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PROCEEDINGS OF THE IEEE, VOL. XX, NO. Y, MONTH 2018 VERSION: V61– JANUARY 1, 2018 1

Message Passing Algorithms forScalable Multitarget Tracking

Florian Meyer, Member, IEEE, Thomas Kropfreiter, Jason L. Williams, Senior Member, IEEE,

Roslyn A. Lau, Student Member, IEEE, Franz Hlawatsch, Fellow, IEEE, Paolo Braca, Senior Member, IEEE,

and Moe Z. Win, Fellow, IEEE

Superior scalability and performance of message passing algorithms enablenew applications involving a large number of inexpensive sensors.

Abstract—Situation-aware technologies enabled by multitargettracking will lead to new services and applications in fields suchas autonomous driving, indoor localization, robotic networks,and crowd counting. In this tutorial paper, we advocate arecently proposed paradigm for scalable multitarget trackingthat is based on message passing or, more concretely, theloopy sum-product algorithm. This approach has advantagesregarding estimation accuracy, computational complexity, andimplementation flexibility. Most importantly, it provides a highlyeffective, efficient, and scalable solution to the probabilistic dataassociation problem, a major challenge in multitarget tracking.This fact makes it attractive for emerging applications requiringreal-time operation on resource-limited devices. In addition, themessage passing approach is intuitively appealing and suitedto nonlinear and non-Gaussian models. We present messagepassing based multitarget tracking methods for single-sensor andmultiple-sensor scenarios, and for a known and unknown numberof targets. The presented methods can cope with clutter, misseddetections, and an unknown association between targets andmeasurements. We also discuss the integration of message passingbased probabilistic data association into existing multitargettracking methods. The superior performance, low complexity,and attractive scaling properties of the presented methods areverified numerically. In addition to simulated data, we usemeasured data captured by two radar stations with overlappingfields-of-view observing a large number of targets simultaneously.

Index Terms—Multitarget tracking, data association, messagepassing, sum-product algorithm, factor graph, data fusion.

Manuscript received XX, 2017; revised XX, 201Y. This work was supportedin part by the Austrian Science Fund (FWF) under grants J3886-N31 andP27370-N30, by the NATO Supreme Allied Command Transformation (ACT)under projects SAC000601 and SAC000608, by the Czech Science Foundation(GACR) under grant 17-19638S, and by the Office of Naval Research (ONR)under Grant N00014-16-1-2141.

F. Meyer and Moe Z. Win are with the Laboratory for Informa-tion and Decision Systems, Massachusetts Institute of Technology, Cam-bridge, MA, USA (e-mail: [email protected], [email protected]). T.Kropfreiter and F. Hlawatsch are with the Institute of Telecommunica-tions, TU Wien, Vienna, Austria and with Brno University of Technology,Brno, Czech Republic (e-mail: [email protected],[email protected]). J. L. Williams is with theNational Security, Intelligence, Surveillance and Reconnaissance Divi-sion, Defence Science and Technology Group, Edinburgh, Australiaand with the School of Electrical Engineering and Computer Sci-ence, Queensland University of Technology, Brisbane, Australia (e-mail: [email protected]). R. A. Lau iswith the Maritime Division, Defence Science and Technology Group,Edinburgh, Australia and with the Research School of ComputerScience, Australian National University, Canberra, Australia (e-mail:[email protected]). P. Braca is with the NATO Cen-tre for Maritime Research and Experimentation (CMRE), La Spezia, Italy(e-mail: [email protected]).

I. INTRODUCTION

A new era of real-time pervasive situational awareness willbe enabled by emerging inexpensive sensors and scalable datafusion algorithms. Situation-aware technologies are key toinnovative products and services that are profoundly changingvarious aspects of our daily life. For example, safe autonomousnavigation is based on a clear understanding of static andmobile objects in the environment. Multitarget tracking (MTT)can infer the states of these objects (or “targets”) frommeasurements provided by one or more sensors even if theirnumber is unknown [1]–[4]. The state of a target usuallyincludes its position and possibly other quantities such as itsvelocity. A complicating factor in many MTT applications isthe measurement origin uncertainty or data association (DA)problem, i.e., the unknown association of measurements withtargets. Since the first MTT methods were introduced foraerospace surveillance some 40 years ago [5], [6], the fieldhas experienced an intense development, and MTT methodshave proven useful in further surveillance scenarios [7]–[9]as well as in biomedical analytics [10]–[12]. Emerging MTTapplications such as autonomous driving [13]–[15], indoorlocalization [16]–[18], robotic networks [19]–[21], and crowdcounting [22]–[24] are especially challenging since they re-quire real-time operation on resource-limited devices.

A. The Message Passing Approach to Multitarget Tracking

The existing MTT methods can be grouped in two broadcategories that may be referred to as “vector-type” and “set-type” methods. The former describe the multitarget statesand measurements as random vectors [1], whereas the latterdescribe the multitarget states and measurements by randomfinite sets (RFSs) [3]. Many of these MTT methods have ahigh complexity and do not scale well in relevant systemparameters. Thus, they are often impractical for applicationsinvolving a large number of inexpensive sensors and/or targets.

Here, we demonstrate that MTT methods with low complex-ity and good scalability are obtained by using the methodologyof message passing, also known as belief propagation, or,more concretely, the sum-product algorithm (SPA). The SPAprovides a principled approximation of optimum Bayesianinference that achieves an attractive performance-complexitycompromise [25]–[33]. It is intuitively appealing, suited tononlinear and non-Gaussian system models, and able to copewith unknown and time-varying hyperparameters.

2 PROCEEDINGS OF THE IEEE, VOL. XX, NO. Y, MONTH 2018 VERSION: V61– JANUARY 1, 2018

In the last 25 years, message passing has been successfullyused in many applications including iterative decoding ofchannel codes [34]–[37], communication receivers [38]–[40],and cooperative localization [41]–[43]. Early work on messagepassing in the context of MTT [44]–[47] focused on thecalculation of “hard” measurement-to-target associations usingthe max-product algorithm. Only recently it was discoveredthat the “loopy” SPA is particularly attractive for MTT [48]–[55]. More specifically, the loopy SPA approach to MTT hasthe following advantages:

• it enables an efficient calculation of association probabil-ities for “soft” measurement-target associations;

• it provides a principled, general technique for derivingMTT methods within the Bayesian inference framework;

• it generalizes previously proposed MTT methods such asthe joint probabilistic data-association (JPDA) filter; and

• it enables the development of scalable MTT methods thatare suitable for scenarios involving a large number oftargets and/or sensors and/or measurements.

B. Contributions, Paper Organization, Notation

This tutorial paper presents SPA-based methods for MTTand discusses their properties and advantages in relation toother MTT methods. We will present three distinct typesof SPA-based MTT methods: (i) vector-type methods for aknown, fixed number of targets; (ii) vector-type methods foran unknown, time-varying number of targets; and (iii) set-typemethods for an unknown, time-varying number of targets.

The paper’s main contributions and organization are asfollows. In the remainder of this section, we introduce somebasic notation and formulate certain assumptions that arecommon to all presented methods. In Section II, we surveyexisting MTT methods. An introduction to factor graphs andthe SPA is given in Section III. Section IV describes a vector-type system model and a corresponding statistical formulationfor the case where the number of targets is fixed and known.The corresponding posterior distributions and their graphicalrepresentation using factor graphs are derived in Section V.Section VI discusses the solution of the probabilistic DAproblem using the loopy SPA. In Section VII, SPA-basedmethods for vector-type MTT are presented for the case ofa known, fixed number of targets. Section VIII describes avector-type system model, statistical formulation, and factorgraph for the case where the number of targets is time-varying and unknown. For that case, new SPA-based methodsfor vector-type MTT are developed in Section IX. Section Xcommences our treatment of set-type MTT methods by givingan introduction to RFSs. In Section XI, we describe a set-type system model and a corresponding statistical formulation.Section XII develops multitarget state propagation relationsfor the set-type system model. Building on these relations,Section XIII derives SPA-based methods for set-type MTT.In Section XIV, we evaluate the performance and complexityof the presented SPA-based MTT methods in comparison tostate-of-the-art methods. In addition to simulated data, we usemeasurement data acquired by two radar stations observing

a large number of targets simultaneously. Our results demon-strate substantial advantages—regarding both performance andcomplexity—of SPA-based methods over other methods.

We will use the following basic notation. Random variablesare displayed in sans serif, upright fonts; their realizations inserif, italic fonts. Vectors and matrices are denoted by boldlowercase and uppercase letters, respectively. For example,x is a random variable and x is its realization, and x is arandom vector and x is its realization. Random sets and theirrealizations are denoted by upright sans serif and calligraphicfont, respectively. For example, X is a random set and Xis its realization. The expectation of a random variable orrandom vector is denoted by E{·}. The probability of an eventis denoted by P{·}. We denote probability density functions(pdfs) as f(·) and probability mass functions (pmfs) as p(·).We write k for the discrete time index, i for the index ofa target, m for the index of a measurement, and s for theindex of a sensor. Integrals are over the entire space of theintegration variable. The symbol ∝ indicates equality up to aconstant factor. A consecutive list of i integers, i.e., 1, 2, . . . , i,is briefly denoted as 1, . . . , i.

C. Basic Assumptions

We consider scenarios where in order to reduce the dataflow of the sensors and the overall complexity, measurementsare produced by a detector performing a thresholding and,possibly, some further preparatory processing of the raw sensordata [1]. The measurement corresponding to a target can bemissed by the detector if its signal-to-noise ratio is below thedetection threshold. Another premise is that both the targetsand the measurements are “points” and, at any particular time,at most one detection can be due to any particular target [1].There is a measurement origin uncertainty, i.e., it is unknownwhich measurement originated from which target, or if ameasurement is a false alarm (due to clutter). This leads toa nontrivial DA problem.

In the following, we state some basic assumptions underly-ing both the vector-type and set-type MTT methods to be pre-sented. At (discrete) time k, there are ik targets i∈ {1, . . . , ik}.

The state x(i)k of the ith target at time k consists of the

position of the target and possibly further parameters, and it is

a random vector of dimension dx, i.e., x(i)k ∈ Rdx. There are

ns sensors s ∈ {1, . . . , ns}. At time k, sensor s produces mk,s

measurements z(m)k,s , m ∈ {1, . . . ,mk,s}. Each measurement is

a random vector of dimension dz , i.e., z(m)k,s ∈ Rdz. We make

the following further assumptions [1]:

A1: The multitarget state (defined as an ordered or unordered

list of all the single-target states x(i)k ) evolves according

to a first-order Markov process.

A2: The single-target states x(i)k evolve independently. The

evolution of state x(i)k is described by the single-target

state-transition pdf f(

x(i)k

∣

∣x(i)k−1

)

.

A3: The measurements z(m)k,s produced by sensor s at time k

have no order, i.e., they are randomly shuffled.

A4: Each measurement z(m)k,s originates from a target or from

clutter (false alarm), and it cannot originate from morethan one target simultaneously. Conversely, at time k,

Meyer et al.: Scalable Multitarget Tracking 3

one target can generate at most one measurement z(m)k,s

at sensor s.A5: The a priori probability that a measurement originates

from target i (i.e., that sensor s “detects target i” at timek) is independent across the targets i and a known func-

tion of the target state x(i)k , denoted as p(s)d

(

x(i)k

)

. Conse-quently, the probability that no measurement originatesfrom target i (i.e., that sensor s “misses target i” at time

k) is 1− p(s)d

(

x(i)k

)

. We assume that 0 ≤ p(s)d

(

x(i)k

)

< 1.A6: The number of clutter measurements at sensor s and time

k is Poisson distributed with mean µ(s)c . It is furthermore

independent across the sensors s and independent of thenumber of targets that are detected at sensor s.

A7: At sensor s and time k, the clutter measurements areindependent of the target-originated measurements.

A8: The clutter measurements at sensor s and time k areindependent and identically distributed (iid) with pdf

f (s)c

(

z(m)k,s

)

= 0.

A9: At sensor s and time k, given all the target states x(i)k , the

target-originated measurements are conditionally inde-pendent of each other and also conditionally independentof all the clutter measurements. Furthermore, a target-

originated measurement z(m)k,s , given the respective target

state x(i)k , is distributed according to f

(

z(m)k,s

∣

∣x(i)k

)

.A10: At time k, the measurements of different sensors s

are conditionally independent given all the target states

x(i)k . (Here, the measurement of sensor s at time k is

defined as a randomly ordered list or a set of all single

measurements z(m)k,s .)

A11: At time k, given all the target states x(i)k , the measure-

ments z(m)k,s are conditionally independent of all the past

and future measurements z(m)k′,s and target states x

(i)k′ ,

k′ = k.

II. SURVEY OF STATE-OF-THE-ART METHODS

The basic setting of most MTT methods is sequentialBayesian estimation (or “filtering”) [56], which consists of aprediction step based on the Chapman-Kolmogorov equation,

f(

xk|z1:k−1

)

=

∫

f(

xk|xk−1

)

f(

xk−1|z1:k−1

)

dxk−1 , (1)

and an update step based on Bayes’ rule,

f(

xk|z1:k)

∝ f(

zk|xk

)

f(

xk|z1:k−1

)

. (2)

Here, xk is the state of a single target and zk is a corre-sponding measurement, both at time k, and z1:k is short for[

zT1 · · · z

Tk

]T. Based on the posterior pdf f

(

xk|z1:k)

, Bayesianstate estimation can be performed, e.g., by means of theminimum mean-square error (MMSE) estimator [57]

xMMSEk !

∫

xk f(

xk|z1:k)

dxk. (3)

The Kalman filter [58] exploits the fact that for linear-Gaussiansystem models, the operations (1) and (2) can be performedin closed form. For nonlinear/non-Gaussian system models,computationally feasible approximate methods for sequentialBayesian estimation include the extended Kalman filter [58],

the unscented Kalman filter [59], the Gaussian sum filter [60],and the particle filter [56], [61]. However, all these methodsassume that the number of targets and the association betweentargets and measurements are perfectly known. Next, we dis-cuss Bayesian methods for MTT that do not require knowledgeof the association between targets and measurements.

A. Vector-Type MTT Methods

Vector-type MTT methods describe the multitarget statesand measurements by random vectors. They are able to explic-itly maintain track continuity, i.e., they associate a state esti-mate with a previous state estimate or declare the appearanceof a new target. Many vector-type methods use heuristics totake into account the appearance and disappearance of targets[1]. Others model the discovery of new targets but not targetdisappearance [6], [62], or use target existence states to capturetarget disappearance while lacking a fully Bayesian approachto initiating tracks based on measurements [63]–[65].

1) Methods Based on Probabilistic DA: Probabilistic DA(PDA) methods for single-target tracking treat the DA pa-rameter as a nuisance variable that is “marginalized out” [1,Sec. 3.4]. The PDA filter subsequently fits a Gaussian pdf tothe resulting posterior target state pdf, whereas other methods(e.g., [66]) retain a more detailed representation. Joint PDA(JPDA) extends PDA to the case of multiple targets [1, Sec.6.2]. The joint association parameters are marginalized outunder the constraint that each measurement relates to at mostone target. Like PDA, JPDA then fits a Gaussian pdf to theposterior state pdf for each target, whereas methods such as[67] retain a Gaussian mixture distribution.

JPDA assumes that the number of targets is known. Thislimitation is removed in joint integrated PDA (JIPDA) [2],[64], which extends JPDA to incorporate a probability of targetexistence. Target existence is modeled as a Markov chain,and new target tracks are initialized based on measurementsthat are not in the neighborhood (the “gate”) of an existingtrack. A similar model along with a particle filter implemen-tation was presented in [65]. A second limitation of JPDA,the high computational complexity of summing over all thejoint associations, has been addressed by various heuristicapproximations [68]–[70], by efficient hypothesis management[71], [72], or by using likely measurement-target associationsto find approximations of the marginal posterior state pdfs[73], [74]. A third limitation, known as coalescence, ariseswhen targets become closely spaced and the filter is no longeraware which target is in which position. As a result, themarginal pdfs become strongly multimodal. Possible solutionsinclude a pruning of the association hypotheses [75], [76] andapproximate calculation of marginal pdfs via optimization ofinformation-theoretic measures [77], [78] or the use of mean-field formulations [79], [80].

Multisensor extensions of MTT filters based on JPDA werepresented in [81], [82]. The measurements of different sensorsare either processed in a sequential manner—known as the“iterated-corrector” approach—or in parallel. The performanceof sequential processing may strongly depend on the order inwhich the measurements are processed.


2) MHT Methods: Multiple hypothesis tracking (MHT)methods seek to find the most likely measurement-target asso-ciation over a sliding window of consecutive time steps [6]. Adecision is made only on the association at the oldest time step.The original MHT formulation [6]—known as hypothesis-oriented MHT—forms an expanding tree of association hy-potheses, where each leaf represents a partitioning of all themeasurements acquired until the current time into subsetsbelieved to correspond to the same target. The computationalcomplexity is problematic but can be reduced by consideringonly likely hypotheses, which are identified by an efficient m-best assignment algorithm [83]–[86]. The more efficient track-oriented MHT methods [87]–[89] represent global associationhypotheses—i.e., for all targets and measurements—implicitlyvia a series of tree structures. Each tree structure representsthe possible association histories for a single target. The mostlikely hypothesis is found by choosing a leaf node for eachsingle-target tree. An enumeration of hypotheses is avoidedthrough combinatorial optimization techniques [90]–[93].

The original MHT formulation in [6] assumes a Poissondistribution of the number of newly detected targets. Calcula-tions are simplified by disregarding targets that have not beendetected so far. This idea is refined in [62], which assumesa fixed but unknown number of targets, with a Poisson prior,and in [94], which accommodates an unknown, time-varyingnumber of targets. We note that MHT methods can also bederived in the set framework [95].

Multisensor MHT methods include iterated-corrector ver-sions performing a sequential processing of the measurements[96], and versions that perform a parallel processing of themeasurements by solving a multidimensional assignment prob-lem [91]. In the multistage MHT approach [96], a first MHTstage is used for each sensor individually to reject cluttermeasurements, and a second MHT stage processes all themeasurements using the iterated-corrector principle.

A key challenge in MHT methods is to retain sufficientlydiverse association hypotheses until clarifying information isreceived. Graph-based methods have been proposed as analternative solution to this problem in circumstances whereassociation likelihoods are well approximated as Markovian[97], [98]. In [99], [100], these methods have been adapted toproblems in which sporadic identity information is available.

B. Set-Type MTT Methods

Set-type MTT methods describe the multitarget states andmeasurements by RFSs [3]. (An introduction to RFSs will begiven in Section X.) The RFS approach facilitates the mod-eling of target appearance and disappearance in a Bayesiansetting, and introduces tools for handling complex, hybridcontinuous/discrete distributions. Some set-type methods areunable to maintain track continuity [3], [78], [101]–[106],others maintain it implicitly [51], [107], and others maintainit explicitly by augmenting the states by distinct labels [108],[109].

1) PHD Filter and CPHD Filter: The probability hy-pothesis density (PHD) filter [106], [110] approximates thepredicted posterior pdf of the target states, f(Xk|Z1:k−1), bya Poisson RFS pdf. The intensity function (aka PHD) of that

Poisson RFS is chosen such that the RFS Kullback-Leiblerdivergence relative to f(Xk|Z1:k−1) is minimized [78]. Thecardinalized PHD (CPHD) filter [111] uses a similar principlebut approximates f(Xk|Z1:k−1) by the pdf of an iid clusterRFS [78]. Multisensor extensions of the PHD and CPHD filters[110] use the iterated-corrector approach, i.e., they sequentiallyperform a separate (C)PHD approximation for each sensor[112], or they perform a single (C)PHD approximation for allsensors jointly [104]. The performance of the latter strategy issuperior and moreover invariant to the ordering of the sensors.(C)PHD filters cannot maintain track continuity.

