Multitarget tracking via joint PHD filtering and multiscan association

Transcription

1 1th International Conference on Inforation Fusion Seattle, WA, USA, July 6-9, 9 Multitarget tracing via joint PHD filtering and ultiscan association F. Papi, G. Battistelli, L.Chisci, S. Morrocchi Dip. Sistei e Inforatica Università di Firenze Firenze, Italy. battistelli,chisci}@dsi.unifi.it A. Farina, A. Graziano Engineering Division SELEX - Sistei Integrati Roe, Italy a.farina,a.graziano}@selex-si.co Abstract A PHD (Probability Hypothesis Density) filter and ultiscan association are cobined in a feedbac fashion in order to provide robust and efficient ultitarget tracing. The resulting hybrid tracer, thans to the feedbac connection, provides rearable perforance iproveents with respect to both an openloop PHD filter with estiate extraction via clustering and a traditional tracer equipped with a trac foration logic. Keywords: Rando set tracing, PHD filtering, ultiscan association, ultitarget tracing. 1 Introduction Multitarget and/or ultisensor tracing represents a challenging estiation proble due to its practical relevance in any air, naval and ground surveillance applications as well as for the well nown difficulty in attributing the available easureents to the true originating sources (i.e. a specific target or clutter). To accoplish thelatter tas, twoainstrea approaches are essentially pursued: a traditional approach [1]-[3] using explicit association of easureents to targets and a ore odern rando set approach [4]-[6] which, on the other hand, regards both easureents and targets as rando sets and avoids explicit association. Aong rando set tracing techniques, the socalled Probability Hypothesis Density (PHD) filter and its Cardinalized (CPHD) variant [6] have gained special attention due to their coputational tractability even for probles involving a high nuber of targets and/or easureents. The pros and cons of PHD/CPHD filters and traditional ultitarget tracers are well nown. In engineering ters, the PHD filter recursively estiates the density of targets in the state space so that the integration of the estiated density over a given region provides the expected nuber of targets in that region. Hence the PHD directly estiates the nuber and spatial distribution of targets. The ain advantage of the PHD is that it does not require any a priori nowledge on the nuber of targets, which is actually estiated. Onthe converse, the PHDdoes notdirectly provideexplicitestiates oftheineatic states ofthe targets; such estiates can be obtained by detecting the peas of the density function, or by clustering particles andthen averaging over clusters if a particle ipleentation of the PHD is adopted, but such state estiates are usually quite inaccurate for low detection probability P d and/or high false alar probability P fa. As a further drawbac, the PHD filter is typically very sensitive to uncertainties on the nowledge of P d and P fa. Traditional tracers, on the other hand, require soe for of initialization of target tracs (e.g. the classical M out of N initialization logic) and this iplies, especially in a highly cluttered scenario, the generation of a lot of teporary tracs with a consequent waste of coputational power for anaging such tracs. The objective of this paper is to cobine the positive features ofphd andtraditional tracing basedon ultiscan association in order to build a powerful tracer capable of jointly estiating the nuber of targets in the scene and providing accurate estiates of the target states, with no prior nowledge on the initial nuber and distribution of targets as well as robustly with respect to the uncertainty on the detection probability and on the clutter density. The idea is to connect a PHD filter and ultiscan association in a feedbac fashion so that both utually correct each other thus providing enhanced robustness and perforance to the overall feedbac tracer. The rest of the paper is organized as follows. Sections and 3 briefly review PHD filtering and, respectively, ultiscan (S-D) association. Next, section4shows how to cobine a PHD filter with SDA (S-D Association) in order to build the proposed hybrid PHD-SDA ultitarget tracer. In section 5, the perforance of the PHD-SDA tracer is assessed by eans of siulation experients in a radar scenario. Finally section 6 s the paper with concluding results ISIF 1163

