The Gaussian Mixture Probability Hypothesis Density Filter
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 4091
The Gaussian Mixture Probability HypothesisDensity Filter
Ba-Ngu Vo and Wing-Kin Ma, Member, IEEE
AbstractA new recursive algorithm is proposed for jointlyestimating the time-varying number of targets and their statesfrom a sequence of observation sets in the presence of data associ-ation uncertainty, detection uncertainty, noise, and false alarms.The approach involves modelling the respective collections oftargets and measurements as random finite sets and applying theprobability hypothesis density (PHD) recursion to propagate theposterior intensity, which is a first-order statistic of the randomfinite set of targets, in time. At present, there is no closed-formsolution to the PHD recursion. This paper shows that under linear,Gaussian assumptions on the target dynamics and birth process,the posterior intensity at any time step is a Gaussian mixture.More importantly, closed-form recursions for propagating themeans, covariances, and weights of the constituent Gaussiancomponents of the posterior intensity are derived. The proposedalgorithm combines these recursions with a strategy for managingthe number of Gaussian components to increase efficiency. Thisalgorithm is extended to accommodate mildly nonlinear targetdynamics using approximation strategies from the extended andunscented Kalman filters.
Index TermsIntensity function, multiple-target tracking, op-timal filtering, point processes, random sets.
I N a multiple-target environment, not only do the states ofthe targets vary with time but also the number of targetschanges due to targets appearing and disappearing. Often, notall of the existing targets are detected by the sensor. Moreover,the sensor also receives a set of spurious measurements (clutter)not originating from any target. As a result, the observation setat each time step is a collection of indistinguishable partial ob-servations, only some of which are generated by targets. Theobjective of multiple-target tracking is to jointly estimate, ateach time step, the number of targets and their states from a se-quence of noisy and cluttered observation sets. Multiple-targettracking is an established area of study; for details on its tech-niques and applications, readers are referred to  and . Upto date overviews are also available in more recent works suchas .
Manuscript received August 18, 2005; revised January 23, 2006. This workwas supported in part by the Australian Research Council under DiscoveryGrant DP0345215. The associate editor coordinating the review of this manu-script and approving it for publication was Dr. Carlos H. Muravchik.
B.-N. Vo is with the Department of Electrical and Electronic Engineering, TheUniversity of Melbourne, Parkville, Victoria 3010, Australia (e-mail: [email protected]).
W.-K. Ma is with the Institute of Communications Engineering and De-partment of Electrical Engineering, National Tsing Hua University, Hsinchu,Taiwan 30013, R.O.C. (e-mail: [email protected]).
Digital Object Identifier 10.1109/TSP.2006.881190
An intrinsic problem in multiple-target tracking is the un-known association of measurements with appropriate targets, , , . Due to its combinatorial nature, the dataassociation problem makes up the bulk of the computationalload in multiple-target tracking algorithms. Most traditionalmultiple-target tracking formulations involve explicit associa-tions between measurements and targets. Multiple hypothesestracking (MHT) and its variations concern the propagation ofassociation hypotheses in time , , . The joint proba-bilistic data association filter (JPDAF) , , the probabilisticMHT (PMHT) , and the multiple-target particle filter , use observations weighted by their association probabilities. Al-ternative formulations that avoid explicit associations betweenmeasurements and targets include symmetric measurementequations  and random finite sets (RFS) , .
The RFS approach to multiple-target tracking is an emergingand promising alternative to the traditional association-basedmethods , , . A comparison of the RFS approach andtraditional multiple-target tracking methods has been given in. In the RFS formulation, the collection of individual tar-gets is treated as a set-valued state, and the collection of indi-vidual observations is treated as a set-valued observation. Mod-elling set-valued states and set-valued observations as RFSs al-lows the problem of dynamically estimating multiple targets inthe presence of clutter and association uncertainty to be cast in aBayesian filtering framework , , . This theoret-ically optimal approach to multiple-target tracking is an elegantgeneralization of the single-target Bayes filter. Indeed, novelRFS-based filters such as the multiple-target Bayes filter, theprobability hypothesis density (PHD) filter , , , andtheir implementations , ,  have generated sub-stantial interest.
The focus of this paper is the PHD filter, a recursion thatpropagates the first-order statistical moment, or intensity, of theRFS of states in time . This approximation was developed toalleviate the computational intractability in the multiple-targetBayes filter, which stems from the combinatorial nature of themultiple-target densities and the multiple integrations on the(infinite dimensional) multiple-target state space. The PHDfilter operates on the single-target state space and avoids thecombinatorial problem that arises from data association. Thesesalient features render the PHD filter extremely attractive.However, the PHD recursion involves multiple integrals thathave no closed-form solutions in general. A generic sequentialMonte Carlo technique , , accompanied by variousperformance guarantees , , , has been proposedto propagate the posterior intensity in time. In this approach,state estimates are extracted from the particles representing the
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posterior intensity using clustering techniques such as -meanor expectation maximization. Special cases of this so-calledparticle-PHD filter have also been independently implementedin  and . Due to its ability to handle the time-varyingnumber of nonlinear targets with relatively low complexity,innovative extensions and applications of the particle-PHDfilter soon followed . The main drawbacks of thisapproach are the large number of particles and the unreliabilityof clustering techniques for extracting state estimates. (Thelatter will be further discussed in Section III-C.)