2) Multi-Bernoulli Filters: Multi-Bernoulli (MB) filters ap-proximate the posterior pdf f(Xk|Z1:k) by an MB RFS pdf.The original MB filter (abbreviated as MeMBer filter) [3] usesapproximations of an RFS representation known as the prob-ability generating functional. The MeMBer filter was foundto have a cardinality bias, which was compensated in [103].The track-oriented marginal MB/Poisson (TOMB/P) filter andthe measurement-oriented marginal MB/Poisson (MOMB/P)filter [51], [113] are two MB filter variants that are based onthe observation that the exact pdf f(Xk|Z1:k) involves an MBmixture. Each MB pdf in this MB mixture corresponds toone of the global association hypotheses in MHT methods.Similar to [62], the TOMB/P and MOMB/P filters modelundetected targets by a Poisson RFS, so that the overallmultitarget state RFS is the union of independent Poissonand MB mixture RFSs. However, in contrast to [62] andother MHT variants, the TOMB/P and MOMB/P filters alsomodel target appearance and disappearance. In the case ofthe TOMB/P filter, a computationally feasible MB filteringalgorithm is obtained by approximating the DA pmf by theproduct of its marginals and by calculating the marginalsusing a scalable SPA-based algorithm. Although the TOMB/Pfilter is based on Bernoulli components without an explicitorder or label, continuity of targets over successive time stepscan be obtained implicitly, similar to the JIPDA filter. Asequential Monte Carlo implementation of the TOMB/P filterwas presented in [107] and applied to static source localizationin [114]. The MeMBer and MOMB/P filters cannot maintaintrack continuity. A multisensor extension of the MB filter waspresented in [105].

3) Labeled RFS-Based Methods: Using labels allows RFS-based MTT filters to explicitly maintain track continuity, i.e.,to obtain entire trajectories of consecutive target states. In theMTT context, the elements of a labeled RFS are the targetstate vectors augmented by a distinct label that identifies therespective target. The δ-GLMB filter [108] models the multi-target state by a generalized labeled MB (GLMB) RFS, whereeach GLMB component corresponds to one possible target-measurement association history. In contrast to the TOMB/Pand MOMB/P filters, the number of hypothesized targets perGLMB component is deterministic, similarly to traditionalMHT methods. To avoid an exponential increase of the numberof GLMB components with time, components with a lowweight are pruned and the generation of new components islimited by an m-best assignment algorithm. The LMB filter[109] is a reduced-complexity approximation of the δ-GLMBfilter. The GLMB RFS is approximated by a labeled MB


(LMB) RFS, which is chosen such that its PHD matches thePHD of the GLMB RFS. This approximation is analogousto that underlying the TOMB/P filter. However, in [109], theDA problem is solved by means of the m-best assignmentalgorithm rather than a scalable SPA-based algorithm.

C. MTT Methods Using Message Passing

In the context of MTT, message passing techniques werefirst studied for DA in sensor networks where each sensor hasa narrow field of view [44]–[47]. Association variables wereused to hypothesize joint association events for all targets andmeasurements within certain nonoverlapping regions and toperform “hard” DA by means of the max-product algorithm.For such multisensor problems, calculating the marginal DAprobabilities using exhaustive hypothesis enumeration is typ-ically infeasible. Related techniques were studied in [115],[116], and a tree-based approximation of DA messages wasdeveloped in [117]. The efficient hypothesis managementmethod proposed in [71], [72] exploits the redundancy presentin many cases to achieve an exact evaluation with reducedcomplexity. This method is effectively a highly tailored versionof the junction tree algorithm [25], [32]. Still, the complexityremains exponential in some extreme scenarios.

Message passing methods provide approximate marginalprobabilities (using the SPA) and maximum a posteriori(MAP) estimates (using the max-product algorithm) withreduced complexity [25], [26]. The max-product algorithmis guaranteed to converge to the optimum MAP solutionin single-scan DA problems (i.e., DA problems where themeasurements of a single sensor and a single time step areconsidered) [118]. Max-product algorithms for multiple-scanDA were considered for MTT [49], [119], [120] and trackassociation [121]. Dual decomposition methods [122], [123]provide a convergent alternative to the max-product algorithm,and were applied to multiple-scan DA in [119].

For estimation of marginal DA probabilities, an SPA-basedmethod involving so-called mutual exclusion constraints [124],[32, Box 12.D] was shown in [49] to be outperformed bythe bipartite SPA-based algorithm proposed in [125], [126]and subsequently studied in [48], [127]. This algorithm wasdeveloped independently and evaluated in the MTT contextin [49], [128]. It was used in set-type MTT methods [51],[78], [107], [129], in vector-type multisensor MTT methods[52]–[55], [79], [80], and in vector-type methods for indoorlocalization [130]–[132]. This bipartite SPA-based algorithmfor probabilistic DA will be discussed in Section VI. Theunderlying bipartite graphical model was also used for grouptracking [133] and batch filtering performing MTT [134] (bothbased on the expectation-maximization algorithm), and formultipath tracking [135], [136].

In [52]–[54], the SPA was used not merely for probabilisticDA but for the entire multisensor MTT problem. These “total-SPA” MTT methods, which will be discussed in Sections VIIand IX, integrate the bipartite SPA-based DA algorithm in theoverall SPA formulation. In [53], a total-SPA MTT methodwas proposed in which target appearance and disappearanceare modeled by “augmented target states” including a binarytarget existence indicator, and a particle implementation of

the method was developed. An adaptive extension in whichthe probabilities of detection for the individual sensors areestimated jointly with the augmented target states was pre-sented in [54]. A reformulation of [53] was used in SPA-basedmethods for indoor localization [130] and for simultaneouslocalization and mapping (SLAM) [131], [132]; these methodsexploit the information provided by multipath components inultra-wideband signals.

III. INTRODUCTION TO THE SUM-PRODUCT ALGORITHM

Many important algorithms—such as sequential Bayesianestimation, the Kalman filter, the particle filter, the forward-backward algorithm, the Viterbi algorithm, the turbo decodingalgorithm, and fast Fourier transform algorithms—can beviewed as instances of the SPA. In addition, the SPA has beenused to develop numerous new, high-performing algorithms ina wide range of applications [34]–[43], [137], [138].

In detection and estimation problems, the SPA can be usedfor an efficient calculation of marginal posterior pdfs [25],[27]. Consider the estimation of random parameter vectors xi,i = 1, . . . , nt from an observation z of a random measure-ment vector z. Most Bayesian estimation methods require theposterior pdfs f(xi|z) [57]. These pdfs are marginal pdfs of

the joint posterior pdf f(x|z), where x ![

xT1 · · · x

Tnt

]T. In

many cases, the SPA is able to calculate the marginal posteriorpdfs f(xi|z), or accurate approximations thereof, at a smallfraction of the computational cost of direct marginalizations. Inparticular, the computational cost of the SPA typically scalessignificantly better with relevant system parameters than thatof direct marginalizations. This advantage makes an SPA-based solution feasible for many large-scale problems whereother solutions are infeasible, including MTT problems witha large number of targets, sensors, and/or measurements.

A. Factor Graphs

The use of the SPA for marginalizing the joint posteriorpdf f(x|z) presupposes that f(x|z) is the product of lower-dimensional factors, i.e.,

f(x|z) ∝∏

l

ψl

(

x(l))

. (4)

Here, each argument x(l) comprises certain parameter vectorsxi, where each xi can appear in several x(l). Note that thefactors ψl

(

x(l))

generally depend also on z. The factoriza-tion (4) can be represented by a factor graph [25], [28].(Alternative graphical models include Markov random fields[139] and Bayesian networks [140].) In a factor graph, eachparameter variable xi is represented by a variable node, andeach factor ψl(·) by a factor node. Variable node “xi” andfactor node “ψl” are adjacent, i.e., connected by an edge, ifxi is an argument of ψl(·). This is visualized in Fig. 1 forthe factorization f(x|z) ∝ ψ1(x1)ψ2(x1,x2)ψ3(x2), where

x=[

xT1 xT

2

]T.

A useful modification of factor graphs is provided bythe principle of “stretching” or “opening” factors [25], [40].Here, one introduces additional variables that may dependdeterministically on certain variables in the original factor


x1 x2

ψ1 ψ3

ψ2

Fig. 1. Factor graph representing the factorization f(x|z) ∝ψ1(x1)ψ2(x1,x2)ψ3(x2). Variable nodes and factor nodes are representedby circles and squares, respectively.

graph. This causes certain factor nodes to be “stretched” intoa larger number of factor nodes. The variable nodes adjacentto these new factor nodes tend to have lower dimensionsthan those adjacent to the original factor nodes. Accordingly,some of the messages in the SPA are replaced with lower-dimensional messages, which results in reduced computationalcomplexity and improved scalability.

B. The SPA

The SPA [25], [29] is a message passing algorithm thatcalculates certain messages for each node and passes each ofthese messages to one of the adjacent nodes. Messages leavingor entering a variable node are functions of the associatedvariable. Let Vl be the set of the indices i of all variablenodes “xi” adjacent to factor node “ψl”, or equivalently, ofall variables xi that appear in the argument of ψl

(

x(l))

in (4).Furthermore, let Fi be the set of the indices l of all factornodes “ψl” adjacent to variable node “xi”, or equivalently, ofall factors ψl

(

x(l))

that involve xi in their argument. Then, themessage passed from factor node “ψl” to an adjacent variablenode “xi”, i ∈ Vl is given by

ζψl→xi(xi) =

∫

ψl

(

x(l))

∏

i′∈Vl\{i}

ηxi′→ψl(xi′) dx∼i . (5)

Here,∫

. . . dx∼i denotes integration with respect to all vectorsxi′ , i′ ∈ Vl except xi. For factorizations involving discretevariables, the integration is replaced with a summation. Fur-thermore, the message ηxi→ψl

(xi) passed from variable node“xi” to an adjacent factor node “ψl”, l ∈ Fi is given by

ηxi→ψl(xi) =

∏

l′∈Fi\{l}

ζψl′→xi(xi) . (6)

In the case of a tree-structured factor graph, such as in Fig.1, the message updates (5) and (6) are performed once foreach variable node and factor node. This process starts at thevariable and factor nodes with only one edge (which pass aconstant message and the corresponding factor, respectively),and proceeds with any node whose incoming messages arealready available. After all messages have been calculated, abelief f(xi) is calculated for each variable node “xi” as

f(xi) = Ci

∏

l∈Fi

ζψl→xi(xi) , (7)

with Ci such that∫

f(xi)dxi = 1. For a tree-structuredfactor graph, it can be shown [25], [32] that each belieff(xi) is exactly equal to the respective marginal posterior pdff(xi|z). In our example (see Fig. 1), the message passed from“ψ2” to “x2” is ζψ2→x2(x2) =

∫

ψ2(x1,x2)ηx1→ψ2(x1)dx1

(here, x(2) =[

xT1 xT

2

]T), the message passed from “x1” to

“ψ2” is ηx1→ψ2(x1) = ζψ1→x1(x1) = ψ1(x1), and the

belief for “x2” is f(x2) = C2 ζψ2→x2(x2)ζψ3→x2(x2) =C2 ζψ2→x2(x2)ψ3(x2), where C2 = 1/

∫

ζψ2→x2(x2)ψ3(x2)×dx2. With ψ1(x1)=f(x1|z1), ψ2(x1,x2)=f(x2|x1), andψ3(x2)=f(z2|x2), it can be verified that f(x2)=f(x2|z1:2).Thus, running the SPA on the factor graph in Fig. 1 is equiv-alent to performing prediction step (1) and update step (2).

In the case of a factor graph with loops, the SPA is appliedin an iterative manner, i.e., the entire message update processis repeated several times. The resulting beliefs f(xi) are thengenerally only approximations of the marginal posterior pdfsf(xi|z). Theoretical analysis [26], [30], [31], [33] showed thatthis “loopy” SPA can be interpreted as a variational approachto approximate inference that corresponds to a constrainedoptimization problem, and that the iterative message updatingprocess converges to a stationary point of that optimizationproblem. Although the optimization problem is typically non-convex, the approximation of the f(xi|z) provided by theloopy SPA has been observed to be very accurate in manyapplications [34]–[36], [38]–[43], [137], [138]. Intuitively, theloopy SPA converges and provides a good approximation ofthe marginal posterior pdfs if the optimization problem islocally convex in a region containing the starting point and theoptimal solution. Alternative optimization problems that areconvex can be constructed [141], [142]. The resulting iterativemessage passing algorithms converge to a global optimumand may lead to more accurate beliefs than the loopy SPA.However, they are typically significantly more complex.

In an iterative loopy SPA, there is no canonical orderof message calculation, and different orders may lead todifferent beliefs. Specifying the order (schedule) and usingthe “stretching factors” principle discussed in Section III-Agives a certain freedom in the design of message passingalgorithms. We will take advantage of this design freedom inour development of SPA-based MTT methods in later sections.

For linear-Gaussian system models, the message passingequations (5)–(7) can be evaluated in closed form (see [27]for details). The resulting equations generalize the Kalmanfilter recursion [58] to more complex factorization structures.For nonlinear/non-Gaussian models, computationally feasibleapproximate implementations of (5)–(7) include nonparamet-ric belief propagation [41] and sigma point belief propagation[143], which generalize the particle filter [56], [61] and theunscented Kalman filter [59], respectively.

IV. VECTOR-TYPE SYSTEM MODEL FOR A

KNOWN, FIXED NUMBER OF TARGETS

In this section, following [1], we present a system modeland a basic statistical formulation for the vector-type DA andMTT methods considered in Sections VI and VII, respectively.We assume that the number of targets is fixed and known. Avector-type MTT system model for an unknown, time-varyingnumber of targets will be presented in Section VIII.

A. State-Transition pdf and Prior Distribution

The vector-type system model is based on the assumptionsA1–A11 listed in Section I-C and on the following additionalassumptions [1]:


Vk1: The number of targets, ik = nt, is fixed and known.Vk2: The target states are ordered (arbitrarily) according

to their arrangement in a joint state vector xk ![

x(1)Tk · · · x(nt)T

k

]T.

Vk3: At time k = 1, the target states x(i)1 are independent and

distributed according to the prior pdfs f(

x(i)1

)

.

Using Vk1 and Vk2 along with A1 and A2, the joint state-transition pdf is obtained as

f(xk|xk−1) =nt∏

i=1

f(

x(i)k

∣

∣x(i)k−1

)

k = 2, 3, . . . . (8)

Similarly, using Vk3, the joint prior pdf at time k = 1 is

f(x1) =nt∏

i=1

f(

x(i)1

)

. (9)

B. Likelihood Function

The measurements produced by sensor s at time k are

described by the vector zk,s ![

z(1)Tk,s · · · z(mk,s)T

k,s

]T, where

the components (subvectors) z(m)k,s , m = 1, . . . ,mk,s have a

random order (cf. A3). The (unknown) association betweenmeasurements and targets can be described by the DA vector

ak,s =[

a(1)k,s · · · a

(nt)k,s

]Twith entries [1]

a(i)k,s !

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

m ∈ {1, . . . ,mk,s} , if at time k, target i generatesmeasurement m at sensor s

0 , if at time k, target i does notgenerate a measurement atsensor s.

(10)

If ak,s was known, we would be able to identify cluttermeasurements and to associate measurements with targets.However, ak,s is unknown and thus considered as a latentrandom variable in the inference problem. It will be convenientto express the constraint A4 by the indicator function

ψ(ak,s) !

⎧

⎪

⎨

⎪

⎩

0, ∃i, i′∈ {1, . . . , nt}

such that i= i′ and a(i)k,s= a(i′)

k,s = 0

1 , otherwise.

Using A3–A6, the prior pmf of the DA vector ak,s and thenumber of measurements mk,s conditioned on the multitargetstate xk is obtained as [1]

p(ak,s,mk,s|xk)

= ψ(ak,s)e−µ(s)

c (µ(s)c )mk,s

mk,s!

(

nt∏

i=1

(

1−p(s)d

(

x(i)k

))

)

×∏

j∈Dak,s

p(s)d

(

x(j)k

)

µ(s)c

(

1−p(s)d

(

x(j)k

))

, (11)

where Dak,sis the set of “detected targets” corresponding to

ak,s, i.e., Dak,s!{

i ∈ {1, . . . , nt} : a(i)k,s = 0

}

. The detailedderivation of expression (11) is reviewed in [144]. Notethat the factor ψ(ak,s) guarantees that p(ak,s,mk,s|xk) = 0for vectors ak,s violating A4 (e.g., if one measurement isassociated with two targets). In addition, ψ(ak,s) introduces a

coupling of the different DA variables a(i)k,s, which implies thatperforming inference independently on single-target states is

suboptimum. Note that p(ak,s,mk,s|xk) in (11) is a valid pmfin the sense that

∑∞mk,s=0

∑

ak,sp(ak,s,mk,s|xk) = 1 for

arbitrary xk; here,∑

ak,sis short for

∑

ak,s∈{0,1,...,mk,s}nt .To gain further insights into the structure of

p(ak,s,mk,s|xk), we next investigate three special cases. Inthe no-detections, no-clutter case, i.e., mk,s= 0 and ak,s= 0,expression (11) reduces to

p(ak,s= 0,mk,s= 0|xk) = e−µ(s)c

nt∏

i=1

(

1−p(s)d

(

x(i)k

))

.

Here, e−µ(s)c is the Poisson pmf of the number of clutter

measurements evaluated at 0, and∏nt

i=1

(

1−p(s)d

(

x(i)k

))

is theprobability that no target is detected by sensor s. In the no-

detections, all-clutter case, i.e., mk,s> 0 and ak,s= 0,

p(ak,s= 0,mk,s|xk) =e−µ(s)

c (µ(s)c )mk,s

mk,s!

nt∏

i=1

(

1−p(s)d

(

x(i)k

))

,

where e−µ(s)c (µ(s)

c )mk,s/mk,s! is the Poisson pmf of thenumber of clutter measurements evaluated at mk,s. Finally,in the all-detections, no-clutter case, i.e., mk,s = nt andak,s= ad

k,s , where adk,s is any DA vector that assigns exactly

one measurement to each target,

p(ak,s=adk,s,mk,s=nt|xk) =

1

nt!e−µ(s)

c

nt∏

i=1

p(s)d

(

x(i)k

)

.

Here, e−µ(s)c is again the Poisson pmf of the clutter measure-

ments evaluated at 0,∏nt

i=1 p(s)d

(

x(i)k

)

is the probability thatall targets are detected by sensor s, and the factor 1/nt! arisesbecause there are nt! different measurement-target associationsand, thus, nt! different ad

k,s .Next, using A7–A9, the dependence of the measurement

vector zk,s on xk, ak,s, and mk,s is described by the pdf [1]

f(zk,s|xk,ak,s,mk,s) =

(mk,s∏

m=1

f (s)c

(

z(m)k,s

)

)

×∏

i∈Dak,s

f (s)(

z(a

(i)k,s

)

k,s

∣

∣

∣x(i)k

)

, (12)

with f (s)(

z(m)k,s

∣

∣x(i)k

)

! f(

z(m)k,s

∣

∣x(i)k

)

/f (s)c

(

z(m)k,s

)

. (Note thatexpression (12) presupposes that ak,s is consistent with A4.)Again, we investigate f(zk,s|xk,ak,s,mk,s) for the three spe-cial cases considered earlier. In the no-detections, no-clutter

case, i.e., mk,s=0 and ak,s= 0, zk,s is not defined; however,an expression of f(zk,s|xk,ak,s,mk,s) replacing (12) can beformally introduced as f(zk,s|xk,ak,s= 0,mk,s= 0) = 1. Inthe no-detections, all-clutter case, i.e., mk,s > 0 and ak,s= 0,expression (12) reduces to

f(zk,s|xk,ak,s= 0,mk,s) =

mk,s∏

m=1

f (s)c

(

z(m)k,s

)

.

