Multiple Model Cardinalized Probability Hypothesis Density Filter

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1 Multiple Model Cardinalized Probability Hypothesis Density Filter Ramona Georgescu a and Peter Willett a a Elec. and Comp. Engineering Department, University of Connecticut, Storrs, CT {ramona, willett}@engr.uconn.edu ABSTRACT The Probability Hypothesis Density PHD filter propagates the first-moment approximation to the multi-target Bayesian posterior distribution while the Cardinalized PHD CPHD filter propagates both the posterior likelihood of an unlabeled target state and the posterior probability mass function of the number of targets. Extensions of the PHD filter to the multiple model MM framework have been published and were implemented either with a Sequential Monte Carlo or a Gaussian Mixture approach. In this work, we introduce the multiple model version of the more elaborate CPHD filter. We present the derivation of the prediction and update steps of the MMCPHD particularized for the case of two target motion models and proceed to show that in the case of a single model, the new MMCPHD equations reduce to the original CPHD equations. Keywords: Multiple model, Cardinalized Probability Hypothesis Density filter, PHD, CPHD, MMPHD, MM- CPHD. 1.1 PHD / CPHD Filters 1. INTRODUCTION Mahler introduced a new approach to multitarget tracking based on finite set statistics FISST, 1 in which target states and measurements are modeled as random finite sets RFS. An RFS is a finite-set valued random variable 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 vx: the cardinality pmf is Poisson with mean N = vxdx and the elements are independent and identically distributed with vx/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. 2 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. 1 In this work, we employ the bin model 3 approach to PHD propagation, in which the PHD surface is discretized into infinitesimally small bins and the probability of each bin containing a target is predicted and updated in turn. 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. 4 The Cardinalized PHD CPHD 5 offers a nice cure by adding memory to the target-number process, and although it introduces an artificial linkage between all targets existence probabilities a significant effect 6 observed as spooky action at a distance, it seems to work very effectively in most applications we have seen. The cardinalized PHD filter is a PHD filter with an extra level of complexity added to its underlying Hidden Markov Model HMM, that is, the cardinality of the Poisson RFS is not restricted to be Poisson distributed, and can be arbitrary. The only restriction for this distribution is that its first moment has to be equal to vxdx, i.e. the expected number of targets. In other words, CPHD approximates the the Bayesian multitarget tracking problem with, so called, a generalized Poisson process. Figure 3 shows the HMM used by the CPHD filter, in which the target number is external to the PHD surface itself and moreover it affects the PHD surface.

2 Closed form PHD/CPHD equations have been derived 7, 2 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. In the GM implementation of the PHD/CPHD filters, 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 filter equations. Sequential Monte Carlo SMC implementations of the PHD/CPHD filters were also proposed we mention only 8, 9 but more exist. 1.2 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 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 the current correct model, posterior probabilities are calculated. 10 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 to satisfactorily 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 corresponding single-model filter. Multiple model extensions for well established tracking algorithms exist, such as the IMMJPDA Interacting MM version of the Joint Probabilistic Data Association filter 11 and IMMMHT IMM version of the Multiple Hypothesis Tracker. 12 When it comes to the recently developed PHD filter, Pasha et al. 13 first proposed PHD filtering for linear jump Markov models. Their approach is a GM implementation and therefore, the recursion has a closed form. They applied it with position measurements and three models, one constant velocity model and two constant turn models differentiated by the direction of the turn clockwise and counterclockwise. Vo et al. 14 introduced a version of the PHD for nonlinear jump Markov models, also implemented using Gaussian Mixtures. Their approach extends the solution to the PHD filter for linear Gaussian jump Markov system multitarget model through the unscented transform. In their simulation, measurements consist of range and bearing; one constant velocity model and one constant turn model are used. Punithakumar et al. 15 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 4 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. 1.3 Notation In the following sections, we use the notation below: d : the dimension of the state vector S : surveillance region. S R d V : the volume of the surveillance region x i : the middle point location of bin i in the state space. x i S ν i : the set of all points inside bin i ν i : volume of bin i, identical for all i U k i : indicator that bin i contains a target at time k, U k i {0, 1} V k i : indicator that an extant target in bin i is detected, V k i {0, 1} P d x i : probability of detection of a target located at x i P s x i : probability of survival of a target located at x i

