The Multiple Model CPHD Tracker

Transcription

1 The Multiple Model CPHD Tracker 1 Ramona Georgescu, Student Member, IEEE, and Peter Willett, Fellow, IEEE Abstract The Probability Hypothesis Density (PHD) is a practical approximation to the full Bayesian multitarget filter. The Cardinalized PHD (CPHD) filter was proposed to deal with the target death problem of the PHD filter. A multiple-model PHD exists; in this work, a multiple model version of the considerably more complex CPHD filter is derived. It is implemented using Gaussian mixtures, and a track management (for display and scoring) strategy is developed. Index Terms Multiple model, Cardinalized Probability Hypothesis Density filter, PHD, CPHD, MMPHD, MMCPHD. I. INTRODUCTION A. Prologue The goal of a multitarget tracker (MTT) is to estimate the states of an unknown and time-varying number of targets from a series of measurements, and a MTT must address measurement-to-track association uncertainty, sensor detection uncertainty, false alarms and noise. Some well established MTT algorithms are the Joint Probabilistic Data Association (JPDA) filter [1], the Multiple Frame Assignment (MFA) tracker [2] and the Multi Hypothesis Tracker (MHT) [3]. A fully Bayesian multitarget tracker [4] predicts and updates (in the sense of Bayes) the probability mass function for the cardinality of the targets and the joint probability density functions (conditioned on cardinality) for the locations of the Copyright (c) 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. R. Georgescu and P. Willett are with the Electrical and Computer Engineering Department, University of Connecticut, Storrs, CT USA, {ramona, willett}@engr.uconn.edu. Manuscript submitted June Revision submitted September 2011.

2 2 targets. This method is optimal, but at least at present seems impractical for most applications as the number of targets increases. B. Probability Hypothesis Density (PHD) Filter Mahler introduced a new approach to multitarget tracking based on finite set statistics (FISST) [4], in which target states and measurements are modeled as random finite sets (RFS). An RFS is characterized by a probability mass function (pmf) of the cardinality of the set and, for a given cardinality, a joint probability density function (pdf) of the elements in the set. Hence, it is an elegant and useful framework for the Bayesian multitarget tracking problem. In particular, a Poisson RFS is completely characterized by its intensity function v(x): the cardinality pmf is Poisson with mean N = v(x)dx (which need not be integral, although that is the more appealing case) and the elements are independent and identically distributed according to v(x)/n. The probability hypothesis density PHD is the intensity function of an RFS and represents the density of the expected number of targets per unit volume [5], [6]. Mahler proposed that the probability hypothesis density (PHD) be propagated as a first moment approximation to the multitarget Bayesian posterior distribution instead of the full posterior distribution [4]. In the bin model [7] approach to the PHD filter which we employ here, the PHD surface is discretized into infinitesimally small bins and the probability of each bin containing a target is predicted and updated. The resulting PHD filter is by nature automatically track-managed. Unlike other trackers such as the MHT, the PHD avoids the explicit enumeration of all possible multi-target multi-detection assignments that leads to the so-called combinatorial disaster: the PHD does not have an exponential complexity. A disadvantage of the PHD, however, is that the Poisson assumption for the target number (cardinality) distribution often leads to large fluctuations in the target-number estimates, and thence to the target death problem [7]. C. Cardinalized Probability Hypothesis Density (CPHD) Filter The Cardinalized PHD (CPHD) filter [8] offers a nice cure to the problem of premature target death through adding memory to the target-number process, and although it introduces an artificial linkage between all targets existence probabilities (a significant effect observed as spooky action at a distance in [9]), it seems to work very effectively in most applications we have seen. It is a PHD filter with a hierarchically supervising Hidden Markov Model (HMM) that describes the number of targets in the scene; that is, the cardinality of the Poisson RFS is not restricted to be Poisson distributed, and can

3 3 in fact be arbitrary. The only restriction for this distribution is that its first moment has to be equal to v(x)dx, i.e., the expected number of targets. In other words, the CPHD filter approximates the Bayesian multitarget tracking problem with an i.i.d cluster process. In the HMM used by the CPHD filter, the target number is external to the PHD surface itself and moreover it affects the PHD surface. D. Gaussian Mixture Implementation of the CPHD Filter (GMCPHD) Closed form PHD/CPHD filter equations have been derived [6], [10] for situations in which linear Gaussian assumptions for the target motion and observation models hold and the probabilities of detection and survival are state independent. The posterior CPHD surface was approximated by a Gaussian Mixture and was shown to remain a Gaussian Mixture after the update step. Hence, the propagation of the whole surface could be replaced by the propagation of the weight, mean and covariance of each mode in the mixture. At each time scan, mode means and covariances are propagated by an Extended Kalman Filter (EKF) while mode weights are calculated using the prediction and update CPHD equations. Sequential Monte Carlo (SMC) implementations of the PHD/CPHD filters were also proposed e.g. in [5], [11]. We have found the GM implementation (the derivation of which can be found in Appendix), labeled the GMCPHD filter, both more intuitive and more effective than the SMC-CPHD filter. We employ the GMCPHD filter with a linear motion model and a nonlinear measurement model in which range r, bearing θ and range rate ṙ (when available), form the measurement. E. Multiple Model Probability Hypothesis Density (MMPHD) Filter Multiple model approaches assume that a system operates according to one of a finite number of models; these models can have different noise, structure, state dimensions and/or unknown inputs. The dynamic MM estimator, used for systems that, with time, transition from one model to another, employs a Bayesian framework: based on prior probabilities of each model being currently in force, posterior probabilities are calculated [12]. By assumption, the model switching, also known as model jumping, is a Markov chain with known model transition probabilities. While a nonmaneuvering target can usually be described well by a single model, a bank of different hypothetical target motion models characterizing possible maneuvers may be needed to operate in parallel in order satisfactorily to track a maneuvering target. MM approaches are attractive because, due to their exploration of the set of viable target motion models, they achieve better performance over the