Here, since all the measurements are clutter measurements,zk,s is independent of xk. Finally, in the all-detections, no-


clutter case, i.e., mk,s=nt and ak,s= adk,s ,

f(zk,s|xk,ak,s= adk,s,mk,s=nt) =

nt∏

i=1

f(

z(ad(i)

k,s)

k,s

∣

∣

∣x(i)k

)

.

Here, each target state x(i)k generates one measurement zk,s,

which is distributed according to f(

z(a

d(i)k,s

)

k,s

∣

∣x(i)k

)

.

The pdf of zk,s, ak,s, and mk,s conditioned on xk is nowobtained as

f(zk,s,ak,s,mk,s|xk) = f(zk,s|xk,ak,s,mk,s)

× p(

ak,s,mk,s

∣

∣xk

)

,

and further, inserting (11) and (12),

f(zk,s,ak,s,mk,s|xk) =e−µ(s)

c

mk,s!

(mk,s∏

m=1

µ(s)c f (s)

c

(

z(m)k,s

)

)

× ψ(ak,s)nt∏

i=1

g(

x(i)k , a(i)k,s; zk,s

)

,

(13)with

g(


)

!

⎧

⎨

⎩

1

µ(s)c

p(s)d

(

x(i)k

)

f (s)(

z(m)k,s

∣

∣x(i)k

)

, a(i)k,s=m∈ {1, . . . ,mk,s}

1−p(s)d

(

x(i)k

)

, a(i)k,s=0 .(14)

In (13) and hereafter, it is assumed that the value of mk,s isconsistent with zk,s, i.e., equal to the number of subvectors

z(m)k,s in zk,s.

Finally, we use A10 and the fact that since the measurementszk,s of different sensors s are conditionally independent givenxk, also the DA vectors ak,s are conditionally independent.We thus obtain for the joint pdf of the all-sensors vectors

zk ![

zTk,1 · · · z

Tk,ns

]T, ak !

[

aTk,1 · · · a

Tk,ns

]T, and mk !

[

mk,1 · · · mk,ns

]Tconditioned on xk:

f(zk,ak,mk|xk) =ns∏

s=1

f(zk,s,ak,s,mk,s|xk). (15)

The global (all-sensors) likelihood function f(zk,mk|xk)follows via marginalization, i.e.,

f(zk,mk|xk) =∑

ak

f(zk,ak,mk|xk), (16)

where the sum is over all ak ∈ {0, 1, . . . ,mk,1}nt × · · ·×{0, 1, . . . ,mk,ns}

nt (note that there are(∏ns

s=1mk,s

)ntdif-

ferent DA vectors ak).

For completeness and future reference, we also present thesingle-sensor likelihood function without ak,s:

f(zk,s,mk,s|xk) =∑

ak,s

f(zk,s,ak,s,mk,s|xk)

=e−µ(s)

c

mk,s!

(mk,s∏

m=1

µ(s)c f (s)

c

(

z(m)k,s

)

)

×∑

ak,s

ψ(ak,s)nt∏

i=1

g(


)

, (17)

where (13) was used. The sum is over all possible DAvectors ak,s ∈ {0, 1, . . . ,mk,s}nt, and thus the computationalcomplexity of evaluating f(zk,s,mk,s|xk) scales exponen-tially with the number of targets nt. The expression (17)

is invariant to a permutation of the subvectors z(m)k,s in zk,s

and of the subvectors x(i)k in xk. This independence of an

assumed order of the targets and the measurements motivatesthe set-type approach considered in Sections XI–XIII. Notethat f(zk,s,mk,s|xk) in (17) is a valid hybrid pdf/pmf in thesense that

∑∞mk,s=0

∫

f(zk,s,mk,s|xk)dzk,s = 1.

V. JOINT POSTERIOR PDFS AND FACTOR GRAPHS FOR A

KNOWN, FIXED NUMBER OF TARGETS

Bayesian estimation of the target states x(i)k typically relies

on the posterior pdfs f(

x(i)k

∣

∣z1:k)

, where z1:k ![

zT1 · · · z

Tk

]T.

The f(

x(i)k

∣

∣z1:k)

are marginals of the joint posterior pdf

f(x1:k|z1:k), with x1:k ![

xT1 · · · x

Tk

]T. However, direct mar-

ginalization of f(x1:k|z1:k) is infeasible since the dimensionof x1:k grows linearly with the number of time steps k and thenumber of targets nt, and thus the computational complexityscales exponentially in k and nt. The exponential scalingin k can be avoided by exploiting statistical independenciesacross the time steps: using Bayes’ rule on f(x1:k|z1:k) =f(x1:k|z1:k,m1:k), where m1:k !

[

mT1 · · · m

Tk

]T, we obtain

f(x1:k|z1:k) =f(z1:k,m1:k|x1:k)f(x1:k)

f(z1:k,m1:k)

∝ f(z1:k,m1:k|x1:k)f(x1:k),

since z1:k (and, hence, also m1:k) are observed and thusconsidered fixed. (Note that knowledge of m1:k is impliedby knowledge of z1:k.) Using A1 and A11, we obtain further

f(x1:k|z1:k) ∝k∏

k′=1

f(xk′ |xk′−1)f(zk′ ,mk′ |xk′). (18)

Here, f(xk|xk−1) and f(zk,mk|xk) are given by (8) and(16), respectively, and we formally introduced f(x1|x0) !f(x1) (cf. (9)). The factorization (18) underlies sequentialBayesian estimation (filtering) using the prediction step (1) andthe update step (2), whose complexity per time step is constant.The factor graph representing the factorization (18) is shownin Fig. 2(a). We note that sequential Bayesian estimation andrelated methods such as the Kalman filter and the particlefilter can be interpreted as running the SPA on this simpletree-structured factor graph.

A. First Stretching Step: Introducing a1:k

The complexity of (1) and (2) still scales exponentiallywith nt. To address this issue and obtain scalable SPA-basedMTT methods, we use the stretching principle from Section

III-A to introduce the DA vector a1:k ![

aT1 · · · a

Tk

]Tand

formally replace the joint posterior pdf f(x1:k|z1:k) byf(x1:k,a1:k|z1:k). Note that

∑

a1:kf(x1:k,a1:k|z1:k) =

f(x1:k|z1:k), and thus the marginal posterior pdfs

f(

x(i)k

∣

∣z1:k)

calculated from f(x1:k,a1:k|z1:k) are equal to


(a) (b)

(c) (d)

s=ns

s=1

fzfz−

x−x= ff− x

k − 2

k − 1

k − 1

k − 1

k − 1

k

k

k

k

a

g1

gnt

x1

x1

x1

xnt

xnt

xnt

x1−

x1−

x1−

xnt−

xnt−

xnt−

f1

f1

f1

fnt

fnt

fnt

ψ

g1g1

gntgnt

h1

h1

hnm

hnm

a1a1

antant

b1b1

bnmbnm

Ψ1,1Ψ1,1

Ψnt,nmΨnt,nm

Ψ1,nmΨ1,nm

Ψnt,1Ψnt,1

Fig. 2. Factor graphs for single-sensor and multisensor MTT, assuming a known, fixed number of targets: (a) Factor graph for sequential Bayesian estimation,corresponding to the factorization (18), (b) factor graph for the single-sensor MTT problem corresponding to the factorization (19) with ns=1, (c) stretchedfactor graph for the single-sensor MTT problem corresponding to the factorization (24) with ns = 1, (d) factor graph for the multisensor MTT problemcorresponding to the factorization (24). Two complete consecutive sections of the factor graph (for times k−1 and k) are shown in (a), and one completesection (for time k) in (b)–(d). Factor nodes in green represent factors related to the state-transition function and factor nodes in red represent factors related to

the likelihood function. The time index k and the sensor index s are omitted, and the following short notations are used: xi ! x(i)k , ai ! a

(i)k,s, bm ! b

(m)k,s ,

nm ! mk,s, x= ! xk−2, x− ! xk−1, xi−

! x(i)k−1, f− ! f

!

xk−1

"

"xk−2#

, f ! f!

xk

"

"xk−1#

, f i ! f!

x(i)k

"

"x(i)k−1

#

, fz

−! f

!

zk−1,mk−1

"

"xk−1#

,

fz ! f!

zk,mk

"

"xk

#

, g i ! g!

x(i)k , a

(i)k,s; zk,s

#

, gi ! g!

x(i)k , a

(i)k,s; zk,s

#

, hm ! h!

b(m)k,s ; z

(m)k,s

#

, ψ! ψ(ak,s), and Ψi,m ! Ψi,m

!

a(i)k,s , b

(m)k,s

#

.

the ones calculated from f(x1:k|z1:k). Suitable modificationof (18) yields the new factorization

f(x1:k,a1:k|z1:k)

∝k∏

k′=1

f(xk′ |xk′−1)f(zk′ ,ak′ ,mk′ |xk′)

=k∏

k′=1

f(xk′ |xk′−1)ns∏

s=1

f(zk′,s,ak′,s,mk′,s|xk′),

where (15) was used in the last step. Inserting (8), (9), and(13), we obtain further

f(x1:k,a1:k|z1:k) ∝k∏

k′=1

(

nt∏

i=1

f(

x(i)k′

∣

∣x(i)k′−1

)

)

ns∏

s=1

ψ(ak′,s)

×nt∏

i′=1

g(

x(i′)k′ , a(i

′)k′,s; zk′,s

)

, (19)

where f(

x(i)1

∣

∣x(i)0

)

! f(

x(i)1

)

. The factor graph representingthis factorization is shown in Fig. 2(b) for the single-sensor

case (ns=1). This factor graph has loops, which are howevernot visible because Fig. 2(b) essentially shows only the partof the factor graph corresponding to time step k.

B. Second Stretching Step: Introducing b1:k

Running the SPA on the factor graph in Fig. 2(b) issignificantly less complex than running it on the factor graphin Fig. 2(a), because the marginalizations with respect tothe multitarget state (the xk−1 integration in (1)) and withrespect to the DA vector (the ak,s summation in (17)) areavoided. However, there are still high-dimensional discretemarginalizations, whose complexity scales exponentially withnt and mk,s. This is because the messages related to the

variable nodes ak,s are functions of all the variables a(i)k,s ,

i = 1, . . . , nt, where a(i)k,s ∈ {0, . . . ,mk,s}. A computation-ally feasible SPA-based MTT method can be obtained by amodification of the factor graph in Fig. 2(b). We again invokethe stretching principle, this time to stretch the factor node“ψ(ak,s)”. Following [49], [50], we introduce the DA vector


bk,s =[

b(1)k,s · · · b

(mk,s)k,s

]Twith entries

b(m)k,s !

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

i ∈ {1, . . . , nt} , if at time k, measurement mat sensor s is generated bytarget i

0 , if at time k, measurement mat sensor s is not generatedby a target.

(20)

Note that bk,s carries the same information as ak,s but in adifferent form. If bk,s was known, ak,s would be known aswell and the DA problem would be solved. However, bk,s isunknown and thus considered as a latent random variable inthe inference problem. The admissibility of a DA hypothesisaccording to the constraint A4 can now be expressed by theindicator function

ψ(ak,s , bk,s) !nt∏

i=1

mk,s∏

m=1

Ψi,m

(

a(i)k,s , b(m)k,s

)

, (21)

with

Ψi,m

(

a(i)k,s , b(m)k,s

)

!

⎧

⎪

⎨

⎪

⎩

0 , a(i)k,s=m, b(m)k,s = i

or b(m)k,s = i, a(i)k,s =m

1 , otherwise.

(22)

We note that∑

bk,sψ(ak,s , bk,s) = ψ(ak,s). Replacing

ψ(ak,s) in (19) by ψ(ak,s , bk,s) and using (21) results in thenew joint posterior pdf

f(x1:k,a1:k, b1:k|z1:k) ∝k∏

k′=1

nt∏

i=1

f(

x(i)k′

∣

∣x(i)k′−1

)

×ns∏

s=1

g(

x(i)k′ , a

(i)k′,s; zk′,s

)

×

mk′,s∏

m=1

Ψi,m

(

a(i)k′,s , b(m)k′,s

)

. (23)

Finally, we perform a last modification of the factorization.This modification does not further reduce the complexity,but it is the basis for a general SPA-based DA algorithm,to be presented in Section VI, which can be used in manydifferent MTT methods. First, we multiply (23) by the constant

c(z1:k) !∏k

k′=1

∏ns

s=1

∏mk,s

m=1 µ(s)c f (s)

c

(

z(m)k′,s

)

. For given kand s, let M0

k,s ⊆ {1, . . . ,mk,s} be the set of all m with

b(m)k,s = 0, i.e., all m indexing clutter measurements. Because

of A4 expressed by∏nt

i=1

∏mk,s

m=1 Ψi,m

(

a(i)k,s, b(m)k,s

)

, for each

m ∈ {1, . . . ,mk,s}\M0k,s, the joint posterior pdf in (23) is

nonzero only if there is exactly one a(i)k,s that equals m. This

means that for a(i)k,s = m ∈ {1, . . . ,mk,s}, the denominator

µ(s)c f (s)

c

(

z(a

(i)k,s

)

k,s

)

of g(


)

(see (14), recalling that

f (s)(

z(m)k,s

∣

∣x(i)k

)

= f(

z(m)k,s

∣

∣x(i)k

)

/f (s)c

(

z(m)k,s

)

) is canceled by

the corresponding factor µ(s)c f (s)

c

(

z(m)k,s

)

in c(z1:k). Equation(23) thus becomes

f(x1:k,a1:k, b1:k|z1:k)

∝k∏

k′=1

(

nt∏

i′=1

f(

x(i′)k′

∣

∣x(i′)k′−1

)

)

ns∏

s=1

(

nt∏

i=1

g(

x(i)k′ , a

(i)k′,s; zk′,s

)

×

mk′,s∏

m′=1

Ψi,m′

(

a(i)k′,s , b(m′)k′,s

)

)

∏

m∈M0k′,s

µ(s)c f (s)

c

(

z(m)k′,s

)

,

where for a(i)k,s ∈ {1, . . . ,mk,s},

g(


)

! g(


)

µ(s)c f (s)

c

(

z(a

(i)k,s

)

k,s

)

= p(s)d

(

x(i)k

)

f(

z(a(i)

k,s)

k,s

∣

∣x(i)k

)

,

and for a(i)k,s = 0, g(

x(i)k , 0; zk,s

)

! g(

x(i)k , 0; zk,s

)

=(

1−

p(s)d

(

x(i)k

))

(cf. (14)). Introducing

h(

b(m)k,s ; z

(m)k,s

)

!

{

µ(s)c f (s)

c

(

z(m)k,s

)

, b(m)k,s = 0

1 , b(m)k,s > 1,

we obtain the final factorization of f(x1:k,a1:k, b1:k|z1:k) as

f(x1:k,a1:k, b1:k|z1:k)

∝k∏

k′=1

(

nt∏

i′=1

f(

x(i′)k′

∣

∣x(i′)k′−1

)

)

ns∏

s=1

(

nt∏

i=1

g(

x(i)k′ , a

(i)k′,s; zk′,s

)

×

mk′,s∏

m′=1

Ψi,m′

(

a(i)k′,s , b(m′)k′,s

)

)mk′,s∏

m=1

h(

b(m)k′,s ; z

(m)k′,s

)

, (24)

with f(

x(i)1

∣

∣x(i)0

)

= f(

x(i)1

)

. Note that f(x1:k,a1:k, b1:k|z1:k) is again consistent with the original jointposterior pdf f(x1:k|z1:k) in (18) in the sense that∑

a1:k

∑

b1:kf(x1:k,a1:k, b1:k|z1:k) = f(x1:k|z1:k).

The factor graph representing the factorization (24) is shownin Fig. 2(c) for the single-sensor case (ns = 1) and in Fig.2(d) for the multisensor case. These factor graphs containone or multiple “bipartite” substructures that are associatedwith probabilistic DA. An interesting interpretation of thesesubstructures is that information on target-originated measure-

ments in the form of g(


)

is connected to target-oriented DA variables and information on clutter-originated

measurements in the form of h(

b(m)k,s ; z

(m)k,s

)

is connected tomeasurement-oriented DA variables. The two factor graphswill be used in Sections VI and VII as a basis for developingscalable SPA-based methods for MTT.

VI. SPA-BASED PROBABILISTIC DA

Next, we describe an efficient loopy SPA-based algorithmfor probabilistic DA [48]–[50]. This algorithm will constitutean important building block of the SPA-based MTT methodsdeveloped in Sections VII, IX, and XIII. We consider nt

targets, where nt is assumed known, and a single sensor (thus,we drop the sensor index s). The assumptions of a fixed,known number of targets and of a single sensor will be liftedin later sections.

The goal of probabilistic DA is to calculate the posterior DA

pmfs p(

a(i)k

∣

∣z1:k)

, where a(i)k was defined in (10). These pmfsare marginals of the joint posterior pdf f(x1:k,a1:k, b1:k|z1:k)in (24). We now use the loopy SPA on the factor graphin Fig. 2(c) to calculate accurate approximations of thesemarginal pmfs. We choose a message calculation schedulesuch that each time k is considered individually; this is


achieved by passing messages only forward in time1 [42].According to (5), the messages passed from the factor nodes

“f(

x(i)k

∣

∣x(i)k−1

)

” to the variable nodes “x(i)k ” are given by

α(i)k

(

x(i)k

)

=

∫

f(

x(i)k

∣

∣x(i)k−1

)

f(

x(i)k−1

)

dx(i)k−1 . (25)

Here, f(

x(i)k−1

)

is the belief approximating the marginal pos-

terior pdf of x(i)k−1; this belief was calculated at time k−1. The

message calculation in (25) is analogous to the prediction stepin sequential filtering (cf. (1)). Similarly, the messages passed

from the factor nodes “g(

x(i)k , a(i)k ; zk

)

” to the variable nodes

“a(i)k ” are given by

β(i)k

(

a(i)k

)

=

∫

g(

x(i)k , a(i)k ; zk

)

α(i)k

(

x(i)k

)

dx(i)k ,

and the messages passed from the factor nodes

“h(

b(m)k ; z(m)

k

)

” to the variable nodes “b(m)k ” are given by

ξ(m)k

(

b(m)k

)

= h(

b(m)k ; z(m)

k

)

. The β(i)k

(

a(i)k

)

and ξ(m)k

(

b(m)k

)

are referred to as association weights [50], [51], [107].For future reference, we note that the belief of the joint DA

vector [aTk bT

k]T can be obtained by interpreting all variable

nodes related to the DA variables a(i)k , i = 1, . . . , nt and

b(m)k , m = 1, . . . ,mk,s on the factor graph in Fig. 2(c) as

one “supernode” and calculating the belief of that supernodeaccording to (7). This belief is obtained as

p(ak , bk) ∝ ψ(

ak, bk)

(

nt∏

i=1

β(i)k

(

a(i)k

)

)

mk∏

m=1

ξ(m)k

(

b(m)k

)

,

(26)

where ψ(

ak, bk)

is given by (21).