In this paper, we propose an analytic solution to the PHD re-cursion for linear Gaussian target dynamics and Gaussian birthmodel. This solution is analogous to the Kalman filter as a so-lution to the single-target Bayes filter. It is shown that whenthe initial prior intensity is a Gaussian mixture, the posteriorintensity at any subsequent time step is also a Gaussian mix-ture. Moreover, closed-form recursions for the weights, means,and covariances of the constituent Gaussian components are de-rived. The resulting filter propagates the Gaussian mixture pos-terior intensity in time as measurements arrive in the same spiritas the Gaussian sum filter of , . The fundamental differ-ence is that the Gaussian sum filter propagates a probability den-sity using the Bayes recursion, whereas the Gaussian mixturePHD filter propagates an intensity using the PHD recursion. Anadded advantage of the Gaussian mixture representation is that itallows state estimates to be extracted from the posterior intensityin a much more efficient and reliable manner than clustering inthe particle-based approach. In general, the number of Gaussiancomponents in the posterior intensity increases with time. How-ever, this problem can be effectively mitigated by keeping onlythe dominant Gaussian components at each instance. Two ex-tensions to nonlinear target dynamics models are also proposed.The first is based on linearizing the model while the second isbased on the unscented transform. Simulation results are pre-sented to demonstrate the capability of the proposed approach.
Preliminary results on the closed-form solution to the PHDrecursion have been presented as a conference paper . Thispaper is a more complete version.
The structure of this paper is as follows. Section II presentsthe random finite set formulation of multiple-target filteringand the PHD filter. Section III presents the main result of thispaper, namely, the analytical solution to the PHD recursionunder linear Gaussian assumptions. An implementation of thePHD filter and simulation results are also presented. Section IVextends the proposed approach to nonlinear models using ideasfrom the extended and unscented Kalman filters. Demonstra-tions with tracking nonlinear targets are also given. Finally,concluding remarks and possible future research directions aregiven in Section V.
II. PROBLEM FORMULATION
This section presents a formulation of multiple-target fil-tering in the random finite set (or point process) framework. Webegin with a review of single-target Bayesian filtering in Sec-tion II-A. Using random finite set models, the multiple-targettracking problem is then formulated as a Bayesian filtering
problem in Section II-B. This provides sufficient backgroundleading to Section II-C, which describes the PHD filter.
A. Single-Target Filtering
In many dynamic state estimation problems, the state is as-sumed to follow a Markov process on the state space ,with transition density , i.e., given a state attime 1, the probability density of a transition to the stateat time is1
This Markov process is partially observed in the observationspace , as modelled by the likelihood function ,i.e., given a state at time , the probability density of re-ceiving the observation is
The probability density of the state at time given allobservations up to time , denoted by
is called the posterior density (or filtering density) at time .From an initial density , the posterior density at time canbe computed using the Bayes recursion
All information about the state at time is encapsulated in theposterior density , and estimates of the state at timecan be obtained using either the minimum mean squared error(MMSE) criterion or the maximum a posteriori (MAP) criterion.2
B. Random Finite Set Formulation of MultiTarget Filtering
Now consider a multiple target scenario. Let be thenumber of targets at time , and suppose that, at time ,the target states are . At the nexttime step, some of these targets may die, the surviving targetsevolve to their new states, and new targets may appear. This re-sults in new states . Note that the orderin which the states are listed has no significance in the RFS mul-tiple-target model formulation. At the sensor, measure-ments are received at time . The originsof the measurements are not known, and thus the order in whichthey appear bears no significance. Only some of these measure-ments are actually generated by targets. Moreover, they are in-distinguishable from the false measurements. The objective ofmultiple-target tracking is to jointly estimate the number of tar-gets and their states from measurements with uncertain origins.
1For notational simplicity, random variables and their realizations are not dis-tinguished.
2These criteria are not necessarily applicable to the multiple-target case.
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Even in the ideal case where the sensor observes all targets andreceives no clutter, single-target filtering methods are not appli-cable since there is no information about which target generatedwhich observation.
Since there is no ordering on the respective collections oftarget states and measurements at time , they can be naturallyrepresented as finite sets, i.e.,
where and are the respective collections of all fi-nite subsets of and . The key in the random finite set for-mulation is to treat the target set and measurement set asthe multiple-target state and multiple-target observation respec-tively. The multiple-target tracking problem can then be posedas a filtering problem with (multiple-target) state spaceand observation space .