4 4 corresponding single-model filter; with some justification the MM filter is sometimes referred to as having adaptive bandwidth. Multiple model extensions for well established tracking algorithms exist, such as the IMMJPDA (Interacting MM version of the Joint Probabilistic Data Association filter) [13] and IMMMHT (IMM version of the Multiple Hypothesis Tracker) [14]. As for the (recently-developed) PHD filter, Pasha et al. [15] first proposed augmented PHD filtering for linear jump Markov models. Their approach is the GM implementation and therefore, the recursion has a closed form. They applied it with position measurements and three models, one with constant velocity and two constant turn models differentiated by the direction of the turn (clockwise and counterclockwise). Moreover, Pasha et al. [15] extended their MMPHD to nonlinear jump Markov models, also implemented using Gaussian Mixtures (an unscented transform handles nonlinearities). In their simulation, measurements consist of range and bearing; a constant velocity model and a constant turn model are used. Punithakumar et al. [16] have developed a multiple model PHD using a particle filter implementation. Its usefulness was demonstrated with bearing-only measurements and used one constant velocity model and one constant turn model. To our knowledge, a multiple model version of the considerably more complex CPHD filter has not been devised, with the challenge lying in describing the interrelation between the models. Figure 1 shows the HMM used by a MMCPHD filter with two models, in which there is crosstalk between the two PHD surfaces and the target number affects them both. This work is an expanded version of conference paper [17], supplemented with full derivation and results on difficult multistatic sonar data. In Section II, we present the derivation of the prediction and update equations in the MMCPHD filter and in Section III, we compare its performance against that of a regular CPHD tracker on a realistic multistatic sonar dataset. We conclude in Section IV. II. MULTIPLE MODEL CARDINALIZED PROBABILITY HYPOTHESIS DENSITY FILTER This derivation of the multiple model version of the CPHD filter is based on the design on an interacting multiple model estimator shown in Section of [12]. It follows closely the bin-occupancy filter CPHD derivation [7], and uses the notation therein. We must make clear at this point that we do not attempt here to overlay multiple models to Mahler s original development (a fully Bayesian filter followed by

5 5 Fig. 1. The underlying hidden Markov model (HMM) of the inference process of the MMCPHD filter. Observe the crosstalk between the two PHD surfaces U k and W k and that the target number N k affects them both. a first moment approximation) of the PHD and CPHD filters. Instead, as in [7], a bin model is used, i.e., we assume the surveillance region is partitioned into bins such that each bin (i.e., partition) has the same volume; and moreover, that these bins are sufficiently small that each bin is potentially occupied by at most one target. We quote from [7], discussing the relationship between the bin-occupancy filter and the PHD: We have provided a bin model of probability (motivated by the physical interpretation of first factorial moment density) and showed that it gives rise (in the limit of infinitesimal bins) to exactly the PHD prediction and update equations. In other words, our development is not a step-by-step interpretation of Mahler s original derivation, but an alternative route that shares the start- and the endpoints of the original. It would seem that the bin-occupancy filter follows from the spirit of Mahler s model, and we will associate with the CPHD filter name the multiple model extension we derive in the this paper. We make several notes: To simplify notation we have derived the prediction and update equations of the MMCPHD filter in the case of two target motion models. The generalization to 3 motion models is straightforward. The development here follows that in [7] closely. The reader familiar with [7] will see many or most of the same steps followed, with the addition of a model variable (usually q). The basic approach is to represent different motion models by different PHD surfaces (a hint of this is in figure 2, although the focus there is track management ), and to allow these surfaces to interact that is, to exchange