3 P{A} : Probability that event A occurs bi : birth probability, that bin i, which did not contain a target on the prior scan, now does bx i is the limiting density, as ν i 0, of bi pi x j : probability that a target located at x j moves to bin i fx i x j : probability density that target moves from x j to x i Z1 k 1 : all measurements from time 1 up to and including time k 1 z s : s th single measurement at time k we suppress the notation k Z k : the set of measurements received at time k. Z k = {z 1, z 2,..., z m } λ : false alarm density in volume V cz j : clutter spatial density fz k : joint probability density function of measurements at time k fz p k 1 k 1 U i p k k 1 U i : probability density function pdf of single measurement z at time k : P{U k 1 i = 1 Z1 k 1 } D k 1 k 1 x i is the limiting density, as ν i 0, of p k 1 k 1 U i : P{U k i = 1 Z1 k 1 } D k k 1 x i is the limiting density, as ν i 0, of p k k 1 U i p k k 1 n p k k 1 c : P{there are c false alarms at time k} : P{N k = n Z k 1 1 } = P{there are n targets at time k} Figure 1. Illustration of the prediction step in the bin model: the target in bin j moves to bin i with probability pi x j. The other possibilities are that the target in bin i dies with probability 1 P s x i and a new-born target appears at bin i. Figure 2. Illustration of the update step for the bin probabilities pu i with measurement z k.

4 Figure 3. The underlying hidden Markov model HMM of the inference process of the CPHD filter. Observe that the target number N k is external to the PHD surface itself and moreover, the target number affects the PHD surface U k. Figure 4. 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.

5 2. MULTIPLE MODEL CARDINALIZED PROBABILITY HYPOTHESIS DENSITY FILTER This work follows closely the CPHD derivation by Erdinc et al. 3 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. Moreover, these bins are sufficiently small that each bin is potentially occupied by at most one target. Note that we have derived the prediction and update equations of the MMCPHD filter in the case of two target motion models; multi-model generalization is straightforward. 2.1 Prediction Equations Given the information about all the occupied 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 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 prediction equation of MMCPHD for the bin probabilities is via the total probability theorem: q=2 P{U k i = 1, r k = q Z1 k 1 } = bi, r k = q + q=1 t qq j P s x j, r k 1 = qpi x j, r k 1 = qp{u k 1 j, r k 1 = q} 1 where r k is the model at time k and t qq is the Markov transition matrix probability that a target switched from model q to model q, where q {1, 2}. Figure 5 is an example of how the PHD surfaces in both models are predicted using Eq. 1. For clarity and only in this figure, we use a particle implementation, ignore particle birth and assume all particles survive between time k 1 and time k. Figure 5.a displays the PHD surfaces after time k 1 and also shows an example of one particle staying in the same model and being propagated in time and another particle transitioning to the other model and being propagated in time. Figure 5.b displays the output of the prediction step, the mixed PHD surfaces predicted from time k 1 to time k. Note that both models have the same particles after the prediction step, but the weights of a particle are different for the two models due to the Markov transition matrix entries. Given the probability mass function pmf of the target birth cardinality, pψ, the prediction step of MM- CPHD 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, some of them survived denoted as i s1 in Eq. 2 and out of the surviving targets, some remained in model r k = 1 denoted as i t1 in Eq. 2; 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 s2 i t2 transitioned to model r k = 1. Hence, the prediction step of MMCPHD for the probability mass function pmf of the number of targets in model r k = 1 is given by: n q=2 p k k 1 n, r k = 1 = p ψ = n i t q, r k = 1 N max is1 i s1 =i t1 N max is2 i s2=i t2 q=2 q=1 i t q=0 t it1 i s1 i 11 tis1 it1 12 t1 t i t2 i s2 i 21 ti s2 i t2 22 t2 N max n 1 =i s1 N max n 2 =i s2 q=1 n 1 n p 1 i k 1 k 1 n s1 1, r k 1 = 11 P ss n 1 i s1 P i s1 ss n 2 n p 2 i k 1 k 1 n s2 2, r k 1 = 21 P ss n 2 is2 Pss is2 2 where ψ is the number of new-born targets at time k and we define the average probability that a target in model q survives as: P ss r k 1 = q = j p k 1 k 1 U j, r k 1 = q P s x j, r k 1 = q p k 1 k 1 U l, r k 1 = q We ignore target spawning for simplicity. However, it is quite easy to include the spawning model into this derivation. stands for multiplication. l 3