2 PHD Filtering In ultitarget and/or ultisensor tracing there are essentially two approaches to cope with the fact that the source (either atarget orclutter) originating agiven easureent is unnown. The traditional approach [1]- [3] consists of tracing each individual target with a separate filter and, thus, requires explicit association of the available easureents to the detected targets. An alternative approach is that of regarding targets and easureents as rando sets [4]-[5], i.e. objects in which randoness is not only in the assued values but also in the nuber of eleents. In this fraewor, the objective is to recursively estiate the rando target set X t = x t,1,x t,,...,x t,nt }, i.e. the set of the states of all targets that are present at tie t, given the rando easureents sets Y = y,1,y,,...,y, }, i.e. the sets of all easureents collected fro all sensors at tie, for all up to tie t. In [6] the ulti-target Bayes filter recursion propagating the probability density function p(x t Y t ), where Y t = Y 1,Y,...,Y t }, has been provided by exploiting rando set statistics [5]. This approach, though theoretically optial, involves integrations inspaces ofincreasing, possiblyunliited, diensionandis, therefore, intractable inost practical applications. To ae the ulti-target Bayes filter coputationally feasible, the so called Probability Hypothesis Density (PHD) filtering approach has been proposed [6]. The idea underlying the PHD filter is to propagate a suitable density function D(x) in the target state-space X IR n (n = di x) such that, for any region S X, the expected nuber of targets in S is given by n(s) = D (x) dx, (1) S i.e. by integration of D( ) over S. Hereafter, D( ) will be referred to as PHD function and the objective of the PHD filter is clearly the tie propagation D t 1 t 1 (x) D t t 1 (x) D t t (x) where D t t 1 ( ) and D t t ( ) denote the PHDs at tie t based on Y t 1 and, respectively, Y t. In order to provide the PHD recursion, let us introduce the following notation: p t+1 t (x ξ): single-target state transition PDF originated by the target dynaics x t+1 = f t (x t ) + w t ; l t (x y): lielihood function originated by the easureent relationship y t = h t (x t ) + v t ; P s,t (x): survival probability of a target at tie t and state x; P d,t (x): detection probability of a target at tie t and state x; b t (x): birth density in the single-target state space; c t (y): clutter (false alar) density in the singleeasureent space Y IR p (p = di y). Then, under reasonable assuptions [6] on the target dynaics as well as on the easureents generation, the PHD recursion taes the following for: D t t 1 (x) = b t (x) + P s,t (x) p t t 1 (x ξ)d t t 1 (ξ)dξ D t t (x) = [1 P d,t (x)]d t t 1 (x) + + P d,t(x)l t (x y)d t t 1(x) c t (y) + P d,t (ξ)l t (ξ y)d t t 1 (ξ)dξ y Y t () Notice that, with respect to the general PHD recursion derived in [6], the target spawning has not been included in (). It is also worth pointing out that the PHD recursion () does not adit, in general, a closed-for solution just lie the nonlinear and/or non-gaussian single-target Bayes filter. The current state-of-art provides two approaches to the ipleentation of the PHD filter, i.e. the Gaussian Mixture PHD (GM-PHD) filter [14] and the Particle Filter PHD (PF-PHD) filter [11]. While the GM-PHD filter provides an analytical solution of the PHD recursion under suitable restrictive assuptions, the PF-PHD filter yields only a nuerical approxiation (via sequential Monte Carlo integration ethods) of such a recursion but has general applicability. In this wor the PF ipleentation of PHD will be adopted. 