6 6 occupancy probabilities via a Markov transition structure as illustrated in figure 1. There are separate hierarchical (hidden) Markov cardinality models for each of these modes/surfaces. We note that [15] uses Campbell s Theorem in its initial steps of the multi-model PHD, and that could provide an alternative starting point for the MMCPHD filter. The data association model used is that the interacting CPHD filters that have access to the same set of measurements. In the update step, each is updated with each measurement from the set of measurements that arrived at the current scan. These are discussed more as they appear and are used. A. Prediction Equations Given the information (probability of occupancy) for all bins at time k 1, D k 1 k 1 (, ), the event that bin i contains a target in model q at current time k is possible in two 1 ways: a new-born target in model q appears in bin i or one of the targets in another bin in the same or in another model survives and moves to bin i (this includes the event that target in bin i in model q stays in bin i in model q). Hence, the MMCPHD prediction equation for the bin probabilities is (via the total probability theorem): P{U k (i)=1 Z 1 k 1 q=2, r k =q}=b(i r k =q)+ q=1 t qq j P s (x j r k 1 = q)f(i, x j r k 1 = q)p{u k 1 (j r k 1 = q)} (1) where r k is the model at time k, b(i r k = q) is the birth probability for a target to appear in bin i under model r k = q, f(i, x j r k 1 = q) is the probability that a target located at x j in model r k = q propagates to bin i and t qq is the Markov transition matrix probability that a target switched from model q to model q, where q {1, 2}. Given the probability mass function (pmf) of the target birth cardinality, p(ψ), the MMCPHD prediction step for the pmf of the number of targets in model r k = 1 is derived by applying the total probability theorem to the following hypotheses, with all hypotheses assumed equally likely to occur: new targets are born in model r k = 1 at time k; given there were n 1 targets in model r k = 1 at time k 1, i s1 of them survived and out of i s1 surviving targets, i t1 remained in model r k = 1; given there were n 2 targets in model r k = 2 at time k 1, i s2 of them survived and out of i s2 surviving targets, i t2 transitioned 1 We ignore target spawning for simplicity, although it is easy to include.

7 7 to model r k = 1. Hence, the MMCPHD prediction step for the probability mass function (pmf) of the number of targets in model r k = 1 is given by Eq. 2. A similar expression for p k k 1 (n r k = 2) can be written down for the second model. Note that, in the case of two possible motion models, t 11 + t 12 = 1 and also t 21 + t 22 = 1. For implementation purposes, the maximum number of targets has to be capped at some value N max. p k k 1 (n r k = 1) = q=2 n q=1 it q=0 p ( q=2 ) ψ = n i t q r k = 1 q=1 N max i N s1 t i t1 11 ti s1 i t1 max 12 n 1 p k 1 k 1 (n i s1=i t1 i s1 i t1 n =i 1 s1 n 1 i 1 r k 1 = 1)(1 P ss ) n i 1 s1 P i s1 ss s1 N max i N s2 t i t2 21 ti s2 i t2 max 22 n 2 p k 1 k 1 (n i s2=i t2 i s2 i t2 n =i 2 s2 n 2 i 2 r k 1 = 2)(1 P ss ) n 2 i s2 Pss is2 (2) s2 where 2 ψ is the number of new-born targets at time k and we define the average probability that a target in model q survives (does not die between scans) as: P ss (r k 1 = q) = j P s (x j r k 1 = q) p k 1 k 1(U j r k 1 = q) p k 1 k 1 (U l r k 1 = q) l (3) Eq. 2 can be rewritten with a more compact notation as: p k k 1 (n r k = q) = q=2 N max q=1 i s q =i t q i s q i s q i t q n q=2 i q=1 t q=0 ( q=2 ) p n i t q r k = q q=1 t it q qq (1 t qq ) is q it q N max n n =i s q n i s q p k 1 k 1 (n r k 1 = q)(1 P ss ) n i s q P is q The generalization to an arbitrary number of models is straightforward but cumbersome. ss (4) In the case of a single model, Eq. 1 and Eq. 4 reduce to the CPHD prediction equations for the bin probabilities derived within the same bin model framework in [7]: P{U k (i) = 1 Z k 1 1 } = b(i) + j P s (x j )f(i, x j )P{U k 1 (j) = 1 Z k 1 1 } (5) 2 stands for multiplication.

8 8 and for the number of targets 3 n N max p k k 1 (n) = p(ψ = n i) n p k 1 k 1 (n )(1 P ss ) n i P i i=0 n =i n ss (6) i where average probability that a target survives is P ss = j P s (x j ) p k 1 k 1(U j ) p k 1 k 1 (U l ) l (7) B. Update Equations The MMCPHD filter propagates the bin probabilities, p k 1 k 1 (U i, ), and the probability mass function (pmf) of the number of targets, p k 1 k 1 (n, ) of each model as shown previously, and then updates them with the observation set received at time k, Z k = {z 1, z 2,..., z m }, via Bayes formula: p k k (U i r k = q) = f(z k, U i Z k 1 1, r k = q) f(z k Z k 1 1 ) p k k (n r k = q) = f(z k n, Z1 k 1, r k = q) f(z k Z1 k 1 p k k 1 (n r k = q) (9) ) Next, the quantities on the right hand side(s) of the above equations are given in more explicit forms. Assuming that the pmf of number of clutter (false alarm) observations p(c) is given and using the total probability theorem, we have: f(z k, U i Z k 1 1, r k = q) = c = c = c = c n n n f(z k, U i, n, c Z k 1 1, r k = q) f(z k U i, n, c, Z k 1 1, r k = q)p(u i, n, c Z k 1 1, r k = q) f(z k U i, n, c, Z k 1 1, r k = q) p(u i n, c, Z k 1 1, r k = q)p k k 1 (n r k = q)p k k 1 (c) n f(z k U i, n, c, Z k 1 1, r k = q) n rk=q j p k k 1(U j r k = q) p k k 1(U i r k = q)p k k 1 (n r k = q)p(c) (10) (8) 3 Eqs. 5 and 6 represent the discrete bin model prediction equations. In the limit when bin volume goes to zero, we obtain the corresponding continuous bin-occupancy filter prediction and update equations which are identical to the original CPHD filter equations of [8]. For details, please see [7].

