Tracking spawning objects

Transcription

1 Published in IET Radar, Sonar and Navigation Received on 10th February 2012 Revised on 10th September 2012 Accepted on 18th December 2012 Tracking spawning objects Ronald P.S. Mahler 1, Vasileios Maroulas 2 1 Lockheed Martin MS2 Tactical Systems, Minnesota, USA 2 Department of Mathematics, University of Tennessee, Knoxville, TN, USA vmaroula@utk.edu ISSN Abstract: Many multi-object tracking scenarios are complicated by the fact that an object of interest may spawn additional objects which, for some period of time, follow roughly the same trajectory as the original object and then fall away. The challenge is then to discriminate the original object from the spawned ancillaries in a timely fashion. This study proposes a solution to this problem based on the increasingly well-known multi-object track-before-detect algorithm called the cardinalised probability hypothesis density (CPHD) filter. Precisely, the authors assume zero false alarms (ZFA) in the CPHD filter, and apply the proposed scheme to linear and non-linear simulation scenarios based on widely used object-trajectory and sensor models. The authors have also demonstrated that a Gaussian mixture implementation of the ZFA-CPHD filter (i) establishes stable estimates of object number, (ii) rapidly eliminates the ancillary objects and (iii) detects and accurately estimates the trajectory of the original object of interest. 1 Introduction The problem addressed in this paper is the rapid detection and localisation of a primary object of interest during and after the point that it spawns a number of ancillary objects. This problem can be challenging because during and after spawning, the trajectory of the original object may be obscured by the ancillary objects. This is because they may follow essentially the same trajectory as the original object, and only subsequently fall away under the force of gravity. A multi-object detection and tracking algorithm of some sort (e.g. [1 3]) is required in order to distinguish the primary object from the ancillaries as quickly as possible, and track it accurately thereafter. We have been able to find only a small number of open-literature papers that address this or related problems. Gardner et al. [4] employ Kalman filters to track spawned objects, but assume that the measurement-to-track association problem has been resolved. That is, given a sequence of measurement, it is known a priori which object generated which measurement. This assumption significantly simplifies the analysis since an important source of uncertainty has been removed. Isaac et al. [5, 6] employ monopulse-radar signal processing techniques, combined with an auxiliary particle filter, to detect and track spawned objects. They assume that, during a sampling period, the number of objects can increase by at most one, which, again, is a significant simplification of a spawning event. In contrast, our approach does not presume any resolution on the difficult combinatorial problem of measurementto-track association. Furthermore, our strategy does not require that the number of objects can increase only by one. In fact, our method proposed in this paper is fundamentally different, and addresses the spawning problem in greater generality than apparently has previously been the case. We focus on primary objects whose measurements are detected by a single sensor. We consider four different scenarios for the primary objects. In the first two cases, we actually focus on a single primary object, which spawns either once with a greater number of ancillaries or twice with a smaller number of ancillaries at each time. The third case focuses on two crossing objects which spawn at two different times, and the fourth scenario considers non-linear models. In any case the unknown trajectories of ancillaries are determined only by the force of gravity. The readers should remark that ancillary objects follow different trajectories from the primary object. Therefore the greater the number of ancillary objects ejected, the larger the number of different trajectories we should expect. As quickly as possible, we have to distinguish the primary object(s) from the spawned objects, and track them thereafter. The key innovation in our approach is to conceptually view the evolving collection of objects as a single set-valued state, and the collection of sensor detection-measurements as a single set-valued observation. The random finite set (RFS) theory [1], described briefly in Section 2.1, provides a unified framework that avoids the simplifying assumptions employed in the earlier approaches just described. The RFS theory generalises the single-object Bayes filtering to a rigorous formulation of Bayesian multi-object detect-before-track filtering. The general multi-object Bayes filter is, however, computationally expensive. Consequently, a number of different approximate implementations of this filter have been devised, with the aim of decreasing the algorithmic cost. In this paper, we consider the generalisation of the PHD filter, which is a statistical IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