3 Multiscan association In this section, the ultiscan association proble over a tie window of S scans (in short S-D Association) is described in detail. To this, let n be the nuber of tracs. It is assued that for each trac i = 1,,...,n an estiate x(i) of the trac state at the beginning of the window [t S + 1,t] is available 1. Given the n tracs and the S sets of easureents Y t, = Yt S+ for = 1,,...,S, the objective of S-D Association is to assign a sequence of S easureents to each trac, where the -th eleent of such a sequence is either taen fro Y t, or represents a isseddetection. Throughoutthis section, forthe sae of copactness, the notation t, = t S+ will be adopted. Aong all feasible assignents, an optial one is found by iniizing a suitably defined cost. In this connection, let c(j 1,j,...,j S ;i) denote the cost of associating a certain sequence (j 1,j,...,j S ) to the trac i = 1,,...,n. Here, each variable j, for = 1,,...,S, taes its value in the set,1,..., t, } and 1 As will be clear in the following, in the proposed fraewor, the estiated nuber of tracs n as well as the initial estiates x(i) are obtained at each tie instant by resorting to the PHD recursion. 1164

3 refers either to the j -th easureent of the set Y t, hen j > ) or to a issed detection hen j = ). The cost taes the additive for c(j 1,j,...,j S ;i) = S c (j 1,j,...,j ;i) where c (j 1,j,...,j ;i) is the cost of adding j to the partial sequence (j 1,j,...,j 1 ). Consider now soe filtering echanis (e.g, the Exted Kalan Filter or a Sequential Monte Carlo filter [11]) that provides an estiate x as a function of a easureent z and an one-step-behind estiate x. The filter is supposed to consist of two parts: a prediction step x + = pred(x ) and a correction step (innovation update) x = update(x +,z). The propagation of the other statistics (e.g., the covariance atrix for the Exted Kalan Filter or the saple distribution for the Sequential Monte Carlo filter) as well as their involveent in the coputation of the estiate x is oitted for the sae of copactness. Then, one can write c (j 1,j,...,j ;i) = α (ˆx (j 1,j,...,j 1 ;i),j ) (3) where ˆx (j 1,j,...,j 1 ;i) is the prediction of the state oftraciat scan onthebasis ofthepartial sequence (j 1,...,j 1 ) and the function α (, ) is defined as [7] α (ˆx,h) = log (Pd l(ˆx y t S+,h )), h > log (1 P d ), h = where both the lielihood function l( ) and the detection probability P d are supposed to be tieinvariant. Given a sequence (j 1,j,...,j ), the prediction ˆx +1 (j 1,j,...,j ;i) can be coputed recursively as ˆx +1 (j 1,j,...,j ;i) = β (ˆx (j 1,j,...,j 1 ;i),j ) where the function β (, ) is defined as β (ˆx,h) = pred(update (ˆx,yt S+,h )), h > pred(ˆx), h = The recursion is initialized fro ˆx 1 (;i) = x(i). i.e. the prediction at the first scan based on the epty sequence. By exploiting the foregoing definitions, it is possible to give a atheatical forulation of the S-D Association proble. For any possible sequence of easureents (j 1,j,...,j S ) and for any trac i, let us define a binary association variable a(j 1,j,...,j S ;i) that taes value 1 if (j 1,j,...,j S ) is associated to trac i and value otherwise. Then, the optial assignents can be obtained by iniizing the loss functional n t,1 t, i=1 j 1= j = t,s j S= subject to the constraints and n t,1 i=1 j 1= t, 1 c(j 1,j,...,j S ;i)a(j 1,j,...,j S ;i) t,+1 j 1 = j +1 = t,s j S= a(j 1,...,j 1,j,j +1...,j S ;i) 1 for j = 1,,..., t,, = 1,,...,S (4) t,1 t, j 1= j = t,s j S= a(j 1,j,...,j S ;i) = 1 for i = 1,,...,n. (5) Condition (4) is needed to ensure that each easureent is assigned to at ost one trac. As to (5), it iposes that exactly one sequence be assigned to each trac. As should be evident, according to such a forulation S-D Association turnsout tobe abinaryinteger prograing proble with O(n S ) variables and O(S) constraints. The difficulty in solving S-D Association is twofold as: (i) the nuber of variables (association hypotheses) increases exponentially with the size S of the association window and (ii) even for