In a single-target system, uncertainty is characterized bymodelling the state and measurement as random vec-tors. Analogously, uncertainty in a multiple-target system ischaracterized by modelling the multiple-target state andmultiple-target measurement as random finite sets. An RFS
is simply a finite-set-valued random variable, which can bedescribed by a discrete probability distribution and a familyof joint probability densities , , . The discretedistribution characterizes the cardinality of , while for agiven cardinality, an appropriate density characterizes the jointdistribution of the elements of .
In the following, we describe an RFS model for the time evo-lution of the multiple-target state, which incorporates target mo-tion, birth and death. For a given multiple-target stateat time 1, each either continues to exist attime with probability3 or dies with probability
. Conditional on the existence at time , theprobability density of a transition from state to is givenby (1), i.e., . Consequently, for a given state
at time 1, its behavior at the next time step ismodelled as the RFS
that can take on either when the target survives or whenthe target dies. A new target at time can arise either by spon-taneous births (i.e., independent of any existing target) or byspawning from a target at time 1. Given a multiple-targetstate at time 1, the multiple-target state at timeis given by the union of the surviving targets, the spawned tar-gets, and the spontaneous births
3Note that p (x ) is a probability parameterized by x .
RFS of spontaneous birth at time ;
RFS of targets spawned at time from a targetwith previous state .
It is assumed that the RFSs constituting the union in (9) are inde-pendent of each other. The actual forms of and areproblem dependent; some examples are given in Section III-D.
The RFS measurement model, which accounts for detectionuncertainty and clutter, is described as follows. A given target
is either detected with probability4 ormissed with probability . Conditional on detec-tion, the probability density of obtaining an observationfrom is given by (2), i.e., . Consequently, at time
, each state generates an RFS
that can take on either when the target is detected orwhen the target is not detected. In addition to the target origi-nated measurements, the sensor also receives a set of falsemeasurements, or clutter. Thus, given a multiple-target stateat time , the multiple-target measurement received at thesensor is formed by the union of target generated measurementsand clutter, i.e.,
It is assumed that the RFSs constituting the union in (11) areindependent of each other. The actual form of is problemdependent; some examples will be illustrated in Section III-D.
In a similar vein to the single-target dynamical model in (1)and (2), the randomness in the multiple-target evolution and ob-servation described by (9) and (11) are, respectively, capturedin the multiple-target transition density and mul-tiple-target likelihood5 , . Explicit expressions for
and can be derived from the un-derlying physical models of targets and sensors using finite setstatistics (FISST)6 , , , although these are not neededfor this paper.
Let denote the multiple-target posterior density.Then, the optimal multiple-target Bayes filter propagates themultiple-target posterior in time via the recursion
where is an appropriate reference measure on ,. We remark that although various applications of pointprocess theory to multiple-target tracking have been reported in
4Note that p (x ) is a probability parameterized by x .5The same notation is used for multiple-target and single-target densities.
There is no danger of confusion since in the single-target case the argumentsare vectors, whereas in the multiple-target case the arguments are finite sets.
6Strictly speaking, FISST yields the set derivative of the belief mass func-tional, but this is in essence a probability density .
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the literature (e.g., ), FISST , ,  is the firstsystematic approach to multiple-target filtering that uses RFSsin the Bayesian framework presented above.
The recursion (12) and (13) involves multiple integrals on thespace , which are computationally intractable. SequentialMonte Carlo implementations can be found in , , ,and . However, these methods are still computationally in-tensive due to the combinatorial nature of the densities, espe-cially when the number of targets is large , . Nonethe-less, the optimal multiple-target Bayes filter has been success-fully applied to applications where the number of targets is small.
C. The Probability Hypothesis Density (PHD) Filter
The PHD filter is an approximation developed to alleviate thecomputational intractability in the multiple-target Bayes filter.Instead of propagating the multiple-target posterior densityin time, the PHD filter propagates the posterior intensity, afirst-order statistical moment of the posterior multiple-targetstate . This strategy is reminiscent of the constant gainKalman filter, which propagates the first moment (the mean) ofthe single-target state.
For an RFS on with probability distribution , its first-order moment is a nonnegative function on , called the in-tensity, such that for each region , 
In other words, the integral of over any region gives theexpected number of elements of that are in . Hence, the totalmass gives the expected number of elements of
. The local maxima of the intensity are points in with thehighest local concentration of expected number of elements, andhence can be used to generate estimates for the elements of .The simplest approach is to round and choose the resultingnumber of highest peaks from the intensity. The intensity is alsoknown in the tracking literature as the probability hypothesisdensity , .