9 9 The last step in Eq. 10 is justified in light of the following assumptions (valid for both models): 1) The number of false alarms, c, and the number of targets, n, are independent. 2) The number of false alarms, c, is independent of the measurement sequence, Z k ) The location of the targets are independent given the number of targets. 4) A target can generate at most one measurement. 5) Each target evolves and generates measurements independently. 6) False alarms are independent from target-originated measurements. The third assumption above permits us to write: p k k 1 (U i n, c, r k = q) = p k k 1 (U i n, r k = q) = n rk=q j p k k 1(U j r k = q) p k k 1(U i r k = q) (11) and follows from the fact that given the PHD surface (the vector of the bin probabilities), and given that there is only one target, the probability that the bin i in model q has the target is: p k k 1 (U i r k = q) j p k k 1(U j r k = q) (12) If there are n such targets, the probability that bin i in model q has one of these targets is given by 1 P{bin i in model q has none of these targets}: ( p k k 1 (U i n, r k = q) = 1 1 p ) k k 1(U i r k = q) nrk =q j p k k 1(U j r k = q) n rk=q j p k k 1(U j r k = q) p k k 1(U i r k = q) (13) Note that this approximation imposes a constraint on the integral of the bin-occupancy probabilities: p k k 1 (U i r k = q) = = = p k k 1 (U i n, r k = q)p k k 1 (n r k = q) n=0 p k k 1 (U i r k = q) n rk =qp k k 1 (n r k = q) n=0 j p k k 1(U j r k = q) E[n r k = q] j p k k 1(U j r k = q) p k k 1(U i r k = q) (14) Eq. 14 implies that the expected number of targets in model q, E[n r k = q], is equal to the integral of the

10 10 PHD surface (i.e. integration over the bin-occupancy probabilities) in model q, j p k k 1(U j r k = q). This property of the single model CPHD filter has been preserved and extended by the MMCPHD filter [7]. In the sequel, we use likelihood ratios that allow us to compare each hypothesis to the null hypothesis that all the measurements consist of false alarms. Therefore, likelihood ratios are formed as: L(Z n, c, r k = q) = f(z n, c, r k = q) f(z all clutter) (15) For clarity, we assume that the probability of detection is not state dependent and is constant across models, i.e. P d (x i r k = q) = P d. We also make the assumption that the clutter distribution is the same in all models 4. Finally, and perhaps most important, the data association is performed separately across MMCPHD modes. As to this last assumption, we admit that there would be for a lumped-target tracker some appeal in performing data association on targets (i.e., after averaging over modes) as opposed to individually on the modes. However, this runs counter to the MMCPHD filter philosophy that we have adopted (that there are interacting separate PHDs for the various modes), and indeed [15] and [16] seem to follow the same approach; as does (say) the IMMPDAF [1]. We define a visibility indicator V k (i), a Bernoulli random variable with probability P d (x i ), which is unity if the target in bin i is detected at time k. Then, the update step for bin probabilities and the cardinality pmf for both models is written for the case of missed detections and also for the case of successful target detections: p(u i Z,r k =q) = L(Z U i, r k = q) p(u i r k = q) L(Z) = (1 P d ) L(Z U i, r k = q, V k (i) = 0) p(u i r k = q) + P d L(Z U i, r k = q, V k (i) = 1) p(u i r k = q) L(Z) L(Z) [ Z n=0 ] n c=0 L(Z U i, n, c,r k =q, V k (i) = 0) rk =q j = (1 P d ) p(u j r k =q) p(n r k=q) p(c) 2q=1 [ Z n=0 ] p(u i r k =q) c=0 L(Z n, c,r k =q)p(n r k =q) p(c)p(r k =q) [ Z n=0 ] n c=0 L(Z U i, n, c,r k =q, V k (i) = 1) rk =q j + P d p(u j r k =q) p(n r k=q) p(c) 2q=1 [ Z n=0 ] p(u i r k =q) (16) c=0 L(Z n, c,r k =q)p(n r k =q) p(c)p(r k =q) 4 This seems reasonable, but there does not appear to be any pressing reason to expect that a more general situation cannot be explored in the same way as we explore here.