a fixed nuber S of scans, unless S = 1, the exact solution of S-D Association requires searching over a decision space that grows exponentially with the nuber of variables. Indeed, S- D Association taes the for of a (S + 1)-diensional assignent proble wherein eleents fro S + 1 sets (one set of tracs and S sets of easureents) have to be atched so that the total atching cost is iniized. With this respect, while two-diensional assignent probles can be solved in polynoial tie using cobinatorial techniques, ulti-diensional assignent probles, even for diension S + 1 = 3, have been shown to be NP-hard[7]. As a consequence, unless very sall instances of the proble are considered, the possibility of solving S-D Association exactly is ruled out by the so-called curse of diensionality (i.e., the exponential growth of the coputational burden) and coputationally tractable approxiation schees should be sought after. Aong the any approaches to the approxiate solution of S-D Association, one of the ost successful is the relaxation algorith of [7, 8] wherein the (S + 1)- diensional assignent proble is solved as a series of -diensional assignent probles via a successive Lagrangian relaxation technique. More recently, in [9, 1] 1165

4 Y t 1:t Y t S:t z 1 t,x t ) } N t t (ˆx PHD t,,pt, PHD ) }ˆnP HD t t PHD clustering SDA ˆn PHD t t t,x t ) }NpˆnSD t t ˆn PHD t t = ˆn SD t t SDA-induced PHD resapling Figure 1: Hybrid PHD-SDA tracer z S (ˆx SD t,,p SD t, ) }ˆnSD t t a novel relaxation technique has been proposed that relies on the idea of representing the association proble as a ulti-coodity (or single-coodity) flow optiization proble on a suitable graph. This latter technique has been adopted in the siulation experients of Section 5. 4 Hybrid PHD-SDA tracer The idea of this wor is that a particle-based PHD filter and SDA can be jointly exploited according to the feedbac schee of fig. 1 for efficient ultitarget tracing. As it can be seen fro the figure, the resulting PHD-SDAtracer essentially consists offour processing blocs. The PHD filter inputs current easureents as well as the resapled particles provided by the SDA-induced PHD Resapling bloc. As a result, it outputs a set of updated particles and the estiated nuber of targets according to the PHD algorith detailed in Appix A. The clustering bloc splits the particles fro the PHD filter into clusters and outputs the centroids of such clusters and the relative saple covariances, according to the K-eans algorith. The SDA bloc inputs, at tie t, the cluster centroids at tie t S where S is the association scanbac and, after the solution of the SDA optiization proble (see [9, 1] for the details), eliinates unliely tracs according to soe crierion and outputs the current state estiates (along with the relative saple covariances) of the reaining tracs. The SDA-Induced PHD Resapling bloc carries out a resapling procedure by generating saples according to soe pre-specified probability distribution (e.g. Gaussian) around the states provided by the SDA bloc so as to re-initalize the PHD filter. 5 Siulation experients The ai of this section is to assess the perforance of the proposed hybrid PHD-SDA ultitarget tracer in ters of estiated nuber of targets, position error and robustness with respect to uncertainties in the paraeters of the scenario, suchas for instance thedetection probability P d. For this purpose, the PHD-SDA algorith has been testedinarealistic scenariolie theonedepictedinfig, and copared with the on-line available Rao Blacwellized Monte Carlo Data Association (RB-MCDA) tracer [1, 13]. RB-MCDA is a rando-set tracer basedonrao-blacwellized particle filtering, propagating in tie particles of the target rando set; fro such particles, estiates ofthe nuber oftargets (i.e., cardinality of the target rando set) and of the target states are obtained at each sapling period according to the MaM (Marginal Multitarget) estiator [5, p. 497]. coordinate y () Target Target 1 Target 3 Target 4 1 Radar sensor coordinate x () Figure : Siulated scenario For each target appearing at a different tie in the scenario, the true state at discrete tie t is x t = [x t, ẋ t, y t, ẏ t ], where (x t,y t ) provides the position and (ẋ t,ẏ t,) the velocity of the target in Cartesian coordinates. Thetarget s otion follows theconstant velocity 1166