2 higher-order multi-object-moment approximation of the multi-object Bayes filter, the so-called CPHD [1, 3, 7]. The CPHD filter propagates both the PHD and the distribution on the number of objects (the cardinality distribution), resulting in increased computational complexity: O(m 3 n), m is the number of measurements and n is the number of monitored objects. However, in many applications, for example, sky-gazing radar, the measurements collected by the sensor will be almost entirely object-generated. Therefore, in this paper, we assume (a valid assumption) that the false alarm rate and ancillary object rate will both be negligible. Under this assumption, the CPHD filter reduces to a novel special case, the zero false alarm (ZFA)-CPHD filter, with complexity O(mn). A Gaussian-mixture implementation of the ZFA-CPHD filter will be used in our simulated scenarios. In the special case of a non-linear environment, we use a modified version of the ZFA-CPHD filter, defined as the extended Kalman (EK) ZFA-CPHD filter. The EK-ZFA-CPHD filter is a modification of the corresponding extended Kalman filter proposed in [3]. The paper is organised as follows: Section 2 discusses briefly the necessary generally needed statistical background. Precisely, Section 2.2 describes the ZFA-CPHD filter, and novel Gaussian implementations of the ZFA-CPHD filter are established in Section for the linear scenarios. The corresponding non-linear case and its approximation are exposed in Section The realistic simulations based on the Gaussian implementations of the ZFA-CPHD filter are described in Section 3. Conclusions of this manuscript can be found in Section 4. 2 Multi-object filtering Single-object motion modelling is relatively simple because only the presumed physical motion of individual objects is encountered, and states and all measurements are vectors of fixed dimensionality. However, the object-spawning problem addressed in this paper intrinsically requires employment of a multi-object detection and tracking filter. We employ a foundational statistical approach. We conceptually view the evolving collection of one or more objects as a single set-valued state, and the collection of sensor detection-measurements as a single set-valued observation. This requires a statistical framework that is capable of dealing with set-valued measurements and states: the theory of RFSs. In this section we briefly summarise this approach, referring the readers to [1] for more details. For clarity, we establish at the outset the following notational conventions: Permutations will be denoted by Pm n ( ) = n!/ (n m)!, with the convention Pm n = 0ifn < m. The cardinality of a set, X, is denoted as X. N( ; μ, P) denotes a Gaussian density with mean μ and covariance P. The transpose of a matrix A is denoted by A T. I m denotes a m m identity matrix. O m denotes a m m zero ( matrix. F R N ) is the collection of all finite subsets of R N. 2.1 Random finite set framework of multi-object filtering Let X t = { x 1 t, x 2 t,..., x n } ( t [ F R t N ) be a collection of object states at time t, n t represents the number of objects at time t. { Moreover, the sensor receives at time t +1, Z = z 1 t, z 2 t,..., z m } t [ F R M measurements, m t +1 is the number of generated measurements at time t + 1. Analogous to the single-object tracking problem, randomness is characterised by modelling the states and the measurements, the stochasticity in a multi-object framework is represented by modelling multi-object states, Ξ t, and multi-object measurements, Σ t, as RFSs on the single-object state and observation spaces R N and R M, respectively. The multi-object dynamics and observations are described as follows: Given a realisation X t of the RFS, Ξ t, at time t, the multi-object state at time t + 1 is modelled by the RFS J = { } < x[xt S (x) G (1) S t +1 t is the RFS representing the objects which survived with probability p s (χ), from the previous time t, and Γ t the RFS which represents the objects