A. SPA-Based DA

Once α(i)k

(

x(i)k

)

and β(i)k

(

a(i)k

)

have been calculated, the

iterative SPA involving the factor nodes “Ψi,m

(

a(i)k , b(m)k

)

”

and the variable nodes “a(i)k ” and “b(m)k ” is performed for all

states i = 1, . . . , nt and all measurements m = 1, . . . ,mk

in parallel. More specifically, at message passing iteration

ℓ ∈ {1, . . . , nit}, a message ϕ[ℓ]

Ψi,m→b(m)k

(

b(m)k

)

is passed from

“Ψi,m

(

a(i)k , b(m)k

)

” to “b(m)k ”, and a message ν[ℓ]

Ψi,m→a(i)k

(

a(i)k

)

is passed from “Ψi,m

(

a(i)k , b(m)k

)

” to “a(i)k ”. Since each factor

node “Ψi,m

(

a(i)k , b(m)k

)

” is connected to only two variablenodes, an outgoing message from such a factor node can beobtained from the incoming message by inserting (6) intothe discrete counterpart of (5). In this way, one obtains thefollowing recursion for the messages ϕ and ν:

ϕ[ℓ]

Ψi,m→b(m)k

(

b(m)k

)

=mk∑

a(i)k

=0

β(i)k

(

a(i)k

)

Ψi,m

(

a(i)k , b(m)k

)

mk∏

m′=1m′=m

ν[ℓ]Ψi,m′→a

(i)k

(

a(i)k

)

(27)

and

ν[ℓ]Ψi,m→a

(i)k

(

a(i)k

)

1We note that SPA-based algorithms for probabilistic DA that considermultiple time steps jointly were introduced in [145].

=nt∑

b(m)k

=0

ξ(m)k

(

b(m)k

)

Ψi,m

(

a(i)k , b(m)k

)

nt∏

i′=1i′=i

ϕ[ℓ−1]

Ψi′,m→b(m)k

(

b(m)k

)

,

(28)

for i = 1, . . . , nt and m = 1, . . . ,mk. This iterative algorithmis initialized by the messages

ϕ[0]

Ψi,m→b(m)k

(

b(m)k

)

=mk∑

a(i)k

=0

β(i)k

(

a(i)k

)

Ψi,m

(

a(i)k , b(m)k

)

. (29)

After the last iteration ℓ = nit, approximations p(

a(i)k

∣

∣z1:k)

of the marginal posterior DA pmfs p(

a(i)k

∣

∣z1:k)

are obtainedaccording to (7), i.e.,

p(

a(i)k

∣

∣z1:k)

= A(i)k β(i)

k

(

a(i)k

)

mk∏

m=1

ν[nit]

Ψi,m→a(i)k

(

a(i)k

)

,

for i = 1, . . . , nt, where the A(i)k are normalization factors.

B. A Scalable SPA-Based DA Algorithm

The messages (27) and (28) can be simplified [50], [118],[128]. Because of the binary consistency constraints expressed

by Ψi,m

(

a(i)k , b(m)k

)

, each message comprises only two differ-

ent values: ϕ[ℓ]

Ψi,m→b(m)k

(

b(m)k

)

in (27) takes on one value for

b(m)k = i and another for all b(m)

k = i, and ν[ℓ]Ψi,m→a(i)

k

(

a(i)k

)

in (28) takes on one value for a(i)k = m and another for all

a(i)k = m. Thus, each message can be represented (up to anirrelevant constant factor) by the ratio of the first value and

the second value, hereafter denoted as ϕ[ℓ](i→m)k or ν[ℓ](m→i)

k .The recursion (27), (28) can then be reformulated as

ϕ[ℓ](i→m)k =

β(i)k (m)

β(i)k (0) +

∑mk

m′=1m′=m

β(i)k (m′)ν[ℓ](m

′→i)k

(30)

ν[ℓ](m→i)k =

ξ(m)k (i)

ξ(m)k (0) +

∑nt

i′=1i′=i

ξ(m)k (i′)ϕ[ℓ−1](i′→m)

k

, (31)

for i = 1, . . . , nt and m = 1, . . . ,mk. This iterative al-

gorithm is initialized by ϕ[0](i→m)k = β(i)

k (m)/β(i)k (0). Note

that this reformulation exploits the fact that, after replacing

ϕ[ℓ]

Ψi,m→b(m)k

(

b(m)k

)

in (27) by ϕ[ℓ](i→m)k and ν[ℓ]

Ψi,m→a(i)k

(

a(i)k

)

in (28) by ν[ℓ](m→i)k , for each term in the sum only one factor

in the product of messages is different from 1. After the

last iteration ℓ = nit, approximations of the p(

a(i)k

∣

∣z1:k)

areobtained as

p(

a(i)k =m∣

∣z1:k)

=β(i)k (m)ν[nit](m→i)

k

β(i)k (0) +

∑mk

m′=1 β(i)k (m′)ν[nit](m′→i)

k

,

for m = 0, 1, . . . ,mk. Here, for m = 0, ν[nit](0→i)k ! 1. In

what follows, this efficient algorithm will be referred to asSum-Product Algorithm for Data Association (SPADA). Notethat one can also obtain approximations of the “measurement-

oriented” DA pmfs p(

b(m)k

∣

∣z1:k)

as

p(

b(m)k = i

∣

∣z1:k)

=ξ(m)k

(

i)

ϕ[nit](i→m)k

ξ(m)k (0) +

∑nt

i′=1 ξ(m)k

(

i′)

ϕ[nit](i′→m)k

,


0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

Target spacing

Ave

rage

max

imum

erro

r

(a)

SPADA (proposed)

MESP

JTree (10−3)

JTree (10−2)

JTree (10−1)

MCMCDA (106)

MCMCDA (105)

MCMCDA (104)

Target spacing

Ave

rage

com

puta

tion

tim

e[m

s]

(b)

0 2 4 6 8 1010−2

10−1

100

101

102

103

104

Fig. 3. Experimental comparison of SPADA with three alternative DA algorithms: (a) Average maximum error of the estimated marginal posterior pmfs

p!

a(i)k

"

"z1:k#

versus the target spacing, (b) average computation time versus the target spacing.

for i = 0, 1, . . . , nt. Here, for i=0, ϕ[nit](0→m)k ! 1.

It was shown in [50], [128] that the recursion (30), (31) isa contraction, which is guaranteed to converge. In addition,it was shown in [127] and in [145, Appendix A-A] that therecursion (30), (31) solves a convex optimization problem,and thus it ultimately converges to the corresponding globaloptimum. Moreover, the number of iterations required to meeta specific convergence criterion is bounded [50]. Finally, thecomplexity of (30), (31) is significantly lower than that of

(27), (28). In each iteration ℓ, ntmk messages ϕ[ℓ](i→m)k

and ntmk messages ν[ℓ](m→i)k are calculated. The associated

complexity is essentially determined by that of calculating the

nt sums∑mk

m′=1,m′=m β(i)k (m′)ν[ℓ](m

′→i)k and the mk sums

∑nt

i′=1,i′=i ξ(m)k

(

i′)

ϕ[ℓ−1](i′→m)k , which scales as O(ntmk).

Hence, the overall complexity of SPADA per iteration isO(ntmk). This is much smaller than the complexity of (27)and (28), which is O(n2

tm3k+n3

tm2k). We note that MATLAB

code for the recursion (30), (31) is provided in [128].

C. Simulation Results

We demonstrate the performance of SPADA for a two-dimensional (2-D) scenario with nt = 6 static targets arrangedon a regular 2×3 grid [50]. The spacing of the targets isvaried between 0 and 10. The measurements are the targetpositions plus circularly symmetric 2-D Gaussian noise ofzero mean and variance 1. The clutter measurements areuniformly distributed over the region of interest with intensity

µcfc

(

z(m)k

)

= 0.01. The region of interest consists of thegates [1] of the six targets with gate threshold 18.4, whichmeans that target-originated measurements are “gated out”with probability 10−4. The probability that a target is detected

by the sensor is pd

(

x(i)k

)

= 0.6. Prior information on the

x(i)k (replacing α(i)

k

(

x(i)k

)

in (25)) is obtained through theprocedure described in [50]. The number of message passingiterations nit is chosen adaptively according to [50].

We compare SPADA with the following three alternativealgorithms (cf. Section II-C): the MESP algorithm, which

performs the SPA on a factor graph that is obtained witha different stretching of the factor node “ψ(ak)” based onelementary mutual exclusion constraints [49], [124] [32, Box12.D]; the JTree algorithm, which performs the junction treealgorithm [25], [32] on that alternative factor graph, withassociation weights thresholded to 10−3, 10−2, or 10−1 toinduce sparsity; and the MCMCDA algorithm [146] using 104,105, or 106 Markov chain Monte Carlo steps. We consideronly a single time step, k = 1, and accordingly, e.g.,

p(

a(i)k

∣

∣z1:k)

= p(

a(i)1

∣

∣z1)

. Fig. 3 shows the average maximum

error of p(

a(i)1

∣

∣z1)

and the average computation time versusthe target spacing. The average maximum error is calculated

by maximizing∣

∣p(

a(i)k =m∣

∣z1)

− p(

a(i)1 =m∣

∣z1)∣

∣ over all mand averaging the result over the six targets (i = 1, . . . , 6)and over 1000 simulation runs. (We do not plot the results ofthe MESP algorithm for very small target spacings becausethe MESP algorithm occasionally failed to converge in thatcase.) SPADA is seen to outperform the MESP algorithm whenthe targets are closely spaced, to outperform the JTree (10−1)algorithm and the MCMCDA (104) algorithm, and to performsimilarly to the MCMCDA (105) algorithm. The computationtime of SPADA is seen to be always lower—partly by severalorders of magnitude—than that of the other algorithms. Thisis true even though the MESP and JTree algorithms wereimplemented using a C++ library [147] whereas SPADA wasimplemented in MATLAB. Further experiments presented in[50] confirm the high accuracy and efficiency of SPADA.

VII. SPA-BASED VECTOR-TYPE MTT METHODS

FOR A KNOWN, FIXED NUMBER OF TARGETS

In this section, we consider the application of the SPAto MTT within the vector-type system model described inSection IV, i.e., assuming a known, fixed number nt of targets.First, we formulate two “total-SPA” methods for single-sensorMTT, where the second method is based on SPADA. Next,we review the JPDA filter and develop a “SPADA-embeddedJPDA filter,” which is a JPDA filter using SPADA for anefficient approximative calculation of the marginal DA pmfs.


Remarkably, the first total-SPA method is equal to the JPDAfilter and the second total-SPA method is equal to the SPADA-embedded JPDA filter. The second total-SPA method is finallyextended to obtain scalable MTT methods for multiple sensors.New variants of these methods for multisensor MTT thatallow for an unknown, time-varying number of targets willbe presented in Section IX.

A. SPA-Based Methods for Single-Sensor MTT

We will now show that the SPA can be used not only for anefficient calculation of the marginal DA pmfs but for the entireMTT problem. The number of targets, nt, is still assumed fixedand known. Furthermore, we still consider the single-sensorcase (ns = 1), and thus again drop the sensor index s.

1) Total-SPA Method for Single-Sensor MTT: In contrastto Section VI, where we calculated the marginal DA pmfs

p(

a(i)k

∣

∣z1:k)

by running the SPA on the bipartite part of thefactor graph in Fig. 2(c), we now calculate the marginal pos-

terior target state pdfs f(

x(i)k

∣

∣z1:k)

by running the SPA on thefactor graph in Fig. 2(b). We again use a message calculationschedule such that messages are passed only forward in timeand hence a noniterative SPA is obtained.

First, the messages α(i)k

(

x(i)k

)

passed from the factor nodes

“f(

x(i)k

∣

∣x(i)k−1

)

” to the variable nodes “x(i)k ” are calculated

according to (25), and the messages β(i)k

(

a(i)k

)

passed from the

factor nodes “g(

x(i)k , a(i)k ; zk

)

” to the variable node “ak” are

calculated according to2 (26) (with g(

x(i)k , a(i)k ; zk

)

replaced

by g(

x(i)k , a(i)k ; zk

)

). Then, the messages passed from “ak” to

“g(

x(i)k , a(i)k ; zk

)

” are calculated according to (6), i.e.,

ω(i)k (ak) = ψ(ak)

nt∏

i′=1i′=i

β(i′)k

(

a(i′)

k

)

.

Next, the messages passed from “g(

x(i)k , a(i)k ; zk

)

” to “x(i)k ”

are calculated according to the discrete counterpart of (5), i.e.,

γ(i)k

(

x(i)k

)

=∑

ak

g(

x(i)k , a(i)k ; zk

)

ω(i)k (ak)

=mk∑

a(i)k

=0

g(

x(i)k , a(i)k ; zk

)

κ(i)k

(

a(i)k

)

, (32)

with

κ(i)k

(

a(i)k

)

!∑

∼a(i)k

ω(i)k (ak), (33)

where∑

∼a(i)k

denotes the summation over all a(i′)

k ∈ {0, . . . ,

mk} for all i′ ∈ {1, · · · , nt}\{i}. Finally, the beliefs for the

x(i)k are obtained according to (7), i.e,

f(

x(i)k

)

∝ α(i)k

(

x(i)k

)

γ(i)k

(

x(i)k

)

. (34)

2Note that this is still consistent with our general notation defined by (5),

because the message β(i)k

!

a(i)k

#

can be interpreted as a function in ak (which

is constant in all a(i′)k except a

(i)k ).

These beliefs approximate the marginal posterior pdfs

f(

x(i)k |z1:k

)

. They can be used for Bayesian state estimation,e.g., by means of the MMSE estimator in (3).

2) A Scalable Total-SPA Method for Single-Sensor MTT:

The complexity of the total-SPA method presented aboveis exponential in nt due to the summation in (33). Thisexponential scaling can be avoided by running the SPA onthe entire factor graph in Fig. 2(c) rather than on the factorgraph in Fig. 2(b) (or on the bipartite part of the factor graph inFig. 2(c) as done for DA in Section VI). The resulting methodwill also be used in Section VII-C to develop scalable MTTmethods for multiple sensors. Due to the additional stretchingstep underlying the factor graph in Fig. 2(c), the summationin (33) can be avoided by using SPADA.

More specifically, after calculating the messages α(i)k

(

x(i)k

)

and β(i)k

(

a(i)k

)

according to (25) and (26), respectively, theiterative SPA (27)–(29) is executed, which yields the messages

ν[nit]

Ψi,m→a(i)k

(

a(i)k

)

in (28). Then, the messages passed from

“a(i)k ” to “g(

x(i)k , a(i)k ; zk

)

” are calculated according to (6),i.e.,

κ(i)k

(

a(i)k

)

=mk∏

m=1

ν[nit]

Ψi,m→a(i)k

(

a(i)k

)

. (35)

Next, the messages γ(i)k

(

x(i)k

)

passed from “g(

x(i)k , a(i)k ; zk

)

”

to “x(i)k ” are calculated according to (32) with κ

(i)k

(

a(i)k

)

replaced by κ(i)k

(

a(i)k

)

. Thereby, the high-dimensional summa-tion in (33) is avoided. Finally, the iterative SPA (27)–(29) is

replaced by SPADA. For this, the messages ν[nit](m→i)k in (31)

are converted into the messages ν[nit]

Ψi,m→a(i)k

(

a(i)k

)

used in (35)

according to ν[nit]

Ψi,m→a(i)k

(

a(i)k

)

= ν[nit](m→i)k for a(i)k = m and

ν[nit]

Ψi,m→a(i)k

(

a(i)k

)

= 1 for a(i)k =m. Finally, evaluation of (34)

yields the beliefs3 f(

x(i)k

)

. This total-SPA method for single-sensor MTT has excellent scalability (namely, O(ntmk)) andis thus suitable also for large tracking scenarios with closelyspaced targets.

B. JPDA Filter with SPA-Based Probabilistic DA

The JPDA filter is a vector-type MTT method that ap-proximates the joint posterior DA pmf by the product ofits marginals. While the original JPDA filter additionallyemploys a Gaussian approximation for the target state pdfs[1], we here use the term JPDA more broadly to refer to amethod that propagates a marginal pdf (without a parametricrepresentation) separately for each single-target state [82]. Inthis subsection, we first review the JPDA filter and then showhow to embed SPADA.

1) Prediction Step: In the prediction step at time k, theChapman-Kolmogorov equation (1) is used to convert the ap-proximate posterior pdf f(xk−1|z1:k−1) calculated at time k−

3These beliefs are not the same approximations of the marginal posterior

pdfs f!

x(i)k |z1:k

#

as in (34) because they are based on loopy SPA for DA.For simplicity, we do not indicate this difference by our notation.


1 into an approximate predicted posterior pdf f(xk|z1:k−1).We assume that f(xk−1|z1:k−1) factors into its marginals, i.e.,

f(xk−1|z1:k−1) =nt∏

i=1

f(

x(i)k−1

∣

∣z1:k−1

)

. (36)

The validity of this assumption will be verified and discussedin Section VII-B2. Note also that for k − 1 = 1, (36) isequivalent to (9). Inserting (36) and (8) into (1) yields

f(xk|z1:k−1) =nt∏

i=1

∫

f(

x(i)k

∣

∣x(i)k−1

)

f(

x(i)k−1

∣

∣z1:k−1

)

dx(i)k−1

=nt∏

i=1

f(

x(i)k

∣

∣z1:k−1

)

, (37)

with

f(

x(i)k

∣

∣z1:k−1

)

=

∫

f(

x(i)k

∣

∣x(i)k−1

)

f(

x(i)k−1

∣

∣z1:k−1

)

dx(i)k−1 ,

(38)for i = 1, . . . , nt. Thus, the prediction step (1)—whichrequired an ntdx-dimensional integration—reduces to nt sep-arate predictions of single-target states, each requiring only adx-dimensional integration.

2) Update Step: In the update step at time k, the ap-proximate posterior pdf f(xk|z1:k) is calculated from theapproximate predicted posterior pdf f(xk|z1:k−1) in (37) andthe current measurement vector zk (cf. (2)). To derive theupdate rule, we first expand the approximate posterior pdf as

f(xk|z1:k) =∑

ak

f(xk|ak, z1:k) p(ak|z1:k), (39)

where the summation is over all ak ∈ {0, . . . ,mk}nt. Using

A11, f(xk|ak, z1:k) is obtained as [144]

f(xk|ak, z1:k) =nt∏

i=1

f(

x(i)k

∣

∣a(i)k , z1:k)

, (40)

where

f(

x(i)k

∣

∣a(i)k , z1:k)

=g(

x(i)k , a(i)k ; zk

)

f(

x(i)k

∣

∣z1:k−1

)

∫

g(

x(i)′k , a(i)k ; zk

)

f(

x(i)′k

∣

∣z1:k−1

)

dx(i)′k

,

(41)

with g(


)

defined in (14). We note that theoriginal derivation of the JPDA filter [1] assumed that theprobability of detection does not depend on the target state

x(i)k , i.e., pd

(

x(i)k

)

= pd. In that case, (41) simplifies as follows:

for a(i)k = m∈ {1, . . . ,mk},

f(

x(i)k

∣

∣m, z1:k)

=f(

z(m)k

∣

∣x(i)k

)

f(

x(i)k

∣

∣z1:k−1

)

∫

f(

z(m)k

∣

∣x(i)′k

)

f(

x(i)′k

∣

∣z1:k−1

)

dx(i)′k

,

and for a(i)k = 0, f(

x(i)k

∣

∣a(i)k =0, z1:k)

= f(

x(i)k

∣

∣z1:k−1

)

.