6 same model Model 1 Model 1 Model 2. Model 2 model switch a U k 1, i.e. PHD surface at time k 1 left. Example of model transitioning and particle propagation right. Model 1 Model 2. b U k k 1, i.e. predicted PHD surface. Note that both models have the same particles after the prediction step, but the weights of a particle are different for the two models due to the Markov transition matrix entries. Figure 5. The prediction step for bin probabilities Eq. 1, here shown with a particle implementation, involves particle propagation, survival, model transition and therefore surface mixing and new particle birth. A generalization to an arbitrary number of models is straightforward but obviously cumbersome. A symmetric 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. In the case of a single model, Eq. 1 and Eq. 2 reduce to the CPHD prediction equations 3 for the bin probabilities: P{U k i = 1 Z1 k 1 } = bi + pi x j P s x j P{U k 1 j = 1 Z1 k 1 } 4 j and for the number of targets [ n Nmax p k k 1 n = pψ = n i i=0 n =i n n i p k 1 k 1 n 1 P ss n i Pss i 5 where the average probability that a target survives is P ss = j P s x j p k 1 k 1 U j l p k 1 k 1 U l. 2.2 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 = fz k, U i, r k = q Z k 1 1 fz k Z k 1 1 6

7 p k k n, r k = q = fz k n, r k = q, Z1 k 1 fz k Z1 k 1 p k k 1 n, r k = q 7 Next, the quantities on the right hand sides of the above equations are given in more explicit forms. Assuming that the pmf of number of clutter false alarm observations pc is given and using the total probability theorem, we have: fz k, U i, r k = q Z k 1 1 = c = c = c n n n fz k, U i, r k = q, n, c Z k 1 1 fz k U i, r k = q, n, c, Z k 1 1 pu i, r k = q, n, c Z k 1 1 fz k U i, r k = q, n, c, Z k 1 1 pu i r k = q, n, c, Z1 k 1 p k k 1 n, r k = qp k k 1 c = fz k U i, r k = q, n, c, Z1 k 1 c n n rk =q j p k k 1U j, r k = q p k k 1U i, r k = qp k k 1 n, r k = qpc 8 The last step in Eq. 8 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 per scan. 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 r k = q, n, c = p k k 1 U i r k = q, n n rk =q = j p k k 1U j, r k = q p k k 1U i, r k = q 9 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 1U j, r k = q 10 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 r k = q, n = 1 1 p k k 1U i, r k = q nrk =q j p k k 1U j, r k = q n rk =q j p k k 1U j, r k = q p k k 1U i, r k = q 11