An important class of RFSs, the Poisson RFSs, are those com-pletely characterized by their intensities. An RFS is Poissonif the cardinality distribution of , , is Poissonwith mean and for any finite cardinality, the elements ofare independently identically distributed according to the proba-bility density , . For the multiple-target problemdescribed in Section II-B, it is common to model the clutter RFS[ in (11)] and the birth RFSs [ and in (9)]as Poisson RFSs.
To present the PHD filter, recall the multiple-target evolutionand observation models from Section II-B with
intensity of the birth RFS at time ;
intensity of the RFS spawned at timeby a target with previous state ;
probability that a target still exists at timegiven that its previous state is ;
probability of detection given a state at time ;
intensity of clutter RFS at time ;
and consider the following assumptions.A.1: Each target evolves and generates observations indepen-
dently of one another.A.2: Clutter is Poisson and independent of target-originated
measurements,A.3: The predicted multiple-target RFS governed by
is Poisson.Remark 1: Assumptions A.1 and A.2 are standard in most
tracking applications (see, for example,  and ) and have al-ready been alluded to in Section II-B. The additional assumptionA.3 is a reasonable approximation in applications where interac-tions between targets are negligible . In fact, it can be shownthat A.3 is completely satisfied when there is no spawning andthe RFSs and are Poisson.
Let and denote the respective intensities associ-ated with the multiple-target posterior density and the mul-tiple-target predicted density in the recursion (12)(13).Under assumptions A.1A.3, it can be shown (using FISST or classical probabilistic tools ) that the posterior intensitycan be propagated in time via the PHD recursion
It is clear from (15) and (16) that the PHD filter completelyavoids the combinatorial computations arising from the un-known association of measurements with appropriate targets.Furthermore, since the posterior intensity is a function on thesingle-target state space , the PHD recursion requires muchless computational power than the multiple-target recursion(12) and (13), which operates on . However, as men-tioned in the introduction, the PHD recursion does not admitclosed-form solutions in general, and numerical integrationsuffers from the curse of dimensionality.
III. THE PHD RECURSION FOR LINEAR GAUSSIAN MODELS
This section shows that for a certain class of multiple-targetmodels, herein referred to as linear Gaussian multiple-targetmodels, the PHD recursion (15) and (16) admits a closed-formsolution. This result is then used to develop an efficient mul-tiple-target tracking algorithm. The linear Gaussian multiple-target models are specified in Section III-A, while the solutionto the PHD recursion is presented in Section III-B. Implementa-tion issues are addressed in Section III-C. Numerical results arepresented in Section III-D, and some generalizations are dis-cussed in Section III-E.
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A. Linear Gaussian Multiple-Target Model
Our closed-form solution to the PHD recursion requires, inaddition to assumptions A.1A.3, a linear Gaussian multiple-target model. Along with the standard linear Gaussian modelfor individual targets, the linear Gaussian multiple-target modelincludes certain assumptions on the birth, death, and detectionof targets. These are summarized below.
A.4: Each target follows a linear Gaussian dynamical modeland the sensor has a linear Gaussian measurement model, i.e.,
where denotes a Gaussian density with mean andcovariance , is the state transition matrix, is theprocess noise covariance, is the observation matrix, andis the observation noise covariance.
A.5: The survival and detection probabilities are state inde-pendent, i.e.,
A.6: The intensities of the birth and spawn RFSs are Gaussianmixtures of the form
where , , , , , are given modelparameters that determine the shape of the birth intensity; simi-larly, , , , , and , ,determine the shape of the spawning intensity of a target withprevious state .
Some remarks regarding the above assumptions are in order.Remark 2: Assumptions A.4 and A.5 are commonly used in
many tracking algorithms , . For clarity in the presentation,we only focus on state-independent and , althoughclosed-form PHD recursions can be derived for more generalcases (see Section III-E).
Remark 3: In assumption A.6, , are thepeaks of the spontaneous birth intensity in (21). These pointshave the highest local concentrations of expected number ofspontaneous births and represent, for example, air bases or air-ports where targets are most likely to appear. The covariancematrix determines the spread of the birth intensity in the
vicinity of the peak . The weight gives the expected
number of new targets originating from . A similar inter-pretation applies to (22), the spawning intensity of a target withprevious state , except that the th peak isan affine function of . Usually, a spawned target is modelledto be in the proximity of its parent. For example, could cor-respond to the state of an aircraft carrier at time 1, while
is the expected state of fighter planes spawnedat time . Note that other forms of birth and spawning intensitiescan be approximated, to any desired accuracy, using Gaussianmixtures .
B. The Gaussian Mixture PHD Recursion
For the linear Gaussian multiple-target model, the followingtwo propositions present a closed-form solution to the PHD re-cursion (15), (16). More concisely, these propositions show howthe Gaussian components of the posterior intensity are analyti-cally propagated to the next time.