11 11.CP HD1.CP HD1.CP HD1.z.M1..M11.M21.M21.M11.CP HD2.CP HD2.CP HD2.M2.M12.M22.z.M12.M22.TM.TM.TM.M3.M2.M1.k 1 k 1.k k 1.k k Fig. 2. Illustration of how the track management surface is obtained by moment matching corresponding GM modes on the Fig. 1. Illustration of how the track management surface is obtained by moment matching corresponding GM modes on the updated PHD surfaces. Modelupdated 1 isphdthe surfaces. high Modelprocess 1 is the high process noise model, 2model the low process 2 the noise one. low process noise one. and p(n Z, r k = q) = L(Z n, r k = q) September 14, 2011 = L(Z) Z p(n r k = q) c=max{ Z n,0} L(Z n, c, r k = q)p(c) 2q=1 [ Z n=0 ] p(n r k =q) (17) c=0 L(Z n, c, r k = q)p(n r k = q) p(c)p(r k =q) In Appendix A, we explicitly derive the likelihood ratios used in the above equations: L(Z U i, r k = q, V k (i) = 1), L(Z U i, r k = q, V k (i) = 0) 5, L(Z n, r k = q) and L(Z). Note that these update equations simplify, in the case of a single model, to the original CPHD filter update equations [7]. The PHD filter is a special case of the CPHD filter, and it can be checked that the MMCPHD equations reduce to the MMPHD equations in [15] and [16]. In Appendix B, we sketch the corresponding MMCPHD equations for the continuous case of our bin occupancy derivation. C. Track Management A multitarget tracker is composed of a filter followed by track management policies. In its original form the GMCPHD filter is not able to provide scoreable tracks, so a track display/management scheme had to be devised [18] [19] in order to make the transition from filter into tracker. This is a set of policies dealing with track initiation, update and deletion, spawning, mode pruning and merging. Note that track management is separate from the operation of the filter. 5 In Appendix A, we drop the subscript (k), and use V i for V k (i).

12 12 In our GMCPHD tracker, modes propagate through their offspring and it is this connectivity over time that allows for track management. Each Gaussian Mixture (GM) mode maintains a record of its father mode, i.e. the mode at the previous scan that was updated and became the current GM mode. It also has the same track identification number (ID) as its father. If the sum of the weight of a mode at the current scan k, the weight of its father at the previous scan k 1 and the weight of its grandfather at scan k 2 is higher than a threshold, a new track is declared. Track deletion is handled similarly, but is prompted by the sum of the weights falling below a threshold. A track is updated with the mode that has the largest weight among the modes at the current scan that share the same track ID. Then, the track is updated backwards, to include the father and deeper ancestors of the mode that updated the track at the current scan. This way, at each scan, the track follows its most likely path. Spawning occurs if two modes A and B at the current scan k with the same track ID have their last common ancestor at a scan earlier than k 4 (tunable). For the mode with smaller weight, let us say B, the connection to its last common ancestor is broken (i.e. the father value for the offspring of the last common ancestor is set to 0) and all the modes in the branch starting with the offspring of the last common ancestor and ending with B receive a new track ID. In common with many other trackers (e.g., MHT), the number of Gaussian modes can increase exponentially with the number of scans, thus pruning, merging, etc. are necessary from a practical point of view. The pruning step discards modes with extremely low weights. The merging step first calculates the Mahalanobis distance between modes; then, if the modes are located very close to each other, a single mode will represent them both. The weight of the new mode is the sum of the weights of the merged modes and the mean and covariance of the new mode are calculated by moment matching. In the case of the MMCPHD filter, the track management scheme described above is applied to an extra display layer, obtained by fusing corresponding Gaussian modes in the two models through moment matching (see Eq ). Hence, track management policies do not interfere with the operation of the two CPHD filters. For example, only the Gaussian modes that compose the extra (third) layer are assigned tracks IDs (through track intialization or through inheritance from their father mode). The process is illustrated in figure 2. Corresponding modes at time k are modes in different models whose fathers at time k 1, also

13 13 in different models, are themselves each other s corresponding mode and were updated with the same measurement received at time k. Note that the correspondence relationship is maintained over time but the corresponding modes will have however different weight, mean and covariance (w, µ, Cov) due to, for example, different process noise in different models. w = w 1 + w 2 (18) µ = w 1µ 1 + w 2 µ 2 w 1 + w 2 (19) Cov = w 1Cov 1 + w 2 Cov 2 w 1 + w 2 (20) A simple example illustrates the concept of corresponding modes: suppose at time k = 1 we have received only one measurement and we initialize the MMCPHD filter by placing mode A in model 1 and an identical mode B in model 2, with a certain weight at the location and with the covariance of our measurement; at time k = 2, suppose we have received two mesurements z 1 and z 2 ; we label mode A1 as the mode obtained by updating mode A with measurement z 1, A2 as the mode obtained by updating mode A with measurement z 2 and the same goes for modes B1 and B2. Now, modes A1 and B1 are corresponding modes (so are modes A2 and B2) and they will be fused into what we choose to label as mode AB1 (respectively mode AB2). The track management scheme operates on the layer defined by modes AB1 and AB2. III. RESULTS A. Sonar Simulator Dataset We compared the single model CPHD filter and the multiple model CPHD filter with two nearly constant velocity models on a simulated dataset. A monostatic sensor with σ tdoa = 0.1 sec and σ bearing = 4 collected measurements from a target on a square trajectory around it with SNR 1km = 50dB (SNR that would be observed from a specular target contact 1 km away from a monostatic sonar) in Rayleigh clutter; the ping period was 60 sec, the total scenario time 3600 sec and the detection threshold was varied. Fifty Monte Carlo simulations were averaged for each detection threshold value. The low process noise variance used was m 2 s 3, the high process noise had variance 0.5 m 2 s 3. The Markov transition matrix was [ ; ].