which entered the scene at time t + 1. Hence, one may conclude that the RFS, Ξ t +1, encapsulates all information of multi-object motion, such as the time-varying number of objects, individual object motion and object birth. Given a realisation X t +1 of Ξ t +1 at time t + 1, the multi-object measurements are modelled via the following RFSs { } = < x[xt Q (x) K (2) Θ t +1 (x) is the RFS of measurements generated by the object x X t, and K t +1 denotes the RFS of ancillary object or false alarms. The RFS Σ t +1 encapsulates all sensor characteristics such as measurement noise, sensor field of view and false alarms. Now, let f t t (X t Z 1:t ) denotes the multi-object posterior density at time step t conditioned on the time sequence Z 1:t =. { } Z 1, Z 2,..., Z t of observation sets accumulated at t. Adopting the ideas from the single-object Bayesian framework, for example, see Chapter 14 of [1], the multi-object Bayes filter propagates the multi-object filtering distribution in time via the recursion ( f X Z 1:t = f X X ) ( f X ) Z 1:t dx (3) f Z f X Z 1:t f +1 X Z 1: = (4) f Z f X Z 1:t dx dx is the set integral, see Definition 10 of [8]; f t +1 t (X X ) is the multi-object transition density associated with (1), and f t +1 (Z t +1 ) is the multi-object likelihood obtained from (2). Using techniques from finite set statistics (FISST), Chapter II.5 of [8], and extending the concept of Radon Nikodym derivative, the multi-object transition density and likelihood are well defined and closed forms are available. 2.2 Zero false alarm CPHD filter The multi-object filter described in (3) and (4) is computationally expensive. Thus, a statistical multi-object first-moment approximation, the probability hypothesis density (PHD) filter, was introduced in [1, 9]. Unfortunately, the PHD s instantaneous estimate of the 2 IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

3 expected number of objects is unstable in the presence of missed detections. Hence, generalisation of the PHD filter, the cardinalised probability hypothesis density (CPHD) filter was introduced in [1, 7]. The CPHD filter jointly produces the PHD, and the cardinality distribution. Definition 1: The CPHD filter recursively propagates a first-order multi-object moment of the multi-object process: the intensity function or PHD D t t (x Z 1:t ). Given any region S # R N, the integral S D k k x Z 1:t dx is the expected number of objects in S. In particular, if S = R N, then N t t = D t t x Z 1:t dx is the total expected number of objects in the scene. The CPHD filter also recursively produces the entire posterior probability distribution, p t t (n Z 1:t ), on object number n and thus also its probability generating function (p.g.f.) is given via G t t (x) = n 0 p ( t t n Z 1:t ) x n. The following theorems show explicitly how the posterior PHD and the posterior cardinality distribution are jointly propagated in time. The readers should refer to Chapter 16 of [1] for more details. Let kh 1, h 2 l = h 1 (x) h 2 (x) dx (5) denote the L 2 inner product of functions h 1 (x), h 2 (x), let f t +1 t (x x ) be the single-object Markov density, and let b t +1 t (x) and p B (n)=p B,t (n) be, respectively, the intensity function and cardinality distribution of the objectappearance model. Let p S (x )=p S,t (x ) be the probability that an object with state x at time-step t will survive into time-step t +1. Theorem 1 (Prediction): At time-step t, assume that the prior PHD is D t t (x)=d t t (x Z 1:t ) and the prior cardinality distribution is p t t (n)=p t t (n Z 1:t ). Then the CPHD filter prediction equations are D (x) = b (x) + p S,t (x )f (x x )D t t (x ) dx (6) p (n) = n p B (n i) 1 l kdt t, 1 p S l l i kd t t, p S l i i=0 l=i i Nt t l p t t (l) (7) Next, assume that at time-step t + 1, a new observation set Z is collected. Let m = Z t +1 be the cardinality of the set Z t +1. Let p D (x)=p D, t +1 (x) be the probability at time-step t + 1 that an object with state x will be detected and L z (x)=f t +1 (z x) be the single-object likelihood function. Take p ( m) tobethe probability distribution on the number m of false alarms, and c t +1 (z) the spatial distribution of the false alarms. For any measurement-set Z ={z 1,, z m } with Z =m, let ( kd, p D L z1 l s i (Z) = s m,i,..., kd ), p D L zm l (8) c z 1 c z m s m,i x 1,..., x m is the elementary symmetric function of degree i with variables x 1,..., x m. Finally, denote the constant p =. N 1 kd, 1 p D l, N = D (x) dx. The theorem below exposes the update equations, and its proof is delegated to Appendix. Theorem 2: (Update): At time-step t + 1, assume that the predicted PHD is D t +1 t (x)=d t +1 t (x Z 1:t ) and the predicted cardinality distribution is p t +1 t (n)=p t +1 t (n Z 1:t ). The predicted p.g.f. is therefore G (x) = 1 n=0 p (n)x n, and let G (i) (x) denotes its ith derivative. Then the CPHD filter measurement-update equations are given below p +1 (n) = D +1 (x) = (1 p D (x))l 0,Z + p D (x) l Z (n) = 1 N n min{m,n} l Z (n)p (n) 1 i=0 l Z (i)p (i) z[z kd, 1 p D l n j s j (Z) (9) L z (x) c (z) l 1,Z {z} D (x) n m 1 p j j (m j) m l 0,Z = (m j)!p (m j)n j 1 G(j+1) (p)s j(z) m i=0 (m i)!p (m i)n i G(i) (p)s i(z) see (13) (10) (11) (12) Remark 1: Equations (6) and (7) follow, respectively, from (16.312) and (16.313) equations of [1]. Similarly, (9) is an algebraic simplification of equation (16.333) in [1], and (10) is a slight rewriting of equation (16.327) in [1]. The CPHD produces stable (low-variance) estimates of object number, as well as better estimates of the states of individual objects [1]. This gain in performance is achieved, however, with increased computational cost. As previously noted in Section 1, if the number of false alarms and/or ancillary object measurements is negligible, the CPHD filter reduces to a less computationally demanding special case. We call this special case the ZFA-CPHD filter. In this subsection we explicitly summarise the equations for the ZFA-CPHD filter. l 1,Z {z} = (m 1 j)!p (m 1 j)n j 1 G(j+1) (p)s j Z {z} m i=0 (m i)!p (m i)n i G(i) (p)s i(z) m 1 (13) IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

4 The prediction step of the ZFA-CPHD filter (Theorem 2) remains the same as for the CPHD filter. The correction step of the ZFA-CPHD filter is simplified, and thus it is more computationally tractable in comparison to the entire CPHD filter [1, 7]. Theorem 3: (Update): At time-step t + 1, assume that the predicted PHD is D t +1 t (x)=d t +1 t (x Z 1:t ) and the predicted cardinality distribution is p t +1 t (n)=p t +1 t (n Z 1:t ). Then the measurement-update equations for the ZFA-CPHD filter are D +1 (x) = L Z (x)d (x) (14) p +1 (n) = ˆl Z (n)p (n) 1 i=m ˆl Z (i)p (i) ( ˆl Z (n) = n ) n kd, 1 p D l m L Z (x) = 1 p D(x) N N G (m+1) (p) G (m) (p) + z[z p D (x)l z (x) kd, p D L z l (15) Proof: Since no false alarms exists, we have that the probability distribution p t +1 (m) on the number m of the false alarms equals to δ 0,m in the corrector equations for the CPHD filter. In this case, m n and since σ m, m (x 1,, x m ) = x 1 x m, (11) is simplified to as follows l Z (n) = 1 N n = 1 N n n kd, 1 p D l n m s m (Z) m n kd m, 1 p D l n m kd, p D L z1 l kd, p D L zm l c (z 1 ) c z m Furthermore, the corresponding parameters of (12) and (13) will respectively equal to (p) G (m) (p) l 0,Z = 1 G (m+1) N l 1,Z {z} = s m 1 Z {z} = s m (Z) c (z) kd, p D L z l The readers should remark that in the denominator and numerator of the fraction that defines p t +1 t +1 (n), the factors D t +1 t,1 p D m and σ m (Z) do not involve n and thus cancel. 2.3 Closed-form solution to the ZFA-CPHD recursion Given the formulation of the ZFA-CPHD filter in Theorems 1 and 3, and motivated by the Gaussian-mixture implementations of the PHD and CPHD filters, respectively [3, 10 13], we implement the analogous closed-form solution to the ZFA-CPHD filter. The Gaussian-mixture implementation is based on the following assumptions: (i) the single-object likelihood function is linear-gaussian; (ii) the single-object Markov transition density is linear-gaussian; (iii) the probability of detection is approximately constant in the region of interest; (iv) the probability of object-survival is constant; and (v) the PHD of the object-appearance process is a Gaussian-mixture. In more detail: Assumption 1: Each object follows a linear-gaussian dynamical model f t t 1 (x z) =. N x; F t 1 z, Q t 1 (transition dynamics) (16) g t (z x) =. N z; H t x, R t (observation dynamics) (17) F t 1 is the state transition matrix, Q t 1 is the process noise covariance, H t is the observation matrix and R t is the observation noise covariance. Assumption 2: The survival and detection probabilities are state-independent, that is, p S (x) ; p S, p D (x) ; p D, x [ R N. Assumption 3: The intensity measure of the birth RFS is a Gaussian-mixture of the form b t (x) = J b,t i=1 w (i) b,t N x; m(i) b,t, Pi b,t (18) w (i) b,t, m(i) b,t, Pi b,t are the weights, means and covariances of the mixture birth intensity. The discussion below involves the main equations of the Gaussian-mixture implementations of the ZFA-CPHD filter. We analyse the linear and