Proposition 1: Suppose that Assumptions A.4A.6 hold andthat the posterior intensity at time is a Gaussian mixtureof the form
Then, the predicted intensity for time is also a Gaussian mix-ture and is given by
where is given in (21)
Proposition 2: Suppose that Assumptions A.4A.6 hold andthat the predicted intensity for time is a Gaussian mixture ofthe form
Then, the posterior intensity at time is also a Gaussian mixtureand is given by
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Propositions 1 and 2 can be established by applying the fol-lowing standard results for Gaussian functions.
Lemma 1: Given , , , , and of appropriate dimen-sions and that and are positive definite
(38)Lemma 2: Given , , , and of appropriate dimensions
and that and are positive definite
Note that Lemma 1 can be derived from Lemma 2, which in turncan be found in  or [44, Section 3.8], though in a slightlydifferent form.
Proposition 1 is established by substituting (17), (19), and(21)(23) into the PHD prediction (15) and replacing integralsof the form (38) by appropriate Gaussians as given by Lemma1. Similarly, Proposition 2 is established by substituting (18),(20), and (31) into the PHD update (16) and then replacing in-tegrals of the form (38) and product of Gaussians of the form(39) by appropriate Gaussians as given by Lemmas 1 and 2, re-spectively.
It follows by induction from Propositions 1 and 2 that if theinitial prior intensity is a Gaussian mixture (including thecase where ), then all subsequent predicted intensities
and posterior intensities are also Gaussian mixtures.Proposition 1 provides closed-form expressions for computingthe means, covariances, and weights of from those of
. Proposition 2 then provides closed-form expressions forcomputing the means, covariances, and weights of from thoseof when a new set of measurements arrives. Proposi-tions 1 and 2 are, respectively, the prediction and update steps ofthe PHD recursion for a linear Gaussian multiple-target model,herein referred to as the Gaussian mixture PHD recursion. Forcompleteness, we summarize the key steps of the Gaussian mix-ture PHD filter in Table I.
Remark 4: The predicted intensity in Proposition 1consists of three terms , , and due, respec-tively, to the existing targets, the spawned targets, and the spon-taneous births. Similarly, the updated posterior intensity inProposition 2 consists of a misdetection term 1and detection terms , one for each measurement
. As it turns out, the recursions for the means andcovariances of and are Kalman predictions,
TABLE IPSEUDOCODE FOR THE GAUSSIAN MIXTURE PHD FILTER
and the recursions for the means and covariances ofare Kalman updates.
Given the Gaussian mixture intensities and , the cor-responding expected number of targets and can beobtained by summing up the appropriate weights. Propositions
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1 and 2 lead to the following closed-form recursions forand .
Corollary 1: Under the premises of Proposition 1, the meanof the predicted number of targets is
Corollary 2: Under the premises of Proposition 2, the meanof the updated number of targets is
In Corollary 1, the mean of the predicted number of targets isobtained by adding the mean number of surviving targets, themean number of spawnings, and the mean number of births.A similar interpretation can be drawn from Corollary 2. Whenthere is no clutter, the mean of the updated number of targets isthe number of measurements plus the mean number of targetsthat are not detected.
C. Implementation Issues
The Gaussian mixture PHD filter is similar to the Gaussiansum filter of  and  in the sense that they both propa-gate Gaussian mixtures in time. Like the Gaussian sum filter,the Gaussian mixture PHD filter also suffers from computationproblems associated with the increasing number of Gaussiancomponents as time progresses. Indeed, at time , the Gaussianmixture PHD filter requires
Gaussian components to represent , where is the numberof components of . This implies that the number of com-ponents in the posterior intensities increases without bound.
A simple pruning procedure can be used to reduce the numberof Gaussian components propagated to the next time step. Agood approximation to the Gaussian mixture posterior intensity
can be obtained by truncating components that have weakweights . This can be done by discarding those withweights below some preset threshold, or by keeping only a cer-tain number of components with strongest weights. Moreover,some of the Gaussian components are so close together thatthey could be accurately approximated by a single Gaussian.Hence, in practice these components can be merged into one.These ideas lead to the simple heuristic pruning algorithmshown in Table II.
Having computed the posterior intensity , the next task isto extract multiple-target state estimates. In general, such a taskmay not be simple. For example, in the particle-PHD filter ,the estimated number of targets is given by the total massof the particles representing . The estimated states are then
TABLE IIPRUNING FOR THE GAUSSIAN MIXTURE PHD FILTER
TABLE IIIMULTIPLE-TARGET STATE EXTRACTION
obtained by partitioning these particles into clusters, usingstandard clustering algorithms. This works well when the pos-terior intensity naturally has clusters. Conversely, when
differs from the number of clusters, the state estimates be-come unreliable.