14 14 B. Metron Dataset The second dataset used for performance comparison was the Metron dataset [20], generated by Kirill Orlov of Metron Inc, Reston VA to provide challenging multistatic sonar data for the evaluation of the tracking algorithms used by the Multistatic Tracking Working Group 6 (MSTWG) participants. Measurements are collected from 25 stationary sensors located as in [20], [21]; all the sensors are receivers with the exception of four which are colocated source/receiver units. Target probability of detection is poor, on average P D = 0.12 per sensor per scan. The error in the time difference of arrival (TDOA) is normally distributed with standard deviation 0.4 sec while bearing error is normally distributed with standard deviation 8.0. The high difficulty of this dataset is due to the extremely large number of contacts per scan coupled with the low quality of the measurements: e.g. a typical scan of data might contain 881 contacts out of which only 24 are detections from a target; the 1-sigma covariance ellipses are very elongated (mostly due to the large uncertainty in the bearing measurements) and can reach 20km in their major axis. The MMCPHD filter employed two nearly constant velocity models to represent target dynamics. The low process noise variance was 10 6 m 2 s 3, the high process noise had variance 10 4 m 2 s 3. The Markov transition matrix was [ ; ]. C. Metrics of Performance To quantify performance we use the metrics of performance (MOPs) agreed upon by the members of the MSTWG. Thus, we report: track detection probability (P D ), i.e., the ratio of the total duration of all true tracks and the total scenario duration; track fragmentation (F RAG); number of false tracks (F T ); and root mean square error (RMSE) evaluated only where tracks exist. Further discussion of these definitions can be found in [22]. D. Tracking Results on Sonar Simulator Dataset On the moderately difficult sonar simulator dataset, the MMCPHD filter performed better than the single model CPHD filter. Table I shows the averaged RMS error over 50 Monte Carlo runs for several 6 The Multistatic Tracking Working Group is an international group of researchers with interest in multistatic sonar tracking. It began in 2004 and now exists as a panel within the International Society for Information Fusion.

15 15 N C CPHD MMCPHD 7.0dB dB dB dB TABLE I RMSE ON THE SONAR SIMULATOR DATASET FOR VARIED DETECTION THRESHOLD LEVELS WITH N C AVERAGE NUMBER OF CONTACTS PER SCAN (MMCPHD TRACKER PERFORMS BETTER THAN THE SINGLE MODEL CPHD TRACKER). detection threshold levels. Note the high density of clutter: N C, the average number of contacts per scan, increases with decreasing detection threshold. The MMCPHD tracker achieves lower RMSE than the CPHD tracker for all detection threshold levels. Furthermore, Figures 3 and 4 display a typical Monte Carlo run at 7dB, in which the CPHD tracker is unable to follow the target around the corners of its trajectory while the MMCPHD tracker can satisfactorily do so. Figure 5 shows the two model probabilities of the MMCPHD tracker on the scenario in Figure 4. The model probability is calculated by integrating the PHD surface of the model and normalizing it with respect to the total PHD surface in both models. A median filter of length 5 has been applied to improve clarity of the trends. Note that soon after the time the maneuvers occur, i.e. scans 15, 30 and 45, the probability of the high process noise model increases; moreover, on the last portion of the trajectory which is a straight line, the low process noise model dominates. Fig. 3. The single model GMCPHD tracker on the sonar simulator dataset is unable to follow the target around corners (ground truth in blue, tracks in green).

16 16 Fig. 4. The multiple model GMCPHD tracker on the sonar simulator dataset successfully follows the target around corners (ground truth in blue, tracks in green). Fig. 5. Model probabilities for the multiple model GMCPHD tracker on the sonar simulator dataset for the Monte Carlo run shown in Figure 4. Note that after maneuvers occur (at times 15, 30, 45), the high process noise model is in effect. E. Tracking Results on Metron Dataset Our first attempt at tracking on the Metron dataset, i.e. running the GMCPHD tracker [7] alone, resulted in unsatisfactory performance. Good tracking results on this dataset were made possible by the addition of a pre-detection fusion step [23] before the tracker. Winnowing involves discarding raw measurements based on a likelihood ratio test on their SNR and if available, Doppler. Predetection fusion [23] is an an efficient way to process the volume of data from large sensor networks of low quality sensors. It consists of a contact sifting procedure followed by an Expectation Maximization step that refines the location of the estimated detections 7. Figures 6 and 7 show results for one of the scenarios in the Metron dataset: the 2 targets present, 7 In this work, we focus on the benefits a multiple model version of the CPHD filter brings. More detail on predetection fusion can be found in [23], [24] and corresponding conference publications [21], [25].