To avoid computations involving functions of the high-dimensional multitarget state xk, the JPDA filter updates

each single-target state x(i)k individually. This is achieved by

approximating the joint posterior DA pmf p(ak|z1:k) by theproduct of its marginals, i.e.,

p(ak|z1:k) ≈nt∏

i=1

p(

a(i)k

∣

∣z1:k)

, (42)

with

p(

a(i)k

∣

∣z1:k)

=∑

∼a(i)k

p(ak|z1:k). (43)

Because the summation in (43) is with respect to all a(i′)

kfor i′ ∈ {1, . . . , nt}\{i}, the computational complexity ofthis summation scales exponentially with nt. We note that(42), (43) is the optimum approximation of p(ak|z1:k) by afully factorizing pmf, where optimality is defined as minimumKullback-Leibler divergence [32, Proposition 8.3]. Inserting(40) and (42) into (39), we obtain

f(xk|z1:k) ≈∑

ak

nt∏

i=1

f(

x(i)k

∣

∣a(i)k , z1:k)

p(

a(i)k

∣

∣z1:k)

=nt∏

i=1

mk∑

a(i)k

=0

f(

x(i)k

∣

∣a(i)k , z1:k)

p(

a(i)k

∣

∣z1:k)

(44)

=nt∏

i=1

f(

x(i)k

∣

∣z1:k)

. (45)

Hence, the calculation of f(xk|z1:k) simplifies to nt sepa-rate calculations of the approximate marginal posterior pdfs

f(

x(i)k

∣

∣z1:k)

. According to (44), these approximate marginalposterior pdfs are given by

f(

x(i)k

∣

∣z1:k)

=mk∑

a(i)k

=0

f(

x(i)k

∣

∣a(i)k , z1:k)

p(

a(i)k

∣

∣z1:k)

, (46)

with f(

x(i)k

∣

∣a(i)k , z1:k)

given by (41) and p(

a(i)k

∣

∣z1:k)

given

by (43). The f(

x(i)k

∣

∣z1:k)

can be used for Bayesian stateestimation, e.g., by means of the MMSE estimator in (3).

It can be shown that approximating the joint posterior DApmfs p(ak|z1:k) according to (42), as is done in the JPDAfilter, results in approximate marginal posterior state pdfs

f(

x(i)k

∣

∣z1:k)

in (46) that are equal to the beliefs f(

x(i)k

)

in (34). Thus, the total-SPA method presented in SectionVII-A1 can be viewed as an SPA-based reformulation of theJPDA filter. Furthermore, we note that the factorization (45)is consistent with our initial factorization assumption in (36).In fact, together with (9), the above derivation provides aninductive proof that the factorization postulated in (42) entails(36), i.e., the factorization of f(xk|z1:k) into the approximate

marginal posterior pdfs f(

x(i)k

∣

∣z1:k)

, for all k=1, 2, . . ..A closed-form implementation of the JPDA filter for linear-

Gaussian models is presented in [1]. A sequential MonteCarlo (particle-based) implementation for nonlinear and non-Gaussian models can be found in [82].

3) Embedding SPADA: While the approximation (42)

makes it possible to update the target states x(i)k individually,

the calculation of the marginal DA pmfs p(

a(i)k

∣

∣z1:k)

in (43)still scales exponentially with nt. The complexity can bereduced by gating [1], which is an additional approximation,but it remains exponential in the case of closely spaced targets.This problem can be solved by using SPADA for an approxi-

mative calculation of the p(

a(i)k

∣

∣z1:k)

. As an input to SPADA,the approximate marginal posterior pdfs at the preceding

time, f(

x(i)k−1

∣

∣z1:k−1

)

, are used, i.e., the f(

x(i)k−1

∣

∣z1:k−1

)

are

substituted for the f(

x(i)k−1

)

in (25). Remarkably, the resulting


“SPADA-embedded JPDA filter” is equivalent to the scalabletotal-SPA method developed in Section VII-A2. Indeed, itcan be shown that the approximate marginal posterior pdfsproduced by the former method are equal to the beliefsproduced by the latter method.

C. SPA-Based Methods for Multisensor MTT

Next, we discuss the extension of the total-SPA reformu-lation of the JPDA filter from Section VII-A1 to the case ofns ≥ 2 sensors. Let us consider the factorization (24) of thejoint posterior pdf f(x1:k,a1:k, b1:k|z1:k). The correspondingfactor graph is shown in Fig. 2(d); it is a generalizationof the single-sensor factor graph in Fig. 2(c) featuring anadditional “outer loop” across the sensors. For this factorgraph, the associated variational inference problem can beshown to be a nonconvex optimization problem [145], andthus the iterative SPA is not guaranteed to converge. Indeed,performing multiple message passing iterations over the outerloop does not necessarily lead to improved performance. Inthe following, we discuss two message passing schedules thatavoid convergence issues and have been observed to resultin good performance in many MTT scenarios. Alternative,iterative approaches based on convexification are studied inforthcoming work [145].

1) Sequential Processing: Following the iterated-correctorstrategy, the target state beliefs are updated sequentially withrespect to the sensors. This corresponds to a message passingschedule consisting of ns successive “sensor update” steps,where in each step the result of the preceding step is used(except for the first step) and messages are passed to and froma part of the factor graph related to one sensor.

First, the messages α(i)k

(

x(i)k

)

are calculated according to

(25). Then, iterated beliefs fs(

x(i)k

)

are calculated sequentiallyfor each sensor s= 1, . . . , ns. At the sth sensor update step,

the messages passed from “g(


)

” to “a(i)k,s” arecalculated as (cf. (26))

β(i)k,s

(

a(i)k,s

)

=

∫

g(


)

fs−1

(

x(i)k

)

dx(i)k , (47)

where the fs−1

(

x(i)k

)

were calculated at the preceding sensor

update step (cf. (49) below) or, for s = 1, f0(

x(i)k

)

=

α(i)k

(

x(i)k

)

. Note that expression (47) involves the measure-

ment zk,s of sensor s. Using the β(i)k,s

(

a(i)k,s

)

as input, theiterative SPA (27)–(29) or SPADA (30), (31) is executed,

which yields the messages ν[nit]

Ψi,m→a(i)k,s

(

a(i)k,s

)

. Then, the mes-

sages passed from “a(i)k,s” to “g(


)

” are calculated

according to (35), i.e., κ(i)k,s

(

a(i)k,s

)

=∏mk,s

m=1 ν[nit]

Ψi,m→a(i)k,s

(

a(i)k,s

)

,

and the messages passed from “g(


)

” to “x(i)k ”

are calculated as (cf. (32))

γ(i)k,s

(

x(i)k

)

=

mk,s∑

a(i)k,s

=0

g(


)

κ(i)k,s

(

a(i)k,s

)

. (48)

Finally, iterated beliefs are obtained according to (7), i.e.,

fs(

x(i)k

)

∝ α(i)k

(

x(i)k

)

s∏

s′=1

γ(i)k,s′(

x(i)k

)

, (49)

which generalizes (34) and can be calculated recursively. Note

that fs(

x(i)k

)

incorporates the sensor measurements zk,s′ for

s′ = 1, . . . , s. According to Fig. 2(d) and (6), fs(

x(i)k

)

is also

the message passed from “x(i)k ” to “g

(

x(i)k , a(i)k,s+1; zk,s+1

)

”at sensor update step s + 1. The final beliefs are given as

f(

x(i)k

)

! fns

(

x(i)k

)

; they take into account the measurementsof all sensors. These beliefs are used for state estimation (e.g.,using the MMSE estimator in (3) with f(xk|z1:k) replaced

by f(

x(i)k

)

) and to calculate the messages α(i)k+1

(

x(i)k+1

)

at thenext time step k + 1 according to (25).

This sequential message passing algorithm is simple and itscomputational complexity is linear in the number of sensorsns. On the other hand, it is not well suited to a parallel or dis-

tributed implementation, and the final beliefs f(

x(i)k

)

dependon the chosen sensor order. We note that a sensor-sequentialprocessing is also used by the sequential multisensor JPDAfilter [81] and by iterated-corrector multisensor extensions ofset-type filters (see Section XIII-B).

2) Parallel Processing: An alternative SPA-based methodfor multisensor MTT is provided by the following “parallel”

message passing schedule. First, the messages α(i)k

(

x(i)k

)

arecalculated according to (25). Then, the following operationsare carried out separately for each sensor s ∈ {1, . . . , ns}.

The messages β(i)k,s

(

a(i)k,s

)

passed from “g(


)

” to

“a(i)k,s” are calculated according to (26), i.e.,

β(i)k,s

(

a(i)k,s

)

=

∫

g(


)

α(i)k

(

x(i)k

)

dx(i)k . (50)

Note that in contrast to (47), the expression (50) does notinvolve a result related to a different sensor. The itera-tive SPA (27)–(29) or SPADA (30), (31) is now executed,

and the messages passed from “a(i)k,s” to “g(


)

”

are calculated according to (35), i.e., κ(i)k,s

(

a(i)k,s

)

=∏mk,s

m=1 ν[nit]

Ψi,m→a(i)k,s

(

a(i)k,s

)

. Next, the messages γ(i)k,s

(

x(i)k

)

passed from “g(


)

” to “x(i)k ” are calculated ac-

cording to (48). After these operations have been done for alls ∈ {1, . . . , ns} separately, beliefs are calculated as

f(

x(i)k

)

∝ α(i)k

(

x(i)k

)

ns∏

s=1

γ(i)k,s

(

x(i)k

)

.

This message passing schedule allows for a parallel imple-mentation and facilitates a distributed implementation, and theresults do not depend on a chosen sensor ordering as in sequen-tial processing. However, in some scenarios, the independentprocessing of the individual sensor measurements may resultin a reduced estimation accuracy. We note that a sensor-parallelprocessing is also used by the parallel multisensor JPDA filter[81] and by the multisensor Monte Carlo JPDA filter [82].

VIII. VECTOR-TYPE SYSTEM MODEL AND FACTOR

GRAPH FOR AN UNKNOWN, TIME-VARYING NUMBER


OF TARGETS

Next, we consider the practically more relevant case ofan unknown, time-varying number ik of targets. Extending[53], [80], [148], we first present a vector-type system modeland an associated factor graph for this case. Correspondingtotal-SPA vector-type methods for MTT will be developed inSection IX. The presented vector-type approach models onlydetected targets, i.e., targets that have so far generated at leastone measurement at any of the sensors. By contrast, the set-type approach presented in Section XI models also undetectedtargets, i.e., targets that potentially exist but did not generateany measurement yet.

Our model is based on an arbitrary ordering s=1, 2, . . . , ns

of the sensors. At time k and sensor s, we distinguish betweenthe following two types of detected targets:

• newly detected targets, which exist at time k and havebeen detected for the first time at time k and sensor s;

• survived targets, which exist at time k and have beendetected previously, either at a previous time k′<k or atthe current time k but at a previous sensor s′< s.

The numbers of newly detected targets and survived targets areunknown. To account for this fact, we introduce the notionof potential targets (PTs). The number of PTs at time k,denoted jk, is the maximum possible number of targets thathave generated a measurement at any of the sensors up totime k. This number depends on the number of measurementsobserved at time k, as explained in Section VIII-B. Theexistence/nonexistence of PT j ∈ {1, . . . , jk} is modeled by

a binary variable r(j)k ∈ {0, 1}, i.e., PT j exists at time k if

and only if r(j)k =1. The state of PT j is denoted x

(j)k , and is

formally considered also if r(j)k = 0. The augmented PT state

is defined as y(j)k !

[

x(j)Tk r

(j)k

]T, and the joint state vector as

yk![

y(1)Tk · · · y(jk)T

k

]T. The states x

(j)k of nonexisting PTs are

obviously irrelevant. Therefore, all pdfs defined for PT states,

f(

y(j)k

)

= f(

x(j)k , r(j)k

)

, are such that

f(

x(j)k , r(j)k = 0

)

= f (j)k fD

(

x(j)k

)

, (51)

where f (j)k ∈ [0, 1] is a constant and fD

(

x(j)k

)

is an arbitrary“dummy pdf.”

A. Assumptions

We will use the following assumptions, which replace Vk1–Vk3:

Vu1: The number of targets, ik, is time-varying and unknown.Vu2: The PT states are ordered (arbitrarily) according to their

arrangement in the joint augmented state vector yk =[

y(1)Tk · · · y(jk)T

k

]T.

Vu3: A PT j that exists at time k − 1 survives (i.e., still

exists at time k) with survival probability ps

(

x(j)k−1

)

and

disappears with probability 1−ps

(

x(j)k−1

)

.Vu4: The number of newly detected targets at time k and

sensor s is a priori (i.e., before the measurements

are observed) Poisson distributed with mean µ(s)n . It

is furthermore independent of the number of cluttermeasurements and of the number of survived targets.

Vu5: The states of newly detected targets at time k and sensors are a priori iid and distributed according to4 fn(xk).

Vu6: At time k, the states of newly detected targets areindependent of the states of survived targets.

Vu7: At time k = 0, there are no PTs, i.e., y0 is an emptyvector and j0 = 0.

Based on these assumptions and the common assumptionsin Section I-C, we will next establish a system model and astatistical formulation and, subsequently, derive a “stretched”version of the joint posterior pdf f(y1:k|z1:k).

B. Legacy PTs and New PTs

Each PT at time k and sensor s is either a “legacy” PT,i.e., a PT that was already established in the past, or a “new”

PT. The corresponding states will be denoted by y(j)k,s and

y(m)k,s , respectively. The new PTs and legacy PTs are related

to, respectively, the newly detected targets and survived targetsmentioned earlier, as described in what follows.

• New PTs: To incorporate in the state space targets that arenewly detected by sensor s at time k, mk,s new PT states

y(m)k,s , m = 1, . . . ,mk,s—one for each measurement

z(m)k,s —are introduced. Accordingly, r(m)

k,s =1 means thatmeasurement m produced by sensor s at time k wasgenerated by a newly detected target. We denote by yk,s

![

y(1)Tk,s · · · y

(mk,s)T

k,s

]Tthe joint vector of all new PT

states. Before the current measurements z(m)k,s are ob-

served, the number mk,s of new PT states y(m)k,s is random.

• Legacy PTs: The legacy PT states y(j)k,s represent survived

targets, i.e., targets that have been detected previously,either at a previous time k′< k or at the current time kbut at a previous sensor s′< s. The target represented by

the new PT state y(m′)k′,s′ introduced due to measurement m′

of sensor s′ at time k′ " k is represented by the legacy PT

state y(j)k,s at time k, with j = jk′−1+

∑s′−1s′′=1 mk,s′′ +m′.

(Note that here, either k′< k and s, s′ arbitrary or k′= kand s′ < s.) Accordingly, r(j)k,s = 1 means that the targetthat was detected the first time via measurement m′ ofsensor s′ at time k′ still exists when the measurementsof sensor s at time k are incorporated. We denote by

yk,s

![

y(1)Tk,s · · · y(jk,s)T

k,s

]Tthe joint vector of all legacy

PT states a time k and sensor s. (The relation betweenjk,s and jk will be explained shortly.)

New PTs become legacy PTs when the nextmeasurements—either of the next sensor or at the nexttime step—are incorporated. In particular, at time k, whenthe measurements of the next sensor, s, are incorporated, thenumber of legacy PTs is updated as

jk,s = jk,s−1 +mk,s−1 ,

with jk,1 = jk−1. Here, jk,s is equal to the number of allmeasurements collected at time k up to sensor s. Note that

4Here and hereafter, with an abuse of notation, xk denotes a generic single-target state vector (whereas in Section VII, it denoted the multitarget statevector).


the vector of all legacy PT states at time k up to sensor s can

be written as yk,s

=[

yTk,s−1

yTk,s−1

]T. The vector of all legacy

PT states at time k, before any sensor measurements at timek are incorporated, is denoted by y

k. (Note that y

k,1= y

kand

thus for j ∈ {1, . . . , jk−1}, y(j)k,s= y

(j)k for all s∈ {1, . . . , ns}.)

We also introduce the joint state of all new PTs introduced at

time k as yk![

yTk,1· · · y

Tk,ns

]T. After the measurements of all

sensors s ∈ {1, . . . , ns} have been incorporated at time k, thetotal number of PT states is

jk = jk,ns+mk,ns = jk−1 +ns∑

s=1

mk,s , (52)

and the vector of all PT states at time k is given by yk =[

yTkyTk

]T=[

yTk,ns

yTk,ns

]T. This comprises jk−1 PTs that have

been introduced at previous time steps and∑ns

s=1mk,s PTsintroduced at time k. Since new PTs are introduced as newmeasurements are incorporated, the number of PT states wouldgrow indefinitely. Thus, for the development of a feasibleMTT method, a suboptimum pruning step removing PTs isemployed; this will be further discussed in Section IX-A6.

With a vector-type model in which the number of targets isunknown, the derivation of a likelihood function of the formf(zk,s,mk,s|yk) is complicated by the fact that the numberof PT states depends on the number of measurements5 mk,s.Thus, contrary to the case of a known number of targets,described in Sections IV and V, we will use a derivationof the joint posterior pdf f(y1:k,a1:k, b1:k|z1:k) that doesnot involve a likelihood function. The joint posterior pdfand the corresponding factor graph will be the basis for thedevelopment of scalable MTT methods in Section IX. We firstestablish some pdfs and pmfs to be used in the derivation ofthe joint posterior pdf.

C. State-Transition pdf

For each PT state y(j)k−1, j ∈ {1, . . . , jk−1} at time k −

1, there is one legacy PT state y(j)k at time k. According to

A1 and A2, the state-transition pdf for legacy PT state yk!

[

y(1)Tk · · · y(jk−1)T

k

]Tfactorizes as

f(yk|yk−1) =

jk−1∏

j=1

f(

y(j)k

∣

∣y(j)k−1

)

, (53)

where the single-target augmented state-transition pdf

f(

y(j)k

∣

∣y(j)k−1

)

= f(

x(j)k , r(j)k

∣

∣x(j)k−1, r

(j)k−1

)

is given as follows.

If PT j does not exist at time k − 1, i.e., r(j)k−1 = 0, then it

does not exist at time k either, i.e., r(j)k =0, and thus its state

pdf is fD

(

x(j)k

)

. This means that

f(

x(j)k , r(j)k

∣

∣x(j)k−1, r

(j)k−1=0

)

=

{

fD

(

x(j)k

)

, r(j)k =0

0 , r(j)k =1.(54)

On the other hand, if PT j exists at time k− 1, i.e., r(j)k−1=1,then, using Vu3, the probability that it still exists at time k, i.e.,

5This issue can be addressed by using a set-type model as discussed inSection XI.

r(j)k =1, is given by the survival probability ps

(

x(j)k−1

)

, and if

it still exists at time k, its state x(j)k is distributed according

to the state-transition pdf f(

x(j)k

∣

∣x(j)k−1

)

. Thus,

f(

x(j)k , r(j)k

∣

∣x(j)k−1, r

(j)k−1=1

)

=

⎧

⎨

⎩

(

1−ps

(

x(j)k−1

))

fD

(

x(j)k

)

, r(j)k =0

ps

(

x(j)k−1

)

f(

x(j)k

∣

∣x(j)k−1

)

, r(j)k =1.(55)

We note that the difference of the state-transition model (53)–(55) from the state-transition model for a known number oftargets in (8) is due to the fact that targets may disappear.

According to Vu7, y0 is empty. Thus, no state transition (53)can be performed at time k=1 and, consequently, y

1is empty

as well. For future use, we formally introduce f(y(j)1

|y(j)0 )! 1

and f(y1|y0)! 1.

D. Conditional pdf of DA Vector, New PT State, and Number

of Measurements

Using A3–A6 and Vu4, the prior pmf of the DA vector

ak,s ![

a(1)k,s · · · a

(jk,s)k,s

]T, the vector of binary existence vari-

ables for new PTs rk,s ![

r(1)k,s · · · r

(mk,s)k,s

]T, and the number

of measurements mk,s conditioned on the vector of legacy PTstates y

k,sis obtained as (see [144] for a derivation)

p(

ak,s, rk,s,mk,s

∣

∣yk,s

)

=1

mk,s!e−µ(s)

n(

µ(s)n

)|Nrk,s|e−µ(s)

c(

µ(s)c

)mk,s−|Dak,s|−|Nrk,s

|

× ψ(ak,s)

(

∏

m∈Nrk,s

Γ(m)ak,s

)(

∏

j∈Dak,s

r(j)k,sp(s)d

(

x(j)k,s

)

)

×∏

j′/∈Dak,s

(

1− r(j)k,sp(s)d

(

x(j′)k,s

))

. (56)

Here, ψ(ak,s) and Dak,sare defined as in Section IV-B but

with i, i′, and nt replaced by j, j′, and jk,s, respectively;j /∈ Dak,s

is short for j ∈ {1, . . . , jk,s}\Dak,s; Nrk,s

is the

set of new PTs that exist at time k, i.e., Nrk,s!{

m ∈

{1, . . . ,mk,s} : r(m)k,s =1

}

; and Γ(m)ak,s is defined as

Γ(m)ak,s

!