In the Gaussian mixture representation of the posterior inten-sity , extraction of multiple-target state estimates is straight-forward since the means of the constituent Gaussian compo-nents are indeed the local maxima of , provided that they arereasonably well separated. Note that after pruning (see Table II),closely spaced Gaussian components would have been merged.Since the height of each peak depends on both the weight andcovariance, selecting the highest peaks of may result instate estimates that correspond to Gaussians with weak weights.This is not desirable because the expected number of targets dueto these peaks is small, even though the magnitudes of the peaksare large. A better alternative is to select the means of the Gaus-sians that have weights greater than some threshold, e.g., 0.5.This state estimation procedure for the Gaussian mixture PHDfilter is summarized in Table III.
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D. Simulation Results
Two simulation examples are used to test the proposedGaussian mixture PHD filter. An additional example can befound in .
Example 1: For illustration purposes, consider a two-di-mensional scenario with an unknown and time varyingnumber of targets observed in clutter over the surveillanceregion [ 1000,1000] [ 1000,1000] (in ). The state
of each target consists ofposition and velocity , while the mea-surement is a noisy version of the position.
Each target has survival probability and followsthe linear Gaussian dynamics (17) with
where and denote, respectively, the identity andzero matrices, s is the sampling period, and(m/s ) is the standard deviation of the process noise. Targetscan appear from two possible locations as well as spawned fromother targets. Specifically, a Poisson RFS with intensity
where , ,and is used to model sponta-neous births in the vicinity of and . Additionally, theRFS of targets spawned from a target with previousstate is Poisson with intensity
Each target is detected with probability , and themeasurement follows the observation model (18) with
, , where m is the standard devia-tion of the measurement noise. The detected measurements areimmersed in clutter that can be modelled as a Poisson RFSwith intensity
where is the uniform density over the surveillance region,m is the volume of the surveillance region, and
m is the average number of clutter returnsper unit volume (i.e., 50 clutter returns over the surveillanceregion).
Fig. 1 shows the true target trajectories, while Fig. 2 plotsthese trajectories with cluttered measurements against time. Tar-gets 1 and 2 are born at the same time but at two different lo-cations. They travel along straight lines (their tracks cross at
s) and at s target 1 spawns target 3.The Gaussian mixture PHD filter, with parameters ,
, and (see Table II for the meanings of theseparameters), is applied. From the position estimates shown inFig. 3, it can be seen that the Gaussian mixture PHD filter pro-vides accurate tracking performance. The filter not only suc-cessfully detects and tracks targets 1 and 2 but also manages todetect and track the spawned target 3. The filter does generate
Fig. 1. Target trajectories. : locations at which targets are born; : loca-tions at which targets die.
Fig. 2. Measurements and true target positions.
anomalous estimates occasionally, but these false estimates dieout very quickly.
Example 2: In this example, we evaluate the performance ofthe Gaussian mixture PHD filter by benchmarking it against theJPDA filter ,  via Monte Carlo simulations. The JPDA filteris a classical filter for tracking a known and fixed number oftargets in clutter. In a scenario where the number of targets isconstant, the JPDA filter (given the correct number of targets)is expected to outperform the PHD filter, since the latter hasneither knowledge of the number of targets, nor even knowledgethat the number of targets is constant. For these reasons, theJPDA filter serves as a good benchmark.
The experiment settings are the same as those of Example1, but without spawning, as the JPDA filter requires a known
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Fig. 3. Position estimates of the Gaussian mixture PHD filter.
and fixed number of targets. The true tracks in this example arethose of targets 1 and 2 in Fig. 1. Target trajectories are fixedfor all simulation trials, while observation noise and clutter areindependently generated at each trial.
We study track loss performance by using the following cir-cular position error probability (CPEP) (see  for and ex-ample)
for some position error radius , where
and is the 2-norm. In addition, we mea-sure the expected absolute error on the number of targets for theGaussian mixture PHD filter
Note that standard performance measures such as the meansquare distance error are not applicable to multiple-target filtersthat jointly estimate the number of targets and their states (suchas the PHD filter).
Fig. 4 shows the tracking performance of the two filters forvarious clutter rates [see (46)] with the CPEP radius fixedat m. Observe that the CPEPs of the two filters arequite close for a wide range of clutter rates. This is rather sur-prising considering that the JPDA filter has exact knowledgeof the number of targets. Fig. 4(a) suggests that the occasionaloverestimation/underestimation of the number of targets is notsignificant in the Gaussian mixture PHD filter.
Fig. 5 shows the tracking performance for various values ofdetection probability with the clutter rate fixed at
m . Observe that the performance gap betweenthe two filters increases as decreases. This is because the
Fig. 4. Tracking performance versus clutter rate. The detection probability isfixed at p = 0:98. The CPEP radius is r = 20 m.