non-linear case separately. Depending on the scenario, the corresponding implementations are adopted in Sections 3.1 and 3.2, respectively Linear models: The proposition below is the Gaussian implementation of Theorem 2 for the prediction step. Since the prediction remains the same under the ZFA-CPHD filter s assumptions, the proof of Proposition 1 is omitted. Proposition 1: (Prediction): Assume that at the recurrent time t the PHD, D t t, and the cardinality distribution, p t t (n) are given. Furthermore, presume that the intensity measure is a Gaussian-mixture of the form D t t (x) = J t i=1 w (i) ( t N x; m (i) t, P t (i) ) Then, D t +1 t is also a Gaussian-mixture, and the ZFA-CPHD prediction equations, (6) and (7), simplify to p (n) = n D (x) = b(x) + D S, (x) (19) p B (n j) 1 l=j l p j ( j S 1 p ) l jpt t S (l) (20) 4 IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

5 Jt b(x) as defined in (18) and D S, = p S i=1 w(i) t N x; m (i) S,, P(i) S,, such that μ S,t +1 t = F t μ t, and P S, = Q t + F t P t Ft T. The two lemmas stated below are critical for establishing Proposition 2. For their proof, the readers should refer to [13] and references therein. Lemma 1: Assume that F, Q are two positive-definite n n matrices. Further, let d, μ be two n 1 vectors, then ( N(x; Fz + d, Q)N(z; m, P)dz = N x; Fm + d, Q + FPF T ) Lemma 2: Assume that H, R, P are n n matrices, and that R, P are positive-definite as well. Furthermore, let μ be a n 1 vector, then N(z; Hx, R)N(x; m, P) = q(z)n x; m, P ( q(z) = N z; Hm, R + HPH T ), m = m + K(z Hm) and P = (I KH)P, K = PH T ( HPH T ) 1 + R Proposition 2: (Update): Suppose that at recurrent time t + 1 the predicted PHD, D t +1 t, and predicted cardinality distribution, p t +1 t, as derived in Proposition 1, are given. Moreover, assume that the predicted intensity is a Gaussian-mixture of the form D (x) = J i=1 w (i) N x; m(i), P(i) Then, D t +1 t +1 is also a Gaussian-mixture, and the ZFA-CPHD update equations, (14) and (15), simplify to D +1 [ 1 ] 1 n=m+1 = q D Pn m+1p (n)q n (m+1) D J 1 i=1 w(i) n=m Pn mp (n)qd n m D (x) + p D z[z J i=1 w (i) p +1 (n) = p (n) (z)n x; m(i) m = Z, w (i) (z) = q (i) (z) = N z;,h(i), R (i) (z), P(i) Pmq n D n m 1 l=m Pl mp (l)q l m D w (i) q(i) (z) J i=1 w(i) q(i) (z), (21) (22) for h (i) = H m (i), and R (i) = R + H P (i) H. T The mean and the covariance matrix are m (i) (z) = m (i) + K(i) z H m (i) [ ], P (i) = I K(i) H P (i), respectively, K (i) = P(i) H T ( R + H P (i) H T ) 1. Proof: Using Assumption 2 one may observe that q D, D t +1 t = q D 1, D t +1 t and thus (22) is established. Furthermore, using all the assumptions stated above, one may apply directly Lemma 2 for the product g t +1 (z x)d t +1 t, Lemma 1 for the inner product D t +1 t, g t +1 ( ), and observe that k1, D l = J i=1 w(i). Thus (21) collapses Non-linear models: In this section, a non-linear extension of the Gaussian ZFA-CPHD filter is discussed. Basically, Assumption 1 is relaxed, and the dynamics of transition and observations are given by x t = f x t 1, v t 1 z t = g x t, 1 t φ t, γ t are non-linear functions, and v t 1, ε t are Gaussian noise processes with covariance matrices Q t 1, R t, respectively. In this work, we consider an EK-ZFA-CPHD filter based on local linearisations of φ, and γ in order to approximate the non-linearities. The EK-ZFA-CPHD filter is a modification of the EK-CPHD filter studied in [3]. First let us consider the predicted step as it was described in Proposition 1. Therein, we replace μ S,t +1 t, and P S,t +1 t by its non-linear analogue P (i) S, = G(i) t m (i) S, = fm(i) t, 0 [ ] T + F (i) Q t G (i) t t [ P t F t (i) ] T F (i) ( ) t = f(x, 0) / x x=m (i) and G (i) (( ( t = f m (i) t, t v))/ v ) v=0. Similarly the update step of Proposition 2 will be modified such that ver non-linearities are considered it will be approximated by using the following equations and h (i) = g m(i) S,, 0 [ ] R (i) T = U (i) R U (i) + H (i) P(i) H (i) U (i) t = g(x, 0) = (i) x x=m S, g m (i) S,, =0 [ ] T H (i) IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