5 odel: x t+1 = Ax t + Bw t 1 T s A = 1 1 T s 1, B = Ts / T s Ts / T s where T s is the sapling tie and w t is a zeroean white process noise with covariance atrix Q = diagσ x,σ y}. It is assued that the sensor is a radar providing easureents of range r t and aziuth θ t, then the easureent equation is given by y t = h(x t ) + v t (6) where y t = [r t, θ t ] is the easureent vector at tie t, the nonlinear function h( ) is defined as [ ] x + y h(x) =, (x + iy) and v t is a zero-ean white easureent noise with covariance atrix R = diagσ r,σ θ }. Inaddition, the radar sensor has non unitprobability detection P d and generates clutter easureents whose nuber is distributed according to a Poisson distribution with ean value n c and positions are uniforly distributed in the radar surveillance area. All the paraeters used in the siulations are reported in table 5. Paraeter Description σ x = σ y =.5 [/s ] speed std. dev. v = 5 [/s] ean speed (x 1,y 1 ) = (4.5,4.5) [K] 1st target pos. (x,y ) = (5.5,4.5) [K] nd target pos. (x 3,y 3 ) = (4.5,5.5) [K] 3rd target pos. (x 4,y 4 ) = (5.5,5.5) [K] 4th target pos. t 1 = [s] 1st target birth t = [s] nd target birth t 3 = 1 [s] 3rd arget birth t 4 = [s] 4th target birth σ r = 3 [] range std. dev. σ θ = [ ] aziuth std. dev. (x,y ) = (,) [] radar pos. P d =.9 detection prob. n c = 4 avg.# clutter eas. S = 3 ass. scanbac T s = 1 [s] sapling period Table 1: Siulation paraeters Figs. 3-6 and 7 display the position root ean square error for each target and, respectively, the estiated nuber of targets for the proposed PHD-SDA as well RMSE for target 1 () Figure 3: RMSE over 5 Monte Carlo runs RMSE for target () Figure 4: RMSE over 5 Monte Carlo runs as for the RB-MCDA algorith, obtained by averaging over 5 indepent Monte Carlo runs. Figs. 3-6 shows how the position RMSE provided by PHD-SDA is significantly lower, while fig. 7 shows that the second level of resapling introduced by association prevents the aintenance of false tracs, at the expense of a negligible delay due to the SDA scanbac. Fig. 8copares thewasserstein distance [15] forthe PHD-SDA algorith and the SDA algorith initialized in two different ways: in particular, Known Initial State refers to the SDA with perfectly nown initial state, while Noisy Initial State refers to the SDA that, for each target, calculates the initial state estiate by applying the polar-to-cartesian transforation to the first two noisy easureents. It is worth pointing out that PHD-SDA does not assue any nowledge on the nuber of targets as well as on the tie instants in which the targets appear, while both SDA schees assue perfect nowledge of such a nuber and of the tie instants of appearance; hence the curve relative to the Noisy Initial State initialization also represents a lower boundfor theperforance achievable bythewell nown M/N trac initialization logic [, p.43] with M = N =. Notice that the transient of PHD-SDA is bounded between the two liiting cases of SDA with 1167

6 True Nuber RMSE for target 3 () Figure 5: RMSE over 5 Monte Carlo runs RMSE for target 4 () Figure 6: RMSE over 5 Monte Carlo runs nown initial state and SDA with noisy initial state; this shows that PHD filtering represents indeed a sensible trac initialization logic for the SDA. Finallyfigs. 