Fig. 5. Tracking performance versus detection probability. The clutter rate isfixed at = 12:5 10 m . The CPEP radius is r = 20 m.
PHD filter has to resolve higher detection uncertainty on top ofuncertainty in the number of targets. When detection uncertaintyincreases ( decreases), uncertainty about the number of tar-gets also increases. In contrast, the JPDA filters exact knowl-edge of the number of targets is not affected by the increase indetection uncertainty.
E. Generalizations to Exponential Mixture and
As remarked in Section III-A, closed-form solutions to thePHD recursion can still be obtained for a certain class of state-dependent probability of survival and probability of detection.
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Indeed, Propositions 1 and 2 can be easily generalized to handleand of the forms
where , , , , , , and ,
, , , , , are given modelparameters such that and lie between zero andone for all .
The closed-form predicted intensity can be ob-tained by applying Lemma 2 to convert into aGaussian mixture, which is then integrated with the transitiondensity using Lemma 1. The closed-form updatedintensity can be obtained by applying Lemma 2 once to
and twice to toconvert these products to Gaussian mixtures. For completeness,the Gaussian mixture expressions for and are givenin the following propositions, although their implementationswill not be pursued.
Proposition 3: Under the premises of Proposition 1, withgiven by (47) instead of (19), the predicted intensity
is given by (24) but with
Proposition 4: Under the premises of Proposition 2, withgiven by (48) instead of (20)
Conceptually, the Gaussian mixture PHD filter implementa-tion can be easily extended to accommodate exponential mix-ture probability of survival. However, for exponential mixtureprobability of detection, the updated intensity contains Gaus-sians with negative and positive weights, even though the up-dated intensity itself (and hence the sum of the weights) is non-negative. Although these Gaussians can be propagated usingPropositions 3 and 4, care must be taken in the implementationto ensure nonnegativity of the intensity function after mergingand pruning.
IV. EXTENSION TO NONLINEAR GAUSSIAN MODELS
This section considers extensions of the Gaussian mixturePHD filter to nonlinear target models. Specifically, the model-ling assumptions A.5 and A.6 are still required, but the state andobservation processes can be relaxed to the nonlinear model
where and are known nonlinear functions and andare zero-mean Gaussian process noise and measurement
noise with covariances and , respectively. Due tothe nonlinearity of and , the posterior intensity can nolonger be represented as a Gaussian mixture. Nonetheless,the proposed Gaussian mixture PHD filter can be adapted toaccommodate nonlinear Gaussian models.
In single-target filtering, analytic approximations of the non-linear Bayes filter include the extended Kalman (EK) filter , and the unscented Kalman (UK) filter , . The EKfilter approximates the posterior density by a Gaussian, which ispropagated in time by applying the Kalman recursions to locallinearizations of the (nonlinear) mappings and . The UKfilter also approximates the posterior density by a Gaussian, butinstead of using the linearized model, it computes the Gaussianapproximation of the posterior density at the next time stepusing the unscented transform. Details for the EK and UK filtersare given in .
Following the development in Section III-B, it can be shownthat the posterior intensity of the multiple-target state propa-gated by the PHD recursions (15) and (16) is a weighted sum ofvarious functions, many of which are non-Gaussian. In the same
VO AND MA: THE GAUSSIAN MIXTURE PROBABILITY HYPOTHESIS DENSITY FILTER 4101
TABLE IVPSEUDOCODE FOR THE EK-PHD FILTER
vein as the EK and UK filters, we can approximate each of thesenon-Gaussian constituent functions by a Gaussian. Adopting thephilosophy of the EK filter, an approximation of the posterior in-tensity at the next time step can then be obtained by applying theGaussian mixture PHD recursions to a locally linearized targetmodel. Alternatively, in a similar manner to the UK filter, theunscented transform can be used to compute the components ofthe Gaussian mixture approximation of the posterior intensity atthe next time step. In both cases, the weights of these compo-nents are also approximations.
Based on the above observations, we propose two nonlinearGaussian mixture PHD filter implementations, namely, theextended Kalman PHD (EK-PHD) filter and the unscentedKalman PHD (UK-PHD) filter. Given that details for the EKand UK filters have been well documented in the literature (see,e.g.,  and ), the developments for the EK-PHDand UK-PHD filters are conceptually straightforward, thoughnotationally cumbersome, and will be omitted. However, forcompleteness, the key steps in these two filters are summarizedas pseudocodes in Tables IV and V, respectively.
Remark 5: Similar to its single-target counterpart, theEK-PHD filter is only applicable to differentiable nonlinear
TABLE VPSEUDOCODE FOR THE UK-PHD FILTER
models. Moreover, calculating the Jacobian matrices may betedious and error-prone. The UK-PHD filter, on the other hand,does not suffer from these restrictions and can even be appliedto models with discontinuities.