8 Note that this approximation imposes a constraint on the integration of the bin-occupancy probabilities: p k k 1 U i, r k = q = p k k 1 U i r k = q, np 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 j p k k 1U j, r k = q n=0 E[n, r k = q j p k k 1U j, r k = q p k k 1U i, r k = q 12 Equation 12 implies that the expected number of targets in model q, E[n, r k = q, is equal to the integral of the PHD surface i.e. integration over the bin-occupancy probabilities within model q, j p k k 1U j, r k = q. This core property of the single model CPHD filter has been preserved and extended by the MMCPHD filter. 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: LZ n, r k = q, c = fz n, r k = q, c fz all clutter For simplicity, we assume that the probability of detection is not state dependent and constant across models, i.e. P d x i, r k = q = P d and we also make the asumption that the clutter distribution is the same in all models. In general, P d and the clutter distribution can vary across models. 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: and pu i,r k =q Z = LZ U i, r k = q pu i, r k = q LZ = 1 P d LZ U i, r k = q, V k i = 0 pu i, r k = q + P d LZ U i, r k = q, V k i = 1 pu i, r k = q LZ LZ [ Z c=0 n=0 LZ U n i,r k =q, n, c, V k i = 0 rk =q j = 1 P d pu j,r k =q pn,r k=q pc 2 [ Z pu i,r k =q q=1 c=0 n=0 LZ n,r k=q, cpn,r k =q pcpr k =q [ Z c=0 n=0 LZ U n i,r k =q, n, c, V k i = 1 rk =q j + P d pu j,r k =q pn,r k=q pc 2 [ Z pu i,r k =q 14 q=1 c=0 n=0 LZ n,r k=q, cpn,r k =q pcpr k =q pn, r k = q Z = LZ n, r k = q pn, r k = q LZ = 2 q=1 Z c=0 Z 13 c=max{ Z n,0} LZ n, r k = q, cpc [ pn, r k =q 15 n=0 LZ n, r k = q, cpn, r k = q pcpr k =q An explicit form for the likelihood ratios used in the above equations can be reached by a similar derivation as in the work of Vo et al. 2 and Erdinc et al. 3 In the case of a missed target detection in bin i in model q, the likelihood ratio is written as: LZ U i, r k = q, V i = 0 = m m j! pc = m j nn 1 n jpn, r k = q m! n=j+1 j=0 1 P d n 1 j 1 l pu P j d l, r k = q σ j L z1 cz 1, L z2 cz 2,, L zm cz m 16

9 while in the case of a successful target detection in bin i in model q, the corresponding likelihood ratio is: LZ U i, r k = q, V i = 1 = m m j! pc = m j m! j=1 m fz s x i, r k = q nn 1 n j 1pn, r k = q1 P d n j cz s=1 s n=j 1 l pu l, r k = q P j 1 Lz1 d σ j 1 cz 1,, L zs 1 cz s 1, L zs+1 cz s+1,, L zm cz m 17 with σ j y 1, y 2,, y m being the elementary symmetric function 16 of order j defined by: σ j y 1, y 2,, y m = y i1 y i2 y ij 18 with σ 0 y 1, y 2,, y m = 1. The normalization term can be expressed as: 0<i 1< <i j m LZ = m m j! pc = m j m! j=0 P j d σ Lz1 j cz 1,, L zm cz m 2 pr k = q nn 1 n j + 1pn, r k = q1 P d n j q=1 n=j 19 The numerator in Eq. 15 describing the update of the target cardinality pmf is: LZ n, r k = q = = m c=max{m n,0} min{n,m} j=0 LZ n, r k = q, cpc pc = m j m j! m! n! n j! P j d 1 P d n j σ j Lz1 cz 1,, L zm cz m 20 Note that the MMCPHD update equations 14 and 15 simplify, in the case of a single model, to the original CPHD filter update equations CONCLUSION In this work, we have applied the MMCPHD tracker to the problem of multiple target tracking with high maneuverability targets. While multiple model approaches to the PHD filter exist, a multiple model approach to the considerably more complex CPHD filter has not been put forth. Here, we have derived a new multiple model approach that allows the CPHD filter to successfully handle target maneuvers. When implemented using the Gaussian Mixtures GM approach which affords a closed form solution, the MMCPHD tracker is expected to achieve smaller RMS error and less track fragmentation over the original CPHD tracker during periods in which the targets maneuver, due to the availability of multiple hypothetical target motion models. ACKNOWLEDGMENT This research was supported by the Office of Naval Research under contract N

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