6 3 Algorithm using the Gaussian-mixture ZFA-CPHD filter Numerous ancillary objects will obscure the trajectory of the primary object of interest, when spawning occurs. The ancillary objects will be extremely close to the corresponding primary object during the initial period following spawning. Consequently, it is difficult to distinguish and track the trajectory of the primary object because of the fact that it is obscured within the ancillary objects. In this section, motivated by Vo et al. [3], we describe our algorithm, and perform simulations in which we try to determine the effectiveness of our approach in distinguishing the primary from the ancillary objects. Step 0: Initialisation: At time t = 0, the initial PHD, D 0x 0, is the sum of J 0 Gaussian distributions. The initial cardinality distribution is set a priori to a single object. Step 1: Time-update: At time t the predicted PHD D t +1 t is a Gaussian-mixture whose weight, mean and covariance matrix can be derived as in (6). The predicted cardinality distribution is given as in (7). Step 2: Measurement-update: At time t + 1 the posterior PHD D t +1 t +1 is a Gaussian-mixture whose weight, mean and covariance matrix can be derived via (14) and the posterior cardinality distribution according to (15). Step 3: Merging and pruning: Gaussian components that are similar are merged, and then Gaussian components with negligible weights are discarded. Step 4: Multi-object state extraction: The number of objects is estimated from the cardinality distribution using a MAP estimator. This is used to extract a multi-object state estimate from the PHD [1]. Step 5: Resolution of the spawning event: During most of a trajectory there will be only a single object. During and after spawning, state extraction provides estimates of the states of the objects, whether primary or ancillary. Immediately after separation, the primary object will have the same velocity, and thus the presumed position of the primary object can be predicted at any time-step. The ancillary objects, on the other hand, will fall to Earth under the force of gravity. The distance between each such ancillary object and the presumed location of the primary object is computed. When this distance exceeds a certain threshold, the corresponding object is declared to be ancillary and is discarded from the list of displayed tracks. This process is repeated until only a single object the primary object of interest remains. The readers should remark that the objects are deleted only in the sense they are deleted from the figures given that they exceed some prespecified threshold. But, they are not deleted from the Gaussian-mixture. 3.1 Numerical results: linear case In this section we determine the performance of our approach by applying it to four realistic simulation scenarios (e.g. see [1, 3, 13] and references therein). An alternative could perhaps be to validate our algorithm on real data. However, data is difficult to obtain and usually proprietary. Moreover, validation on one data collection might lead to biased conclusions. In the first three simulations, the object trajectories are assumed to lie within a three-dimensional space One object of interest: In our first simulation we consider a primary object obscured by two successive spawning events, the first with six ancillary objects and the second with five ancillary objects. In our second simulation, the primary object is obscured by a single spawning event with 19 ancillary objects. Both scenarios are much more difficult, and general than Isaac et al. s approach [5, 6], the number of objects can increase by at most one. Furthermore, Gardner et al. [4] assume which target generates which measurement. This is an impractical assumption. Let us focus now on the first simulation. The state-vector of each object X t =. [ ] T p x,t, ṗ x,t, p y,t, ṗ y,t, p z,t, ṗ z,t consists ( of ) position (p x,t, p y,t, p z,t ) and velocity ṗ x,t, ṗ y,t, ṗ z,t. The single-object transition model is linear-gaussian with transition density as defined in (16), F t 1 = diag(a, A, A), Q t 1 = s 2 v diag(b, B, B) ( A = 1 D ) D 4 D 3, B = D 3 D 2 2 Δ = 1 s is the sampling period and s 2 v = 5 km/s 2 is the standard deviation of the process noise. Each object has survival probability p S,t = The cardinality distributions involved have a maximum of N max = 200 Gaussian components. The object-birth process is a Poisson RFS with PHD defined as in (18), w b = 0.03, m (1) [ ] T, b = m (2) [ b = ] T, m (3) [ ] T, b = m (4) [ ] T, b = and Pb = 100I 6. The probability of detecting an object is p D,t = The single-object measurement model is also linear Gaussian with likelihood as defined in (18), H t = , R t = s 2 1I and σ ε = 10 km is the standard deviation of the measurement noise. Since the ZFA-CPHD filter is being used, there are neither false alarms nor ancillary object detections. The Gaussian-mixture ZFA-CPHD filter uses a weight threshold T =10 5 for the pruning procedure and a threshold U =4m for the merging part of the algorithm. The distance threshold for separating the