4102 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006
Fig. 6. Target trajectories. : locations at which targets are born; : loca-tions at which targets die.
Remark 6: Unlike the particle-PHD filter, where the particleapproximation converges (in a certain sense) to the posteriorintensity as the number of particle tends to infinity , ,this type of guarantee has not been established for the EK-PHDor UK-PHD filter. Nonetheless, for mildly nonlinear problems,the EK-PHD and UK-PHD filters provide good approximationsand are computationally cheaper than the particle-PHD filter,which requires a large number of particles and the additionalcost of clustering to extract multiple-target state estimates.
A. Simulation Results for a Nonlinear Gaussian Example
In this example, each target has a survival probabilityand follows a nonlinear nearly constant turn model  in
which the target state takes the form , where, and is the turn rate. The state
dynamics are given by
s, , m/s , and, rad/s. We assume no spawning, and
that the spontaneous birth RFS is Poisson with intensity
where ,and .
Each target has a probability of detection . Anobservation consists of bearing and range measurements
Fig. 7. Position estimates of the EK-PHD filter.
Fig. 8. Position estimates of the UK-PHD filter.
where with ,rad/s and m. The clutter
RFS follows the uniform Poisson model in (46) over thesurveillance region rad [0, 2000] m, with
rad m (i.e., an average of 20 clutterreturns on the surveillance region).
The true target trajectories are plotted in Fig. 6. Targets 1and 2 appear from two different locations, 5 s apart. They bothtravel in straight lines before making turns at s. Thetracks almost cross at s, and the targets resume theirstraight trajectories after s. The pruning parameters forthe UK-PHD and EK-PHD filters are , , and
. The results, shown in Figs. 7 and 8, indicate thatboth the UK-PHD and EK-PHD filters exhibit good trackingperformance.
In many nonlinear Bayes filtering applications, the UK filterhas shown better performance than the EK filter . The sameis expected in nonlinear PHD filtering. However, this example
VO AND MA: THE GAUSSIAN MIXTURE PROBABILITY HYPOTHESIS DENSITY FILTER 4103
only has a mild nonlinearity, and the performance gap betweenthe EK-PHD and UK-PHD filters may not be noticeable.
Closed-form solutions to the PHD recursion are importantanalytical and computational tools in multiple-target filtering.Under linear Gaussian assumptions, we have shown that whenthe initial prior intensity of the random finite set of targets isa Gaussian mixture, the posterior intensity at any time step isalso a Gaussian mixture. More importantly, we have derivedclosed-form recursions for the weights, means, and covariancesof the constituent Gaussian components of the posterior inten-sity. An implementation of the PHD filter has been proposedby combining the closed-form recursions with a simple pruningprocedure to manage the growing number of components. Twoextensions to nonlinear models using approximation strategiesfrom the extended Kalman filter and the unscented Kalman filterhave also been proposed. Simulations have demonstrated the ca-pabilities of these filters to track an unknown and time-varyingnumber of targets under detection uncertainty and false alarms.
There are a number of possible future research directions.Closed-formed solutions to the PHD recursion for jump Markovlinear models are being investigated. In highly nonlinear, non-Gaussian models, where particle implementations are required,the EK-PHD and UK-PHD filters are obvious candidates for ef-ficient proposal functions that can improve performance. Thisalso opens up the question of optimal importance functions andtheir approximations. The efficiency and simplicity in imple-mentation of the Gaussian mixture PHD recursion also suggestpossible application to tracking in sensor networks.
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Ba-Ngu Vo was born in Saigon, Vietnam, in 1970.He received the B.Sc./B.E. degree (with first-classhonors) from the University of Western Australia in1994 and the Ph.D. degree from Curtin University ofTechnology in 1997.
He is currently a Senior Lecturer in the Departmentof Electrical and Electronic Engineering, Universityof Melbourne. His research interests are stochasticgeometry, random sets, multiple-target tracking, op-timization, and signal processing.
Wing-Kin Ma (M01) received the B.Eng. de-gree in electrical and electronic engineering (withfirst-class honors) from the University of Portsmouth,Portsmouth, U.K., in 1995 and the M.Phil. and Ph.D.degrees in electronic engineering from the ChineseUniversity of Hong Kong (CUHK), Hong Kong, in1997 and 2001, respectively.
Since August 2005, he has been an Assistant Pro-fessor in the Department of Electrical Engineeringand the Institute of Communications Engineering,National Tsing Hua University, Taiwan, R.O.C.
Previously he held research positions at McMaster University, Canada; CUHK;and the University of Melbourne, Australia. His research interests are in signalprocessing for communications and statistical signal processing.
Dr. Mas Ph.D. dissertation was commended to be of very high quality andwell deserved honorary mentioning by the Faculty of Engineering, CUHK, in2001.