed based on the speed envelope information (3). We define the following speed likelihood functions, respectively for each class  if vck1 ≤ 100 [m/s]  0.9, c1 c1 g1 (vk ) = 0.9 − κ1 (vk − 100) , if (100 < vck1 ≤ 300 [m/s])  0.05 if vck1 > 300 [m/s]  if vck2 ≤ 150 [m/s]  0.1, c2 c2 g2 (vk ) = 0.1 + κ2 (vk − 150) , if (150 < vck2 ≤ 650 [m/s])  0.95, if vck2 > 650 [m/s] where κ1 = 0.7/200 and κ2 = 0.85/500. According to the problem formulation, presented in Section 2, feature measurements y k , k = 1, 2, . . . are used for the purposes of classification. In our case we do not have feature measurements. The speed estimates, together with speed likelihood functions, form a virtual “feature measurement” set {Y k }. At each time step k, the filtering algorithm gives a combined state estimate ˆ k . Let us assume that the estimated speed from the previous time step, ˆvk−1 , is a kind x of “feature measurement". The likelihood function is factorized p (ω k | xk , λk , c) = f (z k |xk , λk ) gc (y k ) where y k = ˆvk−1 . Practically, the normalized speed likelihoods represent speed-based class probabilities estimated by the filters. n

(j) (j) MKF for JTC. Consider the set of N compound particles c(j) k , λ k , xk

oN j=1

. The class

variable ck is assumed to be independent of ck−1 . It can take each possible value in C with an equal probability, i.e., P (ck = c) = 1/M, c ∈ C. The indicator variable λk takes values from the set S and evolves according to a Markov chain with transition probability matrix π. The set of samples is initialized according to the known initial class, state and mode distributions. The initial weights have equal probabilities (1/N ). At every time instant k = 1, 2, . . ., for each particle (j), j = 1, . . . , N , first a class vari(j) (j) (j) able ck is drawn. Then, depending on the realization of the class variable, λk and xk are generated in the following way [6]: the MKF scheme runs s KF prediction steps, according to each λ ∈ S. The likelihoods of the predicted states are calculated based on the received measurement z k . They form a trial sampling distribution, according to which

6

/ Joint Target Tracking and Classification (j)

the new λk is selected. Then the KF update step is accomplished only for the selected (j) (j) λk . The weight Wk is updated based on the factorized likelihood of the measurement ωk . The sum of the weights, pertaining to class c, form the likelihood of class c and takes part in the calculation of the posterior class probabilities (5). ˆ ck is updated based on the normalized weights of the particles, corThe state estimate x ˆ k is evalresponding to c, according to (7). Finally, the combined output state estimate x uated according to the explanations in Sec. 2. The sequential update of the pdfs (3)-(4) is finalized. The resampling procedure deals with the elimination of particles with small weights and replicates the particles with higher weights. 4. Simulation results The simulated path of a second class target is shown in Fig. 1 (a). It performs four turn maneuvers with normal accelerations 1g, 2g, −5g, 2g. The third 5g turn provides an insufficient true class information, since the maneuver is of short duration, and the next 2g turn can lead to a misclassification. The target speed of 250 [m/s] provides better conditions for the probability, that the target belongs to class 2, according to the speed constraints. The estimated speed probabilities assist in the proper class identification, as we can see in Fig. 1 (b). The Root-Mean Squared Errors (RMSEs) [9]: on position (both coordinates combined) 70

class # 1 class # 2

65 1 60

55 0.8

y [km]

50

45

0.6

40 0.4 35

30 0.2

START 25

t [scans]

x [km] 20 20

25

30

35

40

45

50

55

60

65

0

70

0

10

Figure 1. (a) Test trajectory 340

20

30

40

50

60

50

60

70

80

(b) Class probabilities 180

combined class # 1 class # 2

300

280

260

combined class # 1 class # 2

160

Speed RMSE [m/s]

Position RMSE [m]

320

140

120

100 240 80 220 60 200 40

180

t [scans] 160

0

10

20

30

40

50

60

70

t [scans] 80

Figure 2. (a) MKF position RMSE [m]

20

0

10

20

30

40

(b) MKF speed RMSE [m/s]

70

80


7

and speed (magnitude of the velocity vector) are presented in Fig. 2. The RMSEs shown are for each separate class, and the combined RMSE obtained after averaging with the class probabilities. The MKF is implemented with N = 200 particles. The results are based on 100 Monte Carlo runs. The experiments show, that the filter provides reliable tracking of intensively maneuvering targets with accelerations up to 5g with acceptable errors. The results confirm the theoretical inferences [7], that the MKF can give better estimation accuracy compared to the particle filter. It was also demonstrated by the detailed studies in [6]. The suggested speed and acceleration constraints ensure a possibility of correct class determination even in the cases of rather complex and conflicting scenarios. The computational complexity of the proposed MKF allows for an on-line implementation. The experiments are performed on PC computer with AMD Athlon processor 1.4 GHz. The MKF computational time in Matlab environment is 0.3 seconds per scan.

5. Concluding remarks We propose a mixture Kalman filter for joint maneuvering target tracking and classification in two classes: commercial and military aircraft. For a proper classification, two kinds of constraints are imposed on the target kinematic parameters: on the acceleration and on the speed. The operation of two multiple model class-dependent MKFs is simulated by a suitably determined random class variable. Thus a relatively simple structure of the algorithm is achieved. The filter performance is analyzed by simulation over typical target trajectories in a plane. The results show reliable tracking and correct class discrimination. The generalization for more target classes is straightforward. References [1] A. Doucet, N. de Freitas, N. Gordon (editors), Sequential Monte Carlo Methods in Practice, Springer-Verlag, New York, 2001. [2] N. Gordon, S. Maskell, T. Kirubarajan, Efficient particle filters for joint tracking and classification, Proc. SPIE Signal and Data Processing of Small Targets, 2002. [3] M. Malick, S. Maskell, T. Kirubarajan, N. Gordon, Littoral Tracking Using Particle Filter, Proc. of the Fifth Int. Conf. Information Fusion, 2002. [4] E. Blasch, C. Yang, Ten Methods to Fuse GMTI and HRRR Measurements for Joint Tracking and Identification, Proc. of the 7th Intl. Conf. on Inf. Fusion, pp.1006-1013, 2004. [5] B. Ristic, N. Gordon, A. Bessell, On Target Classification Using Kinematic Data, Information Fusion, Elsevier Science, 5, pp.15–21, 2004. [6] D.Angelova, L. Mihaylova, Joint Target Tracking and Classification with Particle Filtering and Mixture Kalman Filtering Using Kinematic Radar Information, accepted for publication in Digital Signal Processing, Elsevier Science, 2005. [7] R. Chen, J. Liu, Mixture Kalman Filters, Journal of the Royal Statistical Society B, 62, pp. 493–508, 2000. [8] R. Chen, X. Wang, J. Liu, Adaptive Joint Detection and Decoding in Flat-Fading Channels via Mixture Kalman Filtering, IEEE Trans. on Inform. Theory, 46, pp.493–508, 2000. [9] Y. Bar-Shalom, X.R. Li, Multitarget–Multisensor Tracking: Principles and Techniques, YBS Publishing, Storrs, CT, 1995.