ancillary objects from the primary object is τ = 200 m. In the simulation, the sensor receives one measurement per second. The two spawning events occur at t = 20 and t = 60, respectively, six and five ancillary objects are spawned from the primary object. Our results are described in Figs. 1 and 2. Fig. 1 compares true object number with the ZFA-CPHD filter s estimate. Fig. 2 compares the true object x, y, z-positions with those estimated by the ZFA-CPHD filter. An object is discarded, and its position thus disappears from the graphs, when it is declared to be ancillary. The figure shows that the filter correctly tracks all objects, estimating seven objects at the first spawning event and six objects at the second event. In both cases, 19 measurements are required to eliminate the ancillary objects 6 IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

7 Fig. 1 Output from the ZFA-CPHD filter: cardinality distribution and lock onto the primary object of interest. Note that during the second spawning event, the ancillaries are initially very close to the primary. Analogous results are achieved in our second simulation. A single spawning event occurs at t = 20, spawning 19 new objects. The results are shown in Figs. 3 and 4. Once again, the filter correctly tracks all objects, correctly estimating that 20 objects are present immediately after spawning occurs. As one would expect, the larger number of ancillaries requires the filter to collect more measurements (27 measurements in total) before it can eliminate the final ancillary object Two crossing objects of interest: In the third simulation scenario, two targets of interest are being monitored. For this case, the Step 5 of our algorithm Fig. 2 Comparison between estimates of the ZFA-CPHD filter and the true tracks IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

8 Fig. 3 Output from the ZFA-CPHD filter: cardinality distribution needs to be enhanced such that we do not have an elimination of one of the two objects of interest from the filter while spawning occurs. To this end, we take advantage of the physical fact that at the time of spawning an object of interest and their corresponding ancillaries will have the same velocity (we allow some small perturbation). Therefore, before applying the distance criterion as it was described in the algorithm (see Step 5), we impose another criterion of examining if the velocities of two objects are similar. The spawning of five Fig. 4 Comparison between estimates of the ZFA-CPHD filter and the true tracks 8 IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

9 Fig. 5 Output from the ZFA-CPHD filter for two objects of interest: cardinality distribution ancillaries from the first object of interest occurs at t = 20, and the spawning of seven items from the second object of interest happens at t = 60 (see Fig. 5). It takes about 30 time steps for the filter to estimate accurately the cardinality. However, it is expected because both objects of interests move with close proximity. Given the new enhanced fifth step of our algorithm, the ZFA-CPHD filter delivers accurate estimates of the trajectories of both objects of interest (see Fig. 6). 3.2 Numerical results: non-linear case In this section the object of interest moves in a non-linear fashion, and the observation model consists of the bearing Fig. 6 Comparison between estimates of the ZFA-CPHD filter and the true tracks of two objects of interest IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

10 Fig. 7 Cardinality distribution, and comparison between estimates of the EK-ZFA-CPHD filter and the true tracks of two objects of interest and range coordinates. The same model without considering spawning was considered in [3]. The target state is [ summarised in the vector x t = y T ] T, t, v t [ ] T, y t = p x,t, ṗ x,t, p y,t, ṗ y,t and ω is the turn rate. The state dynamics are propagated via the following recursions y t = F v t 1 yt 1 + Gw t 1 and v t = v t 1 + Du t 1 sin vd 1 cos vd 1 0 v v 0 cos vd 0 sin vd F(v) = 1 cos vd sin vd 0 1 v v 0 sin vd 0 cos vd T 2 2 G = 0 T 0 0 T 0 T 2 2 We consider here that ( the sampling time is T =1s, w t is normally distributed, N ; 0, s 2 ) ( wi, with σw = 15 min/s 2, and u t N ; 0, s 2 ) u, σu = π/180 rad/s. The observations consist of bearing and range measurements as described below p x,t p y,t z t = + 1 t p 2 x,t + p 2 y,t ( 1 t N ; 0, R t, and Rt = diag s 2 u, ) T, s2 r σθ = π/90 rad/s, and σ r = 20 m. The birth RFS, the probability of detection, and survival remain the same as in the linear case. The object of interest is following a non-linear trajectory, and at t = 20, and t = 70, it spawns four ancillaries, respectively (see Fig. 7). Furthermore, the EK-ZFA-CPHD filter accurately estimates the position of the target, the number of objects at the time of spawning, but, as in the linear case, it takes some time to return back to 1. 