17 17 CPHD PD FRAG RMS Target Target MMCPHD PD FRAG RMS Target Target TABLE II MOPS ON THE METRON DATASET (MMCPHD TRACKER ACHIEVES CONSIDERABLY LOWER RMSE AND TRACK FRAGMENTATION THAN THE SINGLE MODEL CPHD TRACKER). a target following a bow-tie path that crosses with a target on a straight line path, are tracked with reasonable performance. As seen in Table II, for both the CPHD and MMCPHD trackers, the tracking performance is good, as little fragmentation is present, the track detection probability is high, while the RMS error is acceptable. Moreover, the number of false tracks is very low: four for the CPHD tracker and a single false track for the MMCPHD tracker. The MMCPHD tracker performs better than the CPHD tracker: it has excellent P D, lower track fragmentation, considerably lower RMSE on average 8 and it generates fewer false tracks. In particular, note the high fragmentation obtained by the CPHD tracker for the bow-tie target while the MMCPHD tracker was able to follow this target without breaking track. Fig. 6. The single model GMCPHD tracker with winnowing and the 2D predetection fusion preprocessing obtains good tracking results on the Metron dataset. 8 Note that while a single-mode tracker needs a level of process noise that is a compromise, a multi-mode tracker can afford to have a nearly constant-velocity mode. In a sense, it is a variable bandwidth filter, and its narrow-bandwidth mode can be made very well-suited to a straight-line track, better even than the ostensibly better-suited single-mode tracker.

18 18 Fig. 7. The multiple model GMCPHD tracker with winnowing and the 2D predetection fusion preprocessing improves on the single model GMCPHD tracking results on the Metron dataset (metrics in Table II also show that the MMCPHD tracker performs better than the regular CPHD tracker). Additionally, Figure 8 shows that the MMCPHD tracker has an advantage over the single model CPHD tracker when estimating target cardinality. In the Metron dataset investigated, two targets are present in the surveillance area; the multiple model CPHD tracker is correctly estimating the number of targets more often than the CPHD tracker. Therefore, it is to be expected that the recently introduced OSPA metric [26], which incorporates both a localization error and a cardinality error, also indicates a gain in performance when using the MMCPHD tracker over the single model CPHD tracker, as seen in Figure 9. The OSPA metric was calculated with p = 2 and c = 5000m, matching the MSTWG metrics of performance computation. Figure 10 shows the model probabilities of the MMCPHD tracker for the bow-tie target of the Metron dataset. Model probabilities for individual targets are calculated by assigning modes close in position (same threshold used as in the metrics of performance calculation) to the ground truth of a particular target to that respective target. Other methods for assignment exist, e.g. k-means clustering [16]. The same median filter was applied as in the case of the sonar simulator results. Maneuvers occur at scans 55, 111, 152 and 200 (last scan). Note that soon afterwards the probability of the high process noise model increases indicating that the target was identified as undergoing a maneuver. These results on the Metron dataset are consistent with the analysis presented on sonar simulator data and reinforce the claim that adding multiple model capability to the CPHD filter is advantageous. It is fair to note that the price for better performance is increased run time. The MMCPHD tracker took 183

19 19 sec to run under Matlab R2009b on an Intel Core 2 Quad CPU machine, while the single-model CPHD tracker took 25 sec: the CPHD filter has O(nm 3 ) complexity, where n is the number of targets and m is the number of measurements; the MMCPHD filter has O(nm 3 r 2 ) 9 complexity, where r is the number of models. In practice, it is common for the number of models to be 2 or 3 so the additional complexity is acceptable when considering the gains in performance Number of targets Number of targets Scan Scan (a) GMCPHD tracker. (b) MMCPHD tracker. Fig. 8. Estimates for the number of targets with the single model GMCPHD and the MMCPHD trackers on Metron data MMCPHD CPHD Ospa Scan Fig. 9. OSPA metric for the single model GMCPHD and for the MMCPHD trackers on the Metron dataset. 9 An r factor is due to computing likelihood ratios r times when running r CPHD filters. The other is due to the prediction step for bin probabilities that doubles (here r = 2) the number of GM modes in a model through the Markov transition matrix.

20 Scan 56 Target 2 (bow tie target) low process noise high process noise Scan 116 Scan 161 Model Probabilities Scan Fig. 10. Model probabilities for the multiple model GMCPHD tracker for the bow tie target of the Metron dataset. Note that soon after maneuvers occur (at scans 55, 111, 152 and 200), the high process noise model goes in effect. IV. CONCLUSION Mahler s probability hypothesis density (PHD) tracker has been extended in a number of ways. Its original particle implementation has been simplified to one in which its surface is represented as a (reproducing) Gaussian mixture (the GMPHD filter); and it has been stabilized by a higher-level HMM to become the Cardinalized PHD filter (let us specifically note the GMCPHD filter). And multiple models have been grafted both to the PHD and to the GMPHD filters. In this work, we have provided the final piece, combining the GM, C and M monickers: we have developed and applied the MMGMCPHD tracker to the problem of multiple target tracking with high maneuverability targets. We have implemented the MMCPHD tracker using the Gaussian Mixture (GM) approach, which permits a closed form solution and efficient implementation, and compared its performance against the single model CPHD filter on simulated and realistic multistatic sonar datasets. On the challenging Metron data, the MMCPHD filter achieved superior tracking results with respect to the investigated performance measures (most noticeably in terms of RMSE). The single model CPHD filter performed well but was less able to adapt to the several target maneuvers present in the dataset; this led to larger RMS errors and sometimes track fragmentation during periods in which the targets maneuvered. The performance of the MM-GMCPHD tracker on this high-fidelity blind challenge dataset, with an unknown number of maneuvering targets observed by a large sensor network of low quality sensors, demonstrates the tracker to be a viable and attractive tracking approach, not just one of academic interest.