9and1otivate thecobination ofthe rando set tracing philosophy (PHD filter) with the traditional tracing philosophy (SDA): the proposed schee, thans to thefeedbac connection ofphd and SDA guarantees a high degree of robustness with respect to the uncertainty on the detection probability P d anda siilar insensitivity is expectedfor other paraeters, e.g. the probability of false alar P fa. Specifically, fig. 9 depicts the estiated nuber of targets (averaged over 5 Monte Carloruns) whenanoverestiated P d (.99 instead of.9) is used. A siilar plot is reported in fig. 9 for an underestiated P d (.8 instead of.9). In both cases, the standard deviation of the estiated nuber of targets is.1 for the PHD-SDA and.9 for the PHD. 6 Conclusions A novel ultitarget tracing technique based on the feedbac connection of a PHD filter and of ultiscan association has been presented. Fro the perspective of traditional tracing, the PHD filter provides a theoretically sound ethod for initializing tracs in alter- Wasserstein Distance () Nubers of Targets Figure 7: Estiated nuber of targets Known Init State Noisy Init State Figure 8: PHD-SDA vs. SDA with a-priori nowledge on the initial state native to heuristic ethods such as, for instance, the M out of N logic. Fro the perspective of PHD filtering, the feedbac connection with ultiscan association aes the PHD filter less sensitive to the uncertainties on the scenario s paraeters (e.g. detection and false alar probabilities). In general, the resulting hybrid tracer sees to positively cobine the erits of thetraditional andrando-set approaches toultitarget tracing while counteracting the negative aspects thans to a echanis of utual correction between PHD and association. Future wor will address the extension to the ultisensor case. Further ideas that will be explored for robustand efficient ultitarget and ultisensor tracing are PHD filtering with ultiscan resapling and Monte Carlo ultiscan association. References [1] Y. Bar-Shalo and T.E. Fortann: Tracing and data association, Acadeic Press, SanDiego, [] Y. Bar-Shalo and X.R. Li: Multitargetultisensor tracing: principles and techniques, 1168

7 Mean nuber of targets PHD True nuber of targets Figure 9: Effect of an overestiated P d 5 [9] G. Battistelli, L. Chisci, F. Papi, A. Benavoli and A. Farina: Multiscan association as a ulticoodity flow optiization proble, Proc. 8 IEEE Radar Conference, Roe, Italy, May 8. [1] G. Battistelli, L. Chisci, F. Papi, A. Benavoli and A. Farina: Multiscan association as a singlecoodity flow optiization proble, Proc. Radar 8 Conference, Adelaide, Australia, Septeber 8. [11] B.N. Vo, S. Singh and A. Doucet: Sequential Monte Carlo ethods for ulti-target filtering with rando finite sets, IEEE Trans. on Aerospace and Electronic Systes, vol. 41, n. 4, pp , 5. Mean nuber of targets PHD True nuber of targets Figure 1: Effect of an underestiated P d YBS Publishing, Storrs, CT, [3] S. Blacan and R. Popoli: Modern tracing systes, Artech House, Norwood, MA, 6. [4] I.R. Goodan, R.P.S. Mahler and H.T. Nguyen: Matheatics of data fusion, Kluwer Acadeic Publishers, Dordrecht, [5] R.P.S. Mahler: Statistical ultisource ultitarget inforation fusion, Artech House, 7. [6] R.P.S. Mahler: Multitarget Bayes filtering via first-orderultitarget oents, IEEE Trans. on Aerospace and Electronic Systes, vol. 39, n. 4, pp , 3. [7] S. Deb, M. Yeddanapudi, K. Pattipati and Y. Bar-Shalo: A generalized S-D assignent algorith for ultisensor-ultitarget state estiation, IEEE Trans. on Aerospace and Electronic Systes, vol. 33, n., pp , [8] A. Poore, S. Lu and B.J. Suchoel: Data Association Using Multiple Frae Assignents, in Handboo of Multisensor Data fusion, D.L. Hall and J. Llinas Eds., CRC Press, [1] S. Särä, A. Vehtari, and J. Lapinen: Rao- Blacwellized Particle Filter for Multiple Target Tracing, Inforation Fusion Journal, vol. 8, n. 1, pp. -15, 7. [13] M. Vihola: Rao-Blacwellised particle filter in rando set ultitarget tracing, IEEE Trans. on Aerospace and Electronic Systes, vol. 43, n., pp , 7. [14] B.N. Vo and W.K. Ma: The Gaussian ixture Probability Hypothesis Density filter, IEEE Trans. on Signal Processing, vol. 54, n. 11, pp , 6. [15] J.R. Hoffan and R.P.S. Mahler: Multitarget iss distance via optial assignent, IEEE Trans. on Systes, Man and Cybernetics - Part A: Systes and Huans, vol. 34, n. 3, pp , 4. Acnowledgents This wor has been partially supported by SELEX- Sistei Integrati. A PHD algorith ThePHDrecursionis propagated byeans ofaparticle Filter algorith. Specifically, at each tie instant t the PHD D t t ( ) is approxiated by a set of particles t,x t ) } N t t where x t X and wt R represent the state and, respectively, the weight of the -th particle. The approxiation is designed so that the cuulative weight of particles in a region S of the state space represents the expected nuber of targets in such a region, i.e., wt D t t (x) dx. x t S S