{

0 , ∃j ∈ {1, . . . , jk,s} such that a(j)k,s=m

1 , otherwise.

Together, the factors ψ(ak,s) and∏

m∈Nrk,sΓ(m)ak,s

en-

force A4, i.e., p(ak,s, rk,s,mk,s|yk,s) = 0 only if

each measurement is associated either with a legacy PTor with a new PT or with clutter, and no measure-ment is associated with more than one PT. Note thatp(

ak,s, rk,s,mk,s

∣

∣yk,s

)

in (56) is a valid pmf in the sense that∑∞

mk,s=0

∑

rk,s∈{0,1}mk,s

∑

ak,sp(

ak,s, rk,s,mk,s

∣

∣yk,s

)

=1 for arbitrary y

k,s. The vector of legacy PTs y

k,smay be empty, i.e., jk,s = 0. In that case, Dak,s

and {1, . . . , jk,s}\Dak,sare empty, and an expression of

p(ak,s, rk,s,mk,s|yk,s) = p(ak,s, rk,s,mk,s) can be obtained

by setting the two products in (56) involving Dak,sto 1.


To better understand the structure of p(ak,s, rk,s,mk,s

|yk,s

), we next consider four special cases. In the no-new-

detections, no-legacy-detections, no-clutter case, i.e., mk,s =0, rk,s= 0, and ak,s= 0, expression (56) reduces to

p(

ak,s= 0, rk,s= 0,mk,s= 0∣

∣yk,s

)

= e−µ(s)n e−µ(s)

c

jk,s∏

j=1

(

1− r(j)k,sp(s)d

(

x(j)k,s

))

.

Here, e−µ(s)n is the Poisson pmf of the number of newly

detected targets evaluated at 0, e−µ(s)c is the Poisson pmf

of the number of clutter measurements evaluated at 0, and∏jk,s

j=1

(

1− r(j)k,sp(s)d

(

x(i)k,s

))

is the probability that no survivedtarget is detected by sensor s. In the no-new-detections, no-

legacy-detections, all-clutter case, i.e., mk,s > 0, rk,s = 0,and ak,s= 0,

p(

ak,s= 0, rk,s= 0,mk,s

∣

∣yk,s

)

= e−µ(s)n

e−µ(s)c (µ(s)

c )mk,s

mk,s!

jk,s∏

j=1

(

1− r(j)k,sp(s)d

(

x(j)k,s

))

,

where e−µ(s)c (µ(s)

c )mk,s/mk,s! is the Poisson pmf of the num-ber of clutter measurements evaluated at mk,s. In the all-new-

detections, no-legacy-detections, no-clutter case, i.e., rk,s= 1

and ak,s= 0,

p(

ak,s= 0, rk,s= 1,mk,s

∣

∣yk,s

)

=e−µ(s)

n (µ(s)n )mk,s

mk,s!e−µ(s)

c

jk,s∏

j=1

(

1− r(j)k,sp(s)d

(

x(j)k,s

))

,

where e−µ(s)n (µ(s)

n )mk,s/mk,s! is the Poisson pmf of the num-ber of newly detected targets evaluated at mk,s. In the no-new-

detections, all-legacy-detections, no-clutter case, i.e., mk,s =

jdk,s, rk,s = 0, and ak,s = ad

k,s , where jdk,s !

∑jk,s

j=1 r(j)k,s is

the number of existing legacy PTs and adk,s is any vector that

assigns exactly one measurement to each existing legacy PT,

p(

ak,s= adk,s, rk,s= 0,mk,s= jd

k,s

∣

∣yk,s

)

=1

jdk,s!

e−µ(s)n e−µ(s)

c

∏

j∈Dak,s

r(j)k,sp(s)d

(

x(j)k,s

)

.

Here, e−µ(s)n and e−µ(s)

c are again the Poisson pmfs of the num-ber of newly detected targets and the number of clutter mea-

surements evaluated at 0, respectively;∏

j∈Dak,sr(j)k,sp

(s)d

(

x(j)k,s

)

is the probability that all existing legacy PTs are detectedby sensor s, and the factor 1/jd

k,s! arises because there are

jdk,s! different measurement-target associations and, thus, jd

k,s!different ad

k,s .

We can express (56) more compactly as

p(ak,s, rk,s,mk,s|yk,s)

= C(mk,s)ψ(ak,s)

( jk,s∏

j=1

q1(

x(j)k,s, r

(j)k,s, a

(j)k,s;mk,s

)

)

×∏

m∈Nrk,s

µ(s)n

µ(s)c

Γ(m)ak,s

, (57)

where C(mk,s) is a normalization factor that depends only on

mk,s and q1(

x(j)k,s, r

(j)k,s, a

(j)k,s;mk,s

)

is defined as

q1(

x(j)k,s, 1, a

(j)k,s;mk,s

)

!

⎧

⎪

⎪

⎨

⎪

⎪

⎩

p(s)d (x(j)k,s)

µ(s)c

, a(j)k,s∈ {1, . . . ,mk,s}

1−p(s)d

(

x(j)k,s

)

, a(j)k,s = 0

q1(

x(j)k,s, 0, a

(j)k,s;mk,s

)

! 1(

a(j)k,s

)

, (58)

where 1(a) denotes the indicator function of the event a= 0(i.e., 1(a) is 1 if a= 0 and 0 otherwise).

Next, using Vu5 and the fact that new PT states with rk,s =1 represent newly detected targets, the prior pdf of the statesxk,s of new PTs conditioned on rk,s and mk,s is obtained as

f(xk,s|rk,s,mk,s) =

(

∏

m′∈Nrk,s

fn

(

x(m′)k,s

)

)

∏

m/∈Nrk,s

fD

(

x(m)k,s

)

,

(59)where m /∈Nrk,s

is short for m∈{1, . . . ,mk,s}\Nrk,s.

Finally, using Vu6, we obtain for the conditional pdf of ak,s,yk,s, and mk,s

f(ak,s,yk,s,mk,s|yk,s)

= f(xk,s|ak,s, rk,s,mk,s,yk,s) p(ak,s, rk,s,mk,s|yk,s

)

= f(xk,s|rk,s,mk,s) p(ak,s, rk,s,mk,s|yk,s).

Inserting (57) and (59), we obtain further

f(ak,s,yk,s,mk,s|yk,s)

= C(mk,s)ψ(ak,s)

( jk,s∏

j=1

q1(

x(j)k,s, r

(j)k,s, a

(j)k,s;mk,s

)

)

×

mk,s∏

m=1

v1(

x(m)k,s , r

(m)k,s ,ak,s

)

, (60)

where v1(

x(m)k,s , r

(m)k,s ,ak,s

)

is defined as

v1(

x(m)k,s , 1,ak,s

)

!

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0 , ∃j ∈ {1, . . . , jk,s}

such that a(j)k,s=mµ(s)

n

µ(s)c

fn

(

x(m)k,s

)

, otherwise(61)

and

v1(

x(m)k,s , 0,ak,s

)

! fD

(

x(m)k,s

)

. (62)

E. Conditional pdf of the Measurements

Using A4, A7, A8, and A9, the dependence of the measure-ment vector zk,s on y

k,s, yk,s, ak,s, and mk,s is described by

the conditional pdf (cf. (12))

f(zk,s|yk,s,yk,s,ak,s,mk,s)

=

(mk,s∏

m=1

f (s)c

(

z(m)k,s

)

)(

∏

j∈Dak,s

f (s)(

z(a

(j)k,s

)

k,s

∣

∣

∣x(j)k,s

)

)

×∏

m′∈Nrk,s

f (s)(

z(m′)k,s

∣

∣x(m′)k,s

)

. (63)


(Note that this expression presupposes that rk,s and ak,s areconsistent with A4.) We can write (63) as

f(zk,s|yk,s,yk,s,ak,s,mk,s)

= C(zk,s)

( jk,s∏

j=1

q2(

x(j)k,s, r

(j)k,s, a

(j)k,s; zk,s

)

)

×

mk,s∏

m=1

v2(

x(m)k,s , r

(m)k,s ; z

(m)k,s

)

, (64)

where C(zk,s) is a normalization factor that depends only

on zk,s (and, thus, also on mk,s), q2(

x(j)k,s, r

(j)k,s, a

(j)k,s; zk,s

)

isdefined as

q2(

x(j)k,s, 1, a

(j)k,s; zk,s

)

!

⎧

⎪

⎨

⎪

⎩

f (s)(

z(m)k,s

∣

∣x(j)k,s

)

, a(j)k,s = m∈ {1,. . . ,mk,s}

1 , a(j)k,s = 0

q2(

x(j)k,s, 0, a

(j)k,s; zk,s

)

! 1, (65)

and v2(

x(m)k,s , r

(m)k,s ; z

(m)k,s

)

is defined as

v2(

x(m)k.s , r

(m)k,s ; z

(m)k,s

)

!

{

f (s)(

z(m)k,s

∣

∣x(m)k,s

)

, r(m)k,s = 1

1 , r(m)k,s = 0.

(66)

The vector of legacy PTs yk,s

may be empty, i.e., jk,s = 0.

In that case, an expression of f(zk,s|yk,s,yk,s,ak,s,mk,s) =

f(zk,s|yk,s,ak,s,mk,s) can be obtained by replacing in (64)the product involving jk,s by 1.

F. Conditional pdf of Measurements, Number of Measure-

ments, DA Vector, and Augmented Target States

The pdf of zk,s, ak,s, mk,s, and yk,s conditioned on yk,s

is

obtained as

f(zk,s,ak,s,mk,s,yk,s|yk,s)

= f(ak,s,yk,s,mk,s|yk,s)f(zk,s|yk,s

,yk,s,ak,s,mk,s), (67)

with f(ak,s,yk,s,mk,s|yk,s) given by (60)

and f(zk,s|yk,yk,s,ak,s,mk,s) given by (64).

Note that f(zk,s,ak,s,mk,s,yk,s|yk,s) in (67)

is a valid hybrid pdf/pmf in the sense that∑∞

mk,s=0

∑

ak,s

∑

rk,s∈{0,1}mk,s

∫ ∫

f(zk,s,ak,s,mk,s,yk,s|yk,s

)dzk,sdxk,s = 1 for arbitray yk,s

. Using A10 and

the fact that the new PT states yk,s′ related to sensorss′=1, . . . , s−1 become legacy PT states at successive sensorss, . . . , ns, the conditional pdf of zk, ak, mk, and yk given y

kis obtained as (cf. (15))

f(zk,ak,mk,yk|yk) =

ns∏

s=1

f(zk,s,ak,s,mk,s,yk,s|yk,s),

(68)where f(zk,s,ak,s,mk,s,yk,s|yk,s

) is given by (67).

Next, we develop the conditional pdf of zk, ak, mk, and ykgiven yk−1. We obtain

f(zk,ak,mk,yk|yk−1)

= f(zk,ak,mk,yk,yk|yk−1)

= f(zk,ak,mk,yk|yk,yk−1)f(yk

|yk−1)

= f(zk,ak,mk,yk|yk)f(y

k|yk−1), (69)

where A11 was used. (Note that A11 implies an analogous

assumption in which the x(i)k′ , k′ = k in A11 are replaced by

y(j)k′ , k′ = k.) In case yk−1 is empty, y

kis empty as well and

(69) reduces to f(zk,ak,mk,yk|yk−1) = f(zk,ak,mk,yk).Using again A11 and the chain rule, the unconditional jointpdf for all times up to k is given by

f(z1:k,a1:k,m1:k,y1:k) =k∏

k′=1

f(zk′ ,ak′ ,mk′ ,yk′ |yk′−1),

(70)

where f(z1,a1,m1,y1|y0) = f(z1,a1,m1,y1) = f(z1,a1,m1,y1).

G. Joint Posterior pdf and Factor Graph

Next, similarly to Section V, we derive the joint poste-rior pdf f(y1:k,a1:k, b1:k|z1:k) and the corresponding fac-tor graph. As in Section V-A, we start by performingthe first stretching step, i.e., we replace f(y1:k|z1:k) byf(y1:k,a1:k|z1:k). With z1:k observed, we obtain for thestretched joint posterior pdf

f(y1:k,a1:k|z1:k) = f(y1:k,a1:k|z1:k,m1:k)

∝ f(z1:k,a1:k,m1:k,y1:k)

=k∏

k′=1

f(zk′ ,ak′ ,mk′ ,yk′ |yk′−1),

where (70) was used. Next, using (69) and subsequently (68)and (67), we obtain further

f(y1:k,a1:k|z1:k)

∝k∏

k′=1

f(yk′|yk′−1)

ns∏

s=1

f(ak′,s,yk′,s,mk′,s|yk′,s)

× f(zk′,s|yk′,s,yk′,s,ak′,s,mk′,s),

where we recall that f(y1|y0) = 1. Finally, inserting (53),

(60), and (64) yields

f(y1:k,a1:k|z1:k)

∝k∏

k′=1

( jk′−1∏

j′=1

f(

y(j′)k′

∣

∣y(j′)k′−1

)

)

ns∏

s=1

ψ(ak′,s)

×

( jk′,s∏

j=1

q(

x(j)k′,s, r

(j)k′,s, a

(j)k′,s; zk′,s

)

)

×

mk′,s∏

m=1

v1(

x(m)k′,s , r

(m)k′,s ,ak′,s

)

v2(

x(m)k′,s , r

(m)k′,s ; z

(m)k′,s

)

, (71)

with f(y(j)1

|y(j)0 ) = 1 and

q(

x(j)k,s, r

(j)k,s, a

(j)k,s; zk,s

)

! q1(

x(j)k,s, r

(j)k,s, a

(j)k,s;mk,s

)

q2(

x(j)k,s, r

(j)k,s, a

(j)k,s; zk,s

)

. (72)

Next, we perform the second stretching step, simi-larly to Section V-B, i.e., we replace f(y1:k,a1:k|z1:k) byf(y1:k,a1:k, b1:k|z1:k). This is done by replacing ψ(ak,s)


in (71) by ψ(ak,s , bk,s) =∏jk,s

j=1

∏mk,s

m=1 Ψj,m

(

a(j)k,s , b(m)k,s

)

(cf. (21)) with Ψj,m

(

a(j)k,s , b(m)k,s

)

given by (22). The fac-torization (71) can now be simplified by the followingmodification. Let ak,s and bk,s be a valid pair of DAvectors in the sense that ψ(ak,s , bk,s) = 1. Then, the

condition in (61), “∃j ∈ {1, . . . , jk,s} such that a(j)k,s =

m”, is equal to “b(m)k,s ∈ {1, . . . , jk,s}”. By inspecting

v1(

x(m)k,s , r

(m)k,s ,ak,s

)

defined in (61) and (62) as well as

v2(

x(m)k,s , r

(m)k,s ; z

(m)k,s

)

defined in (66), one can conclude that

the product v1(

x(m)k,s , r

(m)k,s ,ak,s

)

v2(

x(m)k,s , r

(m)k,s ; z

(m)k,s

)

in (71)

can be replaced by v(

x(m)k,s , r

(m)k,s , b

(m)k,s ; z

(m)k,s

)

defined as

v(

x(m)k,s , 1, b

(m)k,s ; z

(m)k,s

)

!

⎧

⎨

⎩

0 , b(m)k,s ∈ {1, . . . , jk,s}

µ(s)n

µ(s)c

fn

(

x(m)k,s

)

f (s)(

z(m)k,s

∣

∣x(m)k,s

)

, b(m)k,s = 0 (73)

and

v(

x(m)k,s , 0, b

(m)k,s ; z

(m)k,s

)

= fD

(

x(m)k,s

)

. (74)

With this simplification, the stretched marginal posterior pdfreplacing (71) is obtained as

f(y1:k,a1:k, b1:k|z1:k)

∝k∏

k′=1

( jk′−1∏

j′=1

f(

y(j′)k′

∣

∣y(j′)k′−1

)

)

×ns∏

s=1

( jk′,s∏

j=1

q(

x(j)k′,s, r

(j)k′,s, a

(j)k′,s; zk′,s

)

×

mk′,s∏

m′=1

Ψj,m′

(

a(j)k′,s , b(m′)k′,s

)

)

×

mk′,s∏

m=1

v(

x(m)k′,s , r

(m)k′,s , b

(m)k′,s ; z

(m)k′,s

)

. (75)

Here, we recall that j0 = 0 (cf. Vu7). We note thatf(y1:k,a1:k, b1:k|z1:k) is still consistent with the orig-inal joint posterior pdf f(y1:k|z1:k) in the sense that∑

a1:k

∑

b1:kf(y1:k,a1:k, b1:k|z1:k) = f(y1:k|z1:k), which

means that the marginal posterior pdfs f(

y(j)k

∣

∣z1:k)

calculatedfrom f(y1:k,a1:k, b1:k|z1:k) are equal to the ones calculatedfrom f(y1:k|z1:k).

The factorization in (75) can now be represented by a factorgraph. This factor graph is shown for the single-sensor case(ns=1) in Fig. 4. As a difference from the factor graph for aknown, fixed number of targets shown in Fig. 2(c), for each

measurement, an additional variable node “y(m)k,s ” representing

the state of a new PT is introduced.

IX. SPA-BASED VECTOR-TYPE MTT METHODS FOR AN

UNKNOWN, TIME-VARYING NUMBER OF TARGETS

In this section, building on the vector-type system modelpresented in Section VIII and the factor graph shown inFig. 4, we develop total-SPA MTT methods for trackingan unknown, time-varying number of targets. An important

building block of these methods is the “bipartite” formulationof the probabilistic DA problem developed in Section VIII-Gand represented graphically in Fig. 4. This formulation enablesthe application of SPADA, which leads to excellent scalabilityand, thus, suitability even for large tracking scenarios withclosely spaced targets. The proposed SPA-based MTT methodsare improved variants of the total-SPA method presented in[53] and can be seen as SPA-based versions of the JIPDA filter[2], [63]. Differently from the method in [53] and the JIPDAfilter, they do not use a heuristic to model the generation ofnew PTs. A particle-based implementation is described in [53].

A. Single-Sensor MTT

We first consider the single-sensor case (ns = 1) and thusdrop the sensor index s. For MTT with an unknown number of

targets, approximations f(

y(j)k

)

and f(

y(m)k

)

of the marginal

posterior pdfs f(

y(j)k

∣

∣z1:k)

and f(

y(m)k

∣

∣z1:k)

can be obtainedin an efficient way by running the iterative SPA on the loopyfactor graph in Fig. 4. As before, messages are only sentforward in time. The generic SPA rules in Section III-B thenyield the following operations at time k. (With an abuse ofnotation, we denote messages that are analogous to messagesin previously presented methods in the same way, even if thefunctional forms are different.)

1) Prediction: First, the messages αk

(

y(j)k

)

=

αk

(

x(j)k , r(j)k

)

passed from the factor nodes “f(

y(j)k

∣

∣y(j)k−1

)

”

to the variable nodes “y(j)k ” are calculated according to

αk

(

x(j)k , r(j)k

)

=∑

r(j)k−1∈{0,1}

∫

f(

x(j)k , r(j)k

∣

∣x(j)k−1, r

(j)k−1

)

× f(

x(j)k−1, r

(j)k−1

)

dx(j)k−1 .

Here, f(

x(j)k−1, r

(j)k−1

)

was calculated at the previous time k−1.