4 Conclusion In this paper we implemented a Gaussian-mixture type of the ZFA-CPHD filter for the linear case, and we further investigated the EK-ZFA-CPHD filter ver non-linearities are encountered. Based on the two filters, we performed simulations aimed at determining the effectiveness of the ZFA-CPHD filter when applied to the object-spawning problem. Our method did not require any restriction on the number of ancillary objects, and any a priori resolution of the measurement-to-track association. To be effective, any proposed solution must be capable of quickly and accurately distinguishing the primary object of interest from the ancillary objects created during a spawning event. We showed that the ZFA-CPHD and the EK-ZFA-CPHD filters, used in conjunction with a 10 IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013

11 miss-distance criterion, are capable of achieving this goal in the current standard simulation scenarios. 5 Acknowledgments The authors gratefully acknowledge the contribution and assistance of Professors B.-N. Vo and B.-T. Vo for designing and implementing the original version of the ZFA-CPHD filter under Lockheed Martin funding, and the excellent comments of two anonymous reviewers. This paper has been approved for public release by DFOISR. The OFOISF reference number is 10-S References 1 Mahler, R.P.S.: Statistical multisource-multitarget information fusion (Artech House, 2007) 2 Maroulas, V., Stinis, P.: Improved particle filters for multi-target tracking, J. Comput. Phys., 2012, 231, (2), pp Vo, B.-T., Vo, B.-N., Cantoni, A.: Analytic implementations of the cardinalized probability hypothesis density filter, IEEE Trans. Signal Process., 2007, 55, (7), pp Gardner, A., Rollason, M., Everett, N., Salmond, D.: Maintaining track identity for a fragmenting target. The IEE Seminar on Target Tracking: Algorithms and Applications, 2006 (Ref. No. 2006/11359), March 2006, pp Isaac, A., Willett, P., Bar-Shalom, Y.: Detecting and tracking separating objects using direct monopulse measurements IEEE Aerospace Conf., March 2008, pp Isaac, A., Willett, P., Bar-Shalom, Y.: Quickest detection and tracking of spawning targets using monopulse radar channel signals, IEEE Trans. Signal Process., 2008, 56, (3), pp Mahler, R.P.S.: PHD filters of higher order in target number, IEEE Trans. Aerosp. Electron. Syst., 2007, 43, (4), pp Goodman I.R., Mahler R.P.S., Nguyen H.T.: Mathematics of data fusion (Kluwer Academic Publishers, Norwell, MA, 1997) 9 Mahler, R.P.S.: Multitarget Bayes filtering via first-order multitarget moments, IEEE Trans. Aerosp. Electron. Syst., 2003, 39, (4), pp Clark, D., Vo, B.-N.: Convergence analysis of the Gaussian mixture PHD filter, IEEE Trans. Signal Process., 2007, 55, (4), pp Clark, D.E., Panta, K., Vo, B.-N.: The GM-PHD filter multiple target tracker. Ninth Int. Conf. Information Fusion, 2006, July 2006, pp Vo, B.-N., Ma, W.-K.: A closed-form solution for the probability hypothesis density filter. Eighth Int. Conf. Information Fusion, 2005, July 2005, vol. 2, p Vo, B.-N., Ma, W.-K.: The Gaussian mixture probability hypothesis density filter, IEEE Trans. Signal Process., 2006, 54, (11), pp Appendix Below is given a proof of (9) of Theorem 2. Let C(x) ; C (x) = 1 l=0 c (l)x l and C (i) (x) be the p.g.f. and the ith derivative of the p.g.f. of false alarm distribution c t +1 (i). Recall that p =. ( ) 1/ N kqd, D l. According to the formula (16.333) in [1] (p. 639), the cardinality distribution for the CPHD filter corrector is p k+1 k+1 (n) / m C (m j) 1 (0) (n j)! Ĝ (j)(n j) (0) p n j s j (Z) Ĝ (j) (x) =. (( G (j) ) ) (x) / N j. It implicitly presumes that j m and j n and so should, strictly speaking, be written as p k+1 k+1 (n) / s j (Z) min{m,n} C (m j) 1 (0) (n j)! Ĝ (j)(n j) (0) p n j Therefore, performing algebraic procedures Thus p k+1 k+1 (n) / l Z (n) = min{m,n} s j (Z) min{m,n} C (m j) (0) 1 (n j)! Ĝ (j)(n j) (0) p n j s j (Z) min{m,n} = C (m j) 1 (0) (n j)! 1 N j dn j G (j) dx n j (0) p n j s j (Z) min{m,n} = C (m j) 1 (0) (n j)! 1 N j G(n) (0) p n j s j (Z) = p (n) min {m,n} n m 1 m! j j n 1 p (m j) j N j pn j s j (Z) p +1 (n) / l Z (n) p (n) n m m! j j 1 c (m j) n j 1 N j pn j IET Radar Sonar Navig., pp & The Institution of Engineering and Technology 2013