8 Then the estiated nuber of targets at tie t can be obtained as N t t ˆn t t = wt. For the sae of siplicity, heraafter it is supposed that P s,t (x) = P s, b t (x) = b(x) and c t (y) = c. The algorith is initialized by setting ˆn =, Y f = Y, N =. Then, at each tie t + 1, for t =,1,..., the following steps are carried out. PF-PHD algorith step 1 (birth): generate N p Y f t new particles fro the easureents Y f t according to the birth odel b( ); set N t+1 t = N t t + N p Y f t ; step (prediction): for = 1,...,N t+1 t saple x t+1 fro x t according to the single target state transition PDF p( x t ); set w t+1 = P s w t ; step 3 (update): for j = 1,..., t+1 for = 1,...,N t+1 t copute the lielihood λ j = l(x t+1 y t+1,j ); copute the total lielihood Λ j = N t+1 t wt+1λ j ; for = 1,...,N t+1 t set wt+1 = wt+1 1 P d + P d λ j ; c + P d Λ j set Y f t+1 = y t+1,j : Λ j Λ}; step 4 (resapling): set ˆn b t+1 t+1 = N t+1 t =N t t +1 w t+1; set ˆn t+1 t+1 = ˆn t t + ˆn b t+1 t+1 ; set N t+1 t+1 = N t t + N p ˆn b t+1 t+1 ; j: Λ j>λ resaple N t t particles fro t+1,x t+1) } N t t to get t+1,x t+1) } N t t ; resaple N pˆn b t+1 t+1 particles fro t+1,x t+1) } N t+1 t =N t t +1 to get t+1,x t+1) } N t+1 t+1 =N t t +1 ; Soe rears on the proposed PF-PHD algorith are in order. First, note that Y f t+1 represents the set of easureents at tie t + 1 that are far fro all the existing particles. Such a set is obtained by coparing the total lielihood Λ j associated to each easureent y t+1,j with a given threshold Λ (typically such a threshold has the sae order of agnitude of the clutter densityc). Clearly, sucheasureents canrepresenteither clutter or new targets since they are very unliely to be generated by existing targets. Coherently with this consideration, in the proposed algorith the easureents belonging to Y f t+1 are neglected in the update of the existing particles weights and used only for generating new particles according to the birth odel b( ). Notice that the birth of particles in the proxiity of the easureents belonging to Y f t is postponed to the following iteration of the algorith, so that such particles have to be copared also with the easureents Y t+1 before they can give rise to new tracs (thus avoiding the initialization of a new trac for each clutter easureent). Let n b = b(x)dx be the expected nuber of targets enteringthe scenario.the birth of new particles in step 1 of the PF-PHD algorith is perfored as follows. Birth of new particles generate N b new particles ( w t, x t )} N b according to the PDF b( )/n b with weight w t = n b /N b ; for j such that y t,j Yt f for = 1,...,N b copute the lielihood λ j = l( x t y t,j ); copute the total lielihood Λ j = N b w t λ j ; for = 1,...,N b set w t = w t 1 P d + P d λ j ; c + P j d Λ j resaple N p Yt f particles fro ( w t, x t )} N b to get t,x t ) } N t t +N p Y f t ; =N t t +1 As to the resapling step, a sort of stratified resapling is adopted in that previously existing particles (corresponding to the indices 1,...,N t t ) are resapled separately fro the new-born particles (corresponding to the indices N t t + 1,...,N t+1 t ). Notice that the estiated nuber of targets n t+1 t+1 after each PF-PHD iteration corresponds to the initial estiate ˆn t t plus the estiated nuber of new-born targets ˆn b t+1 t+1. Thus, the PHD bloc is used to initialize new tracs but trac deletion is perfored only in the SDA bloc. This choice serves the purpose of itigating the well-nown instability of the PHD filter in estiating the nuber of targets in the presence of issed detections. 117