Inserting (54) and (55), we obtain for r(j)k =1

αk

(

x(j)k , 1

)

=

∫

ps

(

x(j)k−1

)

f(

x(j)k

∣

∣x(j)k−1

)

f(

x(j)k−1, 1

)

dx(j)k−1 ,

(76)

and for r(j)k =0 we have (cf. (51)) αk

(

x(j)k , 0

)

= α(j)k fD

(

x(j)k

)

with

α(j)k = f (j)

k−1 +

∫

(

1−ps

(

x(j)k−1

))

f(

x(j)k−1, 1

)

dx(j)k−1 , (77)

where f (j)k−1 =

∫

f(

x(j)k−1, 0

)

dx(j)k−1. (Note also

that α(j)k =

∫

αk

(

x(j)k , 0

)

dx(j)k ; furthermore,

f(

x(j)k−1, 0

)

= f (j)k−1fD

(

x(j)k−1

)

according to (51).) Because

f(

x(j)k−1, r

(j)k−1

)

is normalized, so is αk

(

x(j)k , r(j)k

)

, i.e.,∑

r(j)k

∈{0,1}

∫

αk

(

x(j)k , r(j)k

)

dx(j)k = 1. Thus, we also have

α(j)k = 1−

∫

αk

(

x(j)k , 1

)

dx(j)k .

2) Measurement Evaluation: Once αk

(

x(j)k , 1

)

and α(j)k

have been calculated, a “measurement evaluation” step isperformed for both the legacy PTs and the new PTs. For the

legacy PTs, the messages β(j)k

(

a(j)k

)

passed from the factor

nodes “q(

x(j)k , r(j)k , a(j)k ; zk

)

” to the variable nodes “a(j)k ” arecalculated according to

β(j)k

(

a(j)k

)


f1−

fnp

−

α1

αnp

γ1

γnp

α1

αnp

β1

βnp

κ1

κnp

ν1,1

νnm,1

ν1,np

νnm,np

ϕ1,1

ϕnp,1

ϕ1,nm

ϕnp,nm

ξ1

ξnm

ι1

ιnm

ς1

ςnm

k−2 k−1 k

y1−

ynp

−

f1−

fnp

−

y1−

ynm

−

q1−

qnp

−

v1−

vnm

−

a1−

anp

−

b1−

bnm

−

y1

ynp

y1=

ynp=

f1

fnp

y 1

ynm

q1

qnp

v1

vnm

a1

anp

b1

bnm

Ψ1,1−

Ψnp,nm

−

Ψ1,nm

−

Ψnp,1−

Ψ1,1

Ψnp,nm

Ψ1,nm

Ψnp,1

Fig. 4. Factor graph for single-sensor MTT with an unknown, time-varying number of targets, corresponding to the factorization (75) for ns = 1. Twocomplete consecutive sections of the factor graph (for times k−1 and k) are shown; the section for time k also depicts the messages passed between adjacentnodes. Factor nodes in green represent factors related to the state-transition function, factor nodes in red represent the remaining factors, and messages are

depicted in blue. The time index k and the sensor index s are omitted, and the following short notations are used: yj= ! y

(j)k−2, nm ! mk−1, np ! jk−2,

yj−

! y(j)k−1, ym

−! y

(m)k−1, a

j−

! a(j)k−1, bm

−! b

(m)k−1, nm ! mk , np ! jk−1, yj ! y

(j)k , ym ! y

(m)k , aj ! a

(j)k , bm ! b

(m)k , f

j−! f

!

y(j)k−1

"

"y(j)k−2

#

,

fj ! f!

y(j)k

"

"y(j)k−1

#

, qj−! q

!

x(j)k−1, r

(j)k−1, a

(j)k−1;zk−1

#

, Ψj,m−

! Ψj,m

!

a(j)k−1, b

(m)k−1

#

, vm−

! v!

x(m)k−1, r

(m)k−1, b

(m)k−1;z

(m)k−1

#

, qj ! q!

x(j)k , r

(j)k , a

(j)k ; zk

#

,

Ψj,m ! Ψj,m

!

a(j)k , b

(m)k

#

, vm ! v!

x(m)k , r

(m)k , b

(m)k ;z(m)

k

#

, fj−

! f!

y(j)k−1

#

, αj ! αk

!

y(j)k

#

, γj ! γ(j)k

!

y(j)k

#

, κj ! κ(j)k

!

a(j)k

#

, βj ! β(j)k

!

a(j)k

#

,

νm,j ! ν[ℓ]

Ψj,m→a(j)k

!

a(j)k

#

, ϕj,m ! ϕ[ℓ]

Ψj,m→b(m)k

!

b(m)k

#

, ξm ! ξ(m)k

!

b(m)k

#

, ιm ! ι(m)k

!

b(m)k

#

, and ςm ! ς(m)k

!

y(m)k

#

.

=∑

r(j)k

∈{0,1}

∫

q(


)

αk

(

x(j)k , r(j)k

)

dx(j)k

=

∫

q(

x(j)k , 1, a(j)k ; zk

)

αk

(

x(j)k , 1

)

dx(j)k + 1

(

a(j)k

)

α(j)k . (78)

In the last expression, we used q(

x(j)k , 0, a(j)k ; zk

)

= 1(

a(j)k

)

,which follows from (72) along with (65) and (58). For the new

PTs, the messages ξ(m)k

(

b(m)k

)

passed from the factor nodes

“v(

x(m)k , r(m)

k , b(m)k ; z(m)

k

)

” to the variable nodes “b(m)k ” are

calculated according to

ξ(m)k

(

b(m)k

)

=∑

r(m)k

∈{0,1}

∫

v(

x(m)k , r(m)

k , b(m)k ; z(m)

k

)

dx(m)k

=

∫

v(

x(m)k , 1, b(m)

k ; z(m)k

)

dx(m)k + 1 , (79)

where (74) was used. Inserting (73), we find that for b(m)k = 0,

expression (79) simplifies to

ξ(m)k (0) =

µn

µc

∫

fn

(

x(m)k

)

f(

z(m)k

∣

∣x(m)k

)

dx(m)k + 1 ,

and for b(m)k = 0, it simplifies to ξ(m)

k

(

b(m)k

)

= 1.

For future reference, we note that the belief of the joint DAvector [aT

k bTk]

T is obtained as

p(ak , bk) ∝ ψ(

ak, bk)

( jk−1∏

j=1

β(j)k

(

a(j)k

)

)

mk∏

m=1

ξ(m)k

(

b(m)k

)

,

(80)where ψ

(

ak, bk)

is given by ψ(ak , bk) =∏jk−1

j=1

∏mk

m=1 Ψj,m

(

a(j)k , b(m)k

)

(cf. (21)).

3) Iterative probabilistic DA: Next, probabilistic DA isperformed using the iterative SPA-based algorithm from Sec-tion VI-A, which yields at each iteration ℓ the messages

ϕ[ℓ]

Ψj,m→b(m)k

(

b(m)k

)

and ν[ℓ]Ψj,m→a(j)

k

(

a(j)k

)

. Here, the messages

β(j)k

(

a(j)k

)

from (78) are used in (27) and (29), and the

messages ξ(m)k

(

b(m)k

)

from (79) are used in (28). After the

last iteration ℓ = nit, the messages passed from “a(j)k ” to

“q(


)

” are obtained as

κ(j)k

(

a(j)k

)

=mk∏

m=1

ν[nit]

Ψj,m→a(j)k

(

a(j)k

)

, (81)

and the messages passed from “b(m)k ” to “v

(

x(m)k , r(m)

k , b(m)k ;

z(m)k

)

” are obtained as

ι(m)k

(

b(m)k

)

=

jk−1∏

j=1

ϕ[nit]

Ψj,m→b(m)k

(

b(m)k

)

. (82)

For an efficient implementation, one can use SPADA. The

messages ν[nit](m→j)k in (31) and ϕ[nit](j→m)

k in (30) then have

to be converted into the messages ν[nit]

Ψj,m→a(j)k

(

a(j)k

)

used in

(81) and ϕ[nit]

Ψj,m→b(m)k

(

b(m)k

)

used in (82), respectively. For

ν[nit](m→j)k , this is done as explained in Section VII-A2. For

ϕ[nit](j→m)k , similarly, ϕ[nit]

Ψj,m→b(m)k

(

b(m)k

)

= ϕ[nit](j→m)k for

b(m)k = j and ϕ[nit]

Ψj,m→b(m)k

(

b(m)k

)

=1 for b(m)k = j.

4) Measurement Update: A “measurement update” step isnow performed for both the legacy PTs and the new PTs.

For the legacy PTs, the messages γ(j)k

(

x(j)k , r(j)k

)

passed from


“q(


)

” to “y(j)k ” are calculated according to

γ(j)k

(

x(j)k , 1

)

=mk∑

a(j)k

=0

q(

x(j)k , 1, a(j)k ; zk

)

κ(j)k

(

a(j)k

)

and γ(j)k

(

x(j)k , 0

)

= γ(j)k with

γ(j)k = κ(j)k (0).

For the new PTs, the messages ς(m)k

(

x(m)k , r(m)

k

)

passed from

“v(

x(m)k , r(m)

k , b(m)k ; z(m)

k

)

” to “y(m)k ” are calculated as

ς(m)k

(

x(m)k , 1

)

=µn

µcfn

(

x(m)k

)

f(

z(m)k

∣

∣x(m)k

)

ι(m)k (0)

and ς(m)k

(

x(m)k , 0

)

= ς(m)k fD

(

x(m)k

)

with

ς(m)k =

jk−1∑

b(m)k

=0

ι(m)k

(

b(m)k

)

.

5) Belief Calculation: Finally, for the legacy PTs, beliefs

f(

y(j)k

)

= f(

x(j)k , r(j)k

)

approximating the marginal posterior

pdfs f(

y(j)k

∣

∣z1:k)

= f(

x(j)k , r(j)k

∣

∣z1:k)

are obtained as

f(

x(j)k , 1

)

=1

C(j)k

αk

(

x(j)k , 1

)

γk(

x(j)k , 1

)

and f(

x(j)k , 0

)

= f (j)k

fD

(

x(j)k

)

with

f (j)k

=1

C(j)k

α(j)k γ(j)k , (83)

where C(j)k !

∫

αk

(

x(j)k , 1

)

γ(j)k

(

x(j)k , 1

)

dx(j)k + α(j)

k γ(j)k .

Similarly, for the new PTs, beliefs f(

y(m)k

)

= f(

x(m)k , r(m)

k

)

approximating the marginal posterior pdfs f(

y(m)k

∣

∣z1:k)

=

f(

x(m)k , r(m)

k

∣

∣z1:k)

are obtained as

f(

x(m)k , 1

)

=1

C(m)k

ς(m)k

(

x(m)k , 1

)

(84)

and f(

x(m)k , 0

)

= f(m)k fD

(

x(m)k

)

with

f(m)k =

1

C(m)k

ς(m)k , (85)

where C(m)k !

∫

ς(m)k

(

x(m)k , 1

)

dx(m)k + ς(m)

k .

6) Target Declaration, State Estimation, and Pruning: For

the legacy PTs, beliefs p(

r(j)k

)

approximating the marginal

posterior pmfs p(

r(j)k

∣

∣z1:k)

of the existence indicators r(j)k are

obtained from the beliefs f(

x(j)k , r(j)k

)

as

p(

r(j)k

)

=

∫

f(

x(j)k , r(j)k

)

dx(j)k .

Target detection—hereafter termed “target declaration” toavoid confusion with the detection performed by the sensors—

is now performed by comparing p(

r(j)k = 1)

to a thresholdPth, i.e., legacy PT j is declared to exist at time k if

p(

r(j)k =1)

>Pth [149, Ch. 2]. For these PTs j, state estimationis then performed, e.g., using (3) with f(xk|z1:k) replaced

by f(

x(j)k , r(j)k = 1

)

/p(

r(j)k = 1)

. For the new PTs, target

declaration and state estimation are done as for the legacy

PTs but with f(

x(j)k , r(j)k

)

replaced by f(

x(m)k , r(m)

k

)

.

Finally, a pruning step is typically performed. According to(52), the number of PTs would grow with time k. Therefore,

legacy and new PTs whose existence beliefs p(

r(j)k = 1)

and p(

r(m)k = 1

)

, respectively are below a threshold Ppr areremoved.

B. Multisensor MTT

For multiple sensors (ns ≥ 2), the factor graph differsfrom that for a single sensor shown in Fig. 4 in that anadditional section of the factor graph is introduced for eachadditional sensor, and for each time k, there is a factornode “f

(

y(j)k

∣

∣y(j)k−1

)

” in the section corresponding to the firstsensor. The total-SPA MTT method for an unknown, time-varying number of targets presented in the previous subsectioncan be extended to the case of multiple sensors by using asensor-sequential or sensor-parallel processing. In either case,the first step is the prediction step, i.e., calculation of the

messages αk

(

x(j)k , r(j)k

)

according to (76) and (77).

1) Sequential Processing: As in Section VII-C1, messagepassing operations are performed sequentially for each sensors = 1, . . . , ns. (However, contrary to MTT methods for aknown number of targets, the sequence of sensors was alreadydefined in the derivation of the factor graph, see SectionVIII.) Let us denote the beliefs—for both the legacy andnew PTs—calculated at the (s−1)th sensor update step (i.e.,

at sensor s − 1) as fs−1

(

x(j)k,s, r

(j)k,s

)

, j = 1, . . . , jk,s. (We

recall from Section VIII-B that yk,s

= [yTk,s−1

yTk,s−1]

T and

jk,s = jk,s−1 + mk,s−1.) These beliefs serve as input to thesth sensor update step (i.e., at sensor s), in which the messagepassing and belief calculation operations described in SectionsIX-A2 through IX-A5 are performed. This sensor-recursive

processing is initialized by f0(

x(j)k,1, r

(j)k,1

)

= αk

(

x(j)k , r(j)k

)

,j = 1, . . . , jk,1, which were calculated in the predictionstep. (Recall that y

k,1= y

kand jk,1 = jk−1.) The beliefs

fns

(

x(j)k,ns

, r(j)k,ns

)

, j =1, . . . , jk,ns and fns

(

x(m)k,ns

, r(m)k,ns

)

, m=1, . . . ,mk,ns calculated at the last sensor s = ns take into

account the measurements of all sensors; they are used fortarget declaration and state estimation as described in SectionIX-A6, and to perform prediction at the next time step k+ 1.

An advantage of this sensor-sequential method over thesensor-parallel method considered next is that it performssensor fusion for new PTs and it executes SPADA only onceper sensor. On the other hand, sequential processing is notwell suited to a parallel or distributed implementation.

2) Parallel Processing: The SPA-based message passingmethod for sensor-parallel MTT can be summarized as fol-

lows. For all sensors s ≥ 2, the messages αk

(

x(j)k,s, r

(j)k,s

)

,j > jk−1 are not available and are thus initialized by

αk

(

x(j)k,s, 0

)

= fD(x(j)k,s) and αk

(

x(j)k,s, 1

)

= 0. (These mes-sages belong to legacy PTs at sensor s and to new PTsat sensors preceding sensor s with respect to the sensorordering defined in Section VIII.) Then, the message pass-ing operations described in Sections IX-A2 through IX-A4are carried out separately for each sensor s ∈ {1, . . . , ns}.


As a result, for each sensor s ∈ {1, . . . , ns}, messages6

γ(j)k,s

(

x(j)k,s, r

(j)k,s

)

= γ(j)k,s

(

x(j)k , r(j)k

)

are available for each

legacy PT j ∈ {1, . . . , jk−1}, and messages ς(m)k,s

(

x(m)k,s , r

(m)k,s

)

are available for each new PT m∈ {1, . . . ,mk,s}. Finally, be-liefs approximating the marginal posterior pdfs for the legacy

PTs j ∈ {1, . . . , jk−1}, f(

y(j)k

∣

∣z1:k)

= f(

x(j)k , r(j)k

∣

∣z1:k)

, arecalculated as (cf. (83) and (83))

f(

x(j)k , r(j)k

)

∝ α(j)k

(

x(j)k , r(j)k

)

ns∏

s=1

γ(j)k,s

(

x(j)k , r(j)k

)

.

Furthermore, beliefs approximating the marginal posterior pdfs

for the new PTs, f(

y(m)k,s

∣

∣z1:k)

= f(

x(m)k,s , r

(m)k,s

∣

∣z1:k)

, for s ∈{1, . . . , ns} and m ∈ {1, . . . ,mk,s}, are directly obtained as(cf. (84) and (85))

f(

x(m)k,s , r

(m)k,s

)

∝ ς(m)k,s

(

x(m)k,s , r

(m)k,s

)

.

This method facilitates a parallel or distributed implemen-tation. On the other hand, its performance may be poorerthan that of the sequential method described previously. Thisis because no sensor fusion is performed for new PTs, i.e.,each sensor attempts to infer new targets individually. Asdemonstrated in [53], the computational complexity of sensor-parallel processing scales strictly linearly with the number ofsensors ns. We note that, similarly to [52], sensor-parallelprocessing can be extended to multiple SPA iterations overan “outer loop” that spans across the different sensors. Inthis way, sensor fusion gains can be leveraged also for theinference of new targets. However, this comes at the cost ofan increased computational complexity since SPADA has tobe executed multiple times for each sensor (once for eachouter-loop iteration).

X. INTRODUCTION TO RFSS

We now turn to the set-type MTT methods. We first give anintroduction to RFSs, which underly the set-type system modeland MTT methods to be presented in Sections XI–XIII.

A. Basic Description

An RFS X = {x(1), . . . , x(n)} (also known as a simple finitepoint process [150], [151]) is a set-valued random variablewhose realizations X are finite sets {x(1), . . . ,x(n)} of vectorsx(1), . . . ,x(n)∈Rdx . Both the number of elements n= |X| ∈N0—the cardinality of X—and the elements x(i) are random,and the elements x(i) are unordered. Adopting the frameworkof finite set statistics (FISST) [3], an RFS can be described byits pdf fX(X ), briefly denoted f(X ). For a realization X ={x(1), . . . ,x(n)} with cardinality |X |=n, the pdf is given by

f(X ) = f({x(1), . . . ,x(n)}) = n! ρ(n)fn(x(1), . . . ,x(n)).

(86)Here, ρ(n) ! P{|X| = n} is the cardinality distribution, i.e.,the pmf of n = |X|, and fn(x(1), . . . ,x(n)) is a pdf of therandom vectors x(1), . . . , x(n) that is invariant to a permutation

6We recall that for j ∈ {1, . . . , jk−1}, y(j)k,s=y

(j)k for all s∈ {1, . . . , ns}.

of its arguments x(i). It will be convenient to define the set

integral of a real-valued set function g(X ) as [3]∫

g(X )δX !∞∑

n=0

1

n!

∫

Rndx

g({x(1), . . . ,x(n)}) dx(1) · · · dx(n).

(87)

For an RFS pdf f(X ), we have∫

f(X )δX = 1.For an RFS X, the probability hypothesis density (PHD)

D(x) : Rdx → R is defined by the property that its integralover a region S⊆Rdx equals the expected number of elementspresent in S, i.e.,

∫

S D(x)dx=E{|X∩S|} [3]. An expressionof D(x) in terms of f(X ) can be found in [3].

B. Special RFSs

Next, we review four types of RFSs that are especiallyrelevant to RFS-based MTT methods. Given the cardinal-ity |X | = n, the elements of a Poisson RFS X are iidwith some “spatial pdf” f(x), i.e., fn(x(1), . . . ,x(n)) =∏n

i=1 f(x(i)) =

∏

x∈X f(x). Furthermore, the cardinality isPoisson distributed with mean µ, i.e., ρ(n) = e−µµn/n! ,n∈ N0. Hence, (86) yields the pdf of X as

f(X ) = e−µ∏

x∈X

µf(x).