21 21 APPENDIX A DERIVATION DETAILS OF MMCPHD UPDATE EQUATIONS To arrive at the explicit forms of the update equations for the bin probabilities Eq. 16 and the pmf of target cardinality Eq. 17, we follow the derivation in [7]. We obtain: L(Z U i, r k = q, V i = 0) = m p(c = m j) j=0 (m j)! ( n(n 1) (n j)p(n r k = q) m! n=j+1 ) (1 P d ) (n 1) j 1 ) l p(u P j d l r k = q) σ j ( Lz1 c(z 1 ), L z2 c(z 2 ),, L zm c(z m ) (21) L(Z U i, r k = q, V i = 1) = m j=1 ( (m j)! p(c = m j) m! n=j m s=1 f(z s x i, r k = q) c(z s ) n(n 1) (n (j 1))p(n r k = q)(1 P d ) n j ) ( ) 1 l p(u l r k = q) P j 1 Lz1 d σ j 1 c(z 1 ),, L zs 1 c(z s 1 ), L zs+1 c(z s+1 ),, L zm (22) c(z m ) L(Z) = m (m j)! 2 p(c = m j) p(r k =q) ( (1 P d ) n j n(n 1) (n j + 1)p(n r k = q) ) m! j=0 q=1 n=j ( ) P j d σ Lz1 j c(z 1 ),, L zm (23) c(z m ) where σ j (y 1, y 2,, y m ) is the elementary symmetric function [27] of order j and L z (r k = q) 10 is the likelihood of a measurement originated from a target in model q given by: L z (r k = q) = j f(z x j, r k = q) p(u j r k = q) l p(u l r k = q) (24) The MMCPHD update equation for bin probabilities is constructed by plugging Eqs into Eq. 16. The MMCPHD update equation for the pmf of the target cardinality is given in Eq. 17 as: p(n Z, r k = q) = L(Z n, r k = q)p(n r k = q) L(Z) 10 In the following, n refers to the number of targets in model r k = q and L z(q) is shortened to L z, where q {1, 2}.

22 22 The denominator term has already been derived in Eq. 23 and the numerator is: L(Z n, r k = q) = min{n,m} j=0 p(c = m j) (m j)! m! n! (n j)! P j d (1 P d) n j σ j ( ) Lz1 c(z 1 ),, L zm c(z m ) (25) Now, all terms in the update equations (Eqs ) of the MMCPHD filter have been accounted for. APPENDIX B CONTINUOUS BIN-OCCUPANCY MMCPHD FILTER Based on Section III.E of [7], in the continuous case, the discrete prediction equation for the bin probabilities (Eq. 1) in both models becomes: q=2 D k k 1 (x i q)=b(x i q)+ q=1 t qq P s (γ q)f(x i, γ q)d k k 1 (γ q)dγ (26) Note that the prediction equation for the cardinality pmf (Eq. 4) in both models stays the same. In the continuous case, the discrete update equations for the bin probabilities (Eq. 16) and the cardinality pmf (Eq. 17) in both models need similar modifications of the likelihood ratios given in Appendix A. For example, the likelihood of a measurement originated from a target in model q (Eq. 24) is now: L z (q) = D k k 1 (γ q) f(z γ, q) dγ (27) Dk k 1 (ω q)dω ACKNOWLEDGMENT This work was supported by the Office of Naval Research under contract number N The authors thank the reviewers for their help in clarifying the model adopted herein. REFERENCES [1] Y. Bar-Shalom and X. R. Li, Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS Publishing, [2] A. Poore and N. Rihavec, A new class of methods for solving data association problems arising from multiple target tracking, in Proc. of the American Control Conference, Boston, MA, [3] S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems. Boston, MA: Artech House, [4] R. Mahler, Multitarget Bayes filtering via first-order multitarget moments, IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 4, pp , [5] B. N. Vo, S. Singh, and A. Doucet, Sequential Monte Carlo methods for multi-target filtering with RFS, IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 4, pp , 2005.

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24 24 [24] R. Georgescu and P. Willett, Predetection fusion with doppler measurements and amplitude information, Submitted to IEEE Journal of Oceanic Engineering, [25], Random finite set Markov Chain Monte Carlo predetection fusion, Chicago, IL, [26] D. Schuhmacher, B. T. Vo, and B. N. Vo, A consistent metric for performance evaluation of multi-object filters, IEEE Trans. Signal Process., vol. 56, no. 8, pp , [27] P. Borwein and T. Erdelyi, Newton s Identities. New York, NY: Springer-Verlag, 1995, sec. I.1.E.2 in Polynomials and Polynomial Inequalities. Ramona Georgescu (S 08) received her B.As in Computer science and Physics from Connecticut College in 2004 and her M.Sc in Electrical Engineering from Boston University in She is currently a Ph.D candidate in the Electrical and Computer Engineering department at the University of Connecticut, working under the direction of Dr. Peter Willett. Her area of interest is statistical signal processing, with an emphasis on estimation and multitarget tracking. Peter Willett Biography text here. PLACE PHOTO HERE