The PHD of the Poisson RFS is given by D(x) = µf(x); itis also known as intensity function.

A Bernoulli RFS either contains one element x with prob-ability of existence r or is empty with probability 1− r. Ifit is nonempty, the element x is distributed according to the“spatial pdf” s(x). Hence, the RFS pdf is given by

f(X ) =

⎧

⎨

⎩

1− r, X = ∅,rs(x), X ={x},0, otherwise.

(88)

We note that∑

j γj f(j)(X ) with normalized γj (i.e.,

∑

j γj =1) and Bernoulli pdfs f (j)(X ) is again a Bernoulli pdf.

A multi-Bernoulli (MB) RFS is the union of nc independentcomponent RFSs X(j), j = 1, . . . , nc, which are BernoulliRFSs with existence probabilities r(j) and spatial pdfs s(j)(x).Let f (j)(X ) denote the pdf of Bernoulli component X(j).To express f(X ) for a realization X = {x(1), . . . ,x(n)} ofcardinality n " nc, we introduce a mapping α of an indexj ∈ {1, . . . , nc} to an index α(j) ∈ {0, . . . , n}. This mappingis such that the set of all α(j) includes {1, . . . , n}, and j1 =j2with α(j1),α(j2) ∈ {1, . . . , n} implies α(j1) =α(j2). In ourcontext, α conveys a mapping of n of the nc Bernoulli com-ponent pdfs f (j)(X ) to single-vector element sets {x(α(j))}and the other nc−n Bernoulli component pdfs to empty sets.Let Pnc,n denote the set of all such “components-to-elements”mappings α; note that |Pnc,n| = nc!/(nc− n)!. The pdf cannow be expressed as [51]

f(X ) =∑

α∈Pnc,n

nc∏

j=1

f (j)(

X (α(j)))

, (89)

where n= |X | and X (α(j)) is {x(α(j))} for α(j) ∈ {1, . . . , n}and ∅ for α(j) = 0.

A labeled MB (LMB) RFS is an MB RFS where each statevariable is augmented by a label [108]. More specifically, a


track-labeling function τ(·) assigns to each Bernoulli com-ponent X(j), j ∈ {1, . . . , nc} a distinct label l ∈ L, whereL = {τ(j) : j ∈ {1, . . . , nc}}. The realization of an LMBRFS is a set X =

{

(x(1), l(1)), . . . , (x(n), l(n))}

of tuples

(x(i), l(i))∈Rdx×L, i∈{1, . . . , n}, with l(i)= l(i′) for i = i′.

The pdf is given by [108]

f(X ) =nc∏

j=1

f (j)(

X (τ(j)))

, (90)

where f (j)(X ) is the pdf of Bernoulli component X(j) andX (τ(j)) is given by X (τ(j)) =

{

(x(i), l(i)) ∈ X : τ(j) = l(i)}

.

For sets X ={

(x(1), l(1)), . . . , (x(n), l(n))}

that have ele-

ments with nondistinct labels, i.e., ∃i, i′ with l(i) = l(i′) =

τ(j), we have |X (τ(j))| > 1 and thus f(X ) = 0 accordingto (88). Note that in the evaluation of f(X ), a sum overall possible components-to-elements mappings as in (89) isavoided since the mapping is fixed, i.e., given by the labelsl(1), . . . , l(n).

XI. SET-TYPE SYSTEM MODEL

The system model for set-type MTT describes all thetarget states and all the measurements of a sensor as time-varying RFSs. Several set-type system models are available.In particular, the “unlabeled” model proposed in [51], [107]describes newborn targets by a Poisson RFS. The posteriorpdf consists of an MB mixture component representing de-

tected targets, which are targets that potentially exist andgenerated at least one measurement so far, and a Poissoncomponent representing undetected targets, which are targetsthat potentially exist but did not generate any measurementyet. (The latter are termed unknown targets in [51], [152].) Anew Bernoulli component is generated for each measurement,representing the hypothesis that this measurement is the firstdetection of a target that was previously undetected andthus was previously modeled by a Poisson component. Thismodel is able to maintain track continuity implicitly basedon information provided by metadata. As an alternative, the“labeled” model proposed in [108], [109] explicitly maintainstrack continuity through labels. Target birth is modeled byan LMB RFS, where each Bernoulli component representsa potential new target and has a distinct label. In cases oflimited prior birth information, one typically uses a heuristicto generate new Bernoulli components based on measurementsfrom the previous time step [109]. Such heuristics can beavoided with the MB–Poisson model [51], [107].

Here, we propose a hybrid “labeled-unlabeled” systemmodel that combines the benefits of the unlabeled and labeledmodels, i.e., generation of new Bernoulli components based onthe Poisson RFS modeling of undetected targets and indexingof already detected targets by a distinct label providing explicittrack continuity. We represent targets that have been detectedby an LMB RFS and targets that remain undetected by anunlabeled Poisson RFS in which each component is formallyaugmented by the same nonunique label lk= 0. When a targetis detected for the first time, this “dummy label” is replaced bya new unique label that identifies the underlying measurement.This is not the result of an update step but an incorporation of

exogenous information; it can be seen as an additional externalmeasurement of the label.7

A. State-Transition pdf and Prior Distribution

1) Assumptions: The entirety of the target states is modeled

as a time-varying RFS Xk !{

(x(1)k , l(1)k ), . . . , (x(ik)k , l(ik)k )}

.Furthermore, we use the following assumptions in addition tothe common assumptions A1–A11 [3], [51]:

S1: The number of targets, ik, is time-varying and unknown.S2: The target states (elements of Xk) are unordered.S3: A target i that exists at time k −1 survives (i.e., still

exists at time k) with survival probability ps

(

x(i)k−1

)

and

disappears with probability 1− ps

(

x(i)k−1

)

. The survival

probability ps

(

x(i)k−1

)

does not depend on the targetlabel. Targets that survive preserve their label.

S4: Each target i is a survived target or a newborn target.S5: The states of the newborn targets at time k are indepen-

dent of the states of the survived targets at time k.S6: The states of the newborn targets at time k are iid. Each

of them has label l(i)k = 0 and is distributed according

to the birth pdf fb

(

x(i)k

)

.S7: The number of newborn targets at time k is Poisson

distributed with mean µb.

S8: At time k = 0, all target states have label l(i)0 = 0(since no targets have been detected); they are iid and

distributed according to a prior pdf fp

(

x(i)0

)

.S9: The number of targets at time k = 0 is Poisson

distributed with mean µp.

The overall multitarget state RFS Xk is the union of theRFSs of the detected and undetected targets, i.e.,

Xk = Xdk ∪ X

uk . (91)

Here, Xdk is a labeled RFS whose elements have unique

nonzero labels, and Xuk is an unlabeled RFS whose elements

have the nonunique dummy label lk=0. The label of a detectedtarget is a unique identifier of the first measurement generatedby that target. Undetected targets become detected targets assoon as they produce a measurement; conversely, detectedtargets cannot become undetected targets.

2) Detected Targets: Let X dk−1 !

{(

x(1)k−1, l

(1)k−1

)

, . . . ,(

x(id

k−1)k−1 , l

(idk−1)

k−1

)}

comprise the states of the detected targetsthat existed at time k−1. Because of A1, A2, and S3, eachdetected target j that existed at time k−1 gives rise to a—possibly nonexistent—“survived target” at time k. The stateof this survived target is modeled as a Bernoulli RFS Xd

k,j

(cf. (88)) with r = ps

(

x(j)k−1

)

and s(xk) = f(

xk

∣

∣x(j)k−1

)

. Theassociated multitarget (RFS) state-transition pdf is thus

f(

X dk,j

∣

∣

{(

x(j)k−1, l

(j)k−1

)})

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

1−ps

(

x(j)k−1

)

, X dk,j= ∅,

ps

(

x(j)k−1

)

f(

xk

∣

∣x(j)k−1

)

, X dk,j=

{(

xk, l(j)k−1

)}

,

0, otherwise.

(92)

7If the label is viewed as a proxy for the trajectory, the formulation in [153]provides a similar result directly via a Bayes update.


In view of A2, S1, and S2, the overall multitarget stateRFS of the detected targets at time k is the LMB RFS

Xdk =

{(

x(1)k , l(1)

)

, . . . ,(

x(idk)k , l(i

dk))}

given by

Xdk =

idk−1⋃

j=1

Xdk,j . (93)

Here, because of A2, the Bernoulli components Xdk,j are

conditionally independent given Xdk−1.

The joint evolution of the detected target states is de-scribed by the multitarget state-transition pdf f(X d

k |Xdk−1),

where X dk =

{(

x(1)k , l(1)

)

, . . . ,(

x(id

k)k , l(i

dk))}

and X dk−1 =

{(

x(1)k−1, l

(1))

, . . . ,(

x(id

k−1)k−1 , l(i

dk−1)

)}

. (Note that idk " id

k−1,

where idk = id

k−1 if all survived targets exist at time k and

idk < id

k−1 otherwise.) Using (93), one obtains [108]

f(X dk |X

dk−1) =

idk−1∏

j=1

f(

X (τ(j))k

∣

∣

{(

x(j)k−1, l

(j)k−1

)})

, (94)

where f(

X (τ(j))k

∣

∣

{(

x(j)k−1, l

(j)k−1

)})

is f(

X dk,j

∣

∣

{(

x(j)k−1, l

(j)k−1

)})

in (92) evaluated at X dk,j = X (τ(j))

k ={

(x(i)k , l(i)) ∈ X d

k :τ(j) = l(i)

}

. (We recall that τ(·) is the track-labeling functionof the LMB RFS introduced in Section X-B.) As discussedafter (90), for sets X d

k with nondistinct labels, f(X dk |X

dk−1) =

0. At k=0, we have X d0 = ∅.

3) Undetected Targets: Let X uk−1 !

{(

x(1)k−1, 0

)

, . . . ,(

x(iu

k−1)k−1 , 0

)}

comprise the states of the undetected targetsthat existed at time k − 1. Because of A1 and S3, eachundetected target j that existed at time k − 1 gives rise toa—possibly nonexistent—survived target at time k. Again,the state of this (undetected) legacy target is modeled as a

Bernoulli RFS Xuk,j with pdf f

(

X uk,j

∣

∣

{(

x(j)k−1, 0

)})

given by(92) (with obvious modifications). Furthermore, there may betargets newly appearing at time k. The entirety of the states ofthese newborn targets will be described by an RFS Γk. Becauseof S6 and S7, Γk is a Poisson RFS with mean µb and spatial pdffb(xk), and all elements have label lk=0. Thus, the intensityfunction of Γk is λb(xk, lk) = µbfb(xk)1(lk). With S4 andS5, the overall multitarget state RFS of the undetected targets

at time k, Xuk =

{(

x(1)k , 0), . . . , (x

(iuk)k , 0

)}

, is then obtained as

Xuk =

( iuk−1⋃

j=1

Xuk,j

)

∪ Γk . (95)

Here, because of A2 and S5, the components Xuk,j and Γk are

all conditionally independent given Xuk−1.

The joint evolution of the undetected target states is de-scribed by the multitarget state-transition pdf f(X u

k |Xuk−1),

where X uk =

{(

x(1)k , 0

)

, . . . ,(

x(iu

k)k , 0

)}

. Let α be a mappingof an index j ∈ {1, . . . , iu

k−1} to an index α(j) ∈ {0, . . . , iuk}.

This mapping is such that j1 = j2 with α(j1),α(j2) ∈{1, . . . ,m} implies α(j1) =α(j2). Furthermore, let Miu

k−1,iuk

be the set of all possible mappings α. Using (95), (92), and(89) together with S6 and S7, one obtains [3, Ch. 13]

f(X uk |X

uk−1)

= e−µb

( iuk∏

i=1

µbfb

(

x(i)k

)

)( iuk−1∏

j=1

(

1−ps

(

x(j)k−1

))

)

×∑

α∈Miuk−1,i

uk

∏

j′:α(j′)>0

ps

(

x(j′)k−1

)

f(

x(α(j′))k

∣

∣x(j′)k−1

)

µb

(

1−ps

(

x(j′)k−1

))

fb

(

x(α(j′))k

)

.

(96)

An analysis of this expression shows that for α(j′)> 0, target

j′ survives and its state x(j′)k−1 is mapped to x

(α(j′))k , whereas

for α(j′) = 0, target j does not survive. At k=0, due to S8and S9, Xu

0 is a Poisson RFS with mean µp and spatial pdffp(x0), and all elements have label l0 =0.

B. Likelihood Function

The entirety of the measurements of sensor s at time k is

modeled as a time-varying RFS Zk,s !{

z(1)k,s , . . . , z

(mk,s)k,s

}

.

Let Xk ={(

x(1)k , l(1)k

)

, . . . ,(

x(ik)k , l(ik)k

)}

be the set of thestates of the targets at time k. Based on A5 and A9, a—possibly nonexistent—measurement zk,s originating from tar-get j is modeled as a Bernoulli RFS Zk,s,j (cf. (88)) with

r= p(s)d

(

x(j)k

)

and s(zk,s) = f(

zk,s∣

∣x(j)k

)

. Both r and s(zk,s)do not dependent on the target label. The associated RFSlikelihood function is thus given by

f(

Zk,s,j

∣

∣

{

x(j)k

})

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

1−p(s)d

(

x(j)k

)

, Zk,s,j= ∅,

p(s)d

(

x(j)k

)

f(

zk,s∣

∣x(j)k

)

, Zk,s,j={zk,s},

0, otherwise.(97)

Because of A4 and A9, the entirety of the target-originatedmeasurements then forms the MB RFS

⋃ikj=1Zk,s,j . Further-

more, the entirety of the clutter-originated measurements isdescribed by an RFS Λk,s. Because of A6 and A8, Λk,s is a

Poisson RFS with mean µ(s)c and spatial pdf f (s)

c (zk,s) and,

thus, intensity function λ(s)c (zk,s) = µ(s)c f (s)

c (zk,s). Using A3,A4, A8, and A11, the overall measurement RFS at sensor sand time k, Zk,s =

{

z(1)k,s , . . . , z

(mk,s)k,s

}

, is then obtained as

Zk,s =

(

ik⋃

j=1

Zk,s,j

)

∪ Λk,s . (98)

Here, because of A7, A8, and A9, the components Zk,s,j andΛk,s are all conditionally independent given Xk .

The dependence of Zk,s on the multitarget state RFS

Xk={(

x(1)k , l(1)k

)

, . . . ,(

x(ik)k , l(ik)k

)}

is described by the single-sensor multitarget likelihood function f(Zk,s|Xk). Using (98)and (97) together with A6–A9, one obtains [3, Ch. 12]

f(Zk,s|Xk) = e−µ(s)c

(mk,s∏

m=1

µ(s)c f (s)

c

(

z(m)k,s

)

)

×∑

α∈Miuk,mk,s

ik∏

j=1

g(

x(j)k ,α(j); zk,s

)

, (99)

where g(

x(j)k ,α(j); zk,s

)

is as in (14) with a(i)k,s replacedby α(j). An analysis of this expression shows that for

α(j) > 0, target j (with state x(j)k ) generates measurement


z(α(j))k,s , whereas for α(j) = 0, target j does not generate a

measurement at sensor s. Note that due to missed detectionsand clutter measurements, the number of measurements mk,s

may also be smaller or larger than the number of targetsik. According to (99), the measurements do not provideinformation about the target labels. (If the labels could bemeasured as well, there would not be a DA uncertainty.)Comparing (99) with (17), it can be shown that for ik fixed,8

f(Zk,s|Xk) = mk,s! f(zk,s,mk,s|xk). This is not surprisingas the assumptions underlying the vector-type and set-typemeasurement models (i.e., A3–A9) are equal.

Finally, using A10, the multisensor multitarget likelihoodfunction is obtained as

f(Zk|Xk) =ns∏

s=1

f(Zk,s|Xk), (100)

where Zk !(

Zk,1, . . . ,Zk,ns

)

is an ordered list of the sets

Zk,s. For later use, we also introduce the ordered list Z1:k !(Z1, . . . ,Zk).

XII. SET-TYPE MULTITARGET STATE PROPAGATION

In this section, extending [113] and [51], we developprediction and update steps for the set-type system model fromSection XI. The prediction step and an approximation of theupdate step will be used in Section XIII to devise a hybridlabeled/unlabeled variant of the TOMB/P filter proposed in[51], [113]. We consider the single-sensor case (ns = 1) andthus drop the sensor index s. We note that two alternativederivations of our results are provided in [51] and [152].

According to (91), the multitarget state RFS Xk is the unionof the detected target RFS Xd

k and the undetected target RFSXuk. It can be shown [51] that the posterior pdfs of Xd

k and Xuk

can be propagated in parallel, in both cases by carrying out aprediction step and an update step. While the two predictionsteps are performed completely separately, the update step forXdk involves the prediction results for both Xd

k and Xuk. We

note that explicit propagation of the pdf of the undetectedtargets Xu

k enables the consideration of targets that were bornat an earlier time but so far have not been detected by anysensor. This ability is beneficial especially if the detectionprobabilities pd(xk) are low, time-varying, or nonuniform (i.e.,different for different values of xk).

In the following development of the prediction and updatesteps, we use the fact that the posterior pdf of the overallmultitarget state RFS Xk−1 at time k−1 factorizes as

f(Xk−1|Z1:k−1) = f(X dk−1|Z1:k−1)f(X

uk−1|Z1:k−1). (101)

Here, according to Section X-B, Xdk−1 is an LMB RFS

consisting of |Lk−1| Bernoulli components, where Lk−1

is a set of nonzero labels. Furthermore, Xuk−1 is a Pois-

son RFS with mean parameter µuk−1, spatial pdf fu(xk−1),

dummy label lk−1 (which is 0) and, thus, intensity functionλuk−1(xk−1, lk−1) = µu

k−1fu(xk−1)1(lk−1). The sets X dk−1

and X uk−1 in (101) are given by X d

k−1 = {(xk−1, lk−1) ∈

8Here, the factor mk,s! is related to the fact that f(Zk,s|Xk) integrates toone using the set integral (87) whereas f(zk,s,mk,s|xk) integrates to oneusing conventional integration.

Xk−1 : lk−1 = 0} and X uk−1 = {(xk−1, lk−1)∈Xk−1 : lk−1=

0}. Indeed, for k− 1 = 0, the form in (101) is a consequenceof the initial condition defined by S8 and S9, which impliesthat at k= 0, X0 = Xu

0 is a Poisson RFS with intensity functionλu0(x0, l0) = µpfp(x0)1(l0). For k − 1 # 1, it can be shown

that the form in (101) is preserved by the prediction and updatesteps presented in what follows, and therefore it is valid forall values of k − 1. (We note that a partial proof of this factis provided in [51], [152].)

For simplicity of notation, we will index the existenceprobabilities and spatial pdfs of the Bernoulli componentsdirectly by their labels (assigned upon first detection), i.e.,

we write them as r(l)k−1 and s(l)k−1(xk−1) rather than r(τ−1(l))

k−1

and s(τ−1(l))

k−1 (xk−1), respectively, with l∈Lk−1.

A. Prediction Step

In the prediction step, the preceding posterior pdff(Xk−1|Z1:k−1) is converted into a predicted posteriorpdf f(Xk|Z1:k−1) via the RFS version of the Chapman-Kolmogorov equation (cf. (1))

f(Xk|Z1:k−1) =

∫

f(Xk|Xk−1)f(Xk−1|Z1:k−1)δXk−1 .

The posterior pdf of the detected targets, f(X dk |Z1:k), and

that of the undetected targets, f(X uk |Z1:k) = f(X u

k ), can bepredicted separately. (Note that Xu

k is independent of Z1:k,but impacted by the characteristics of the detection process—in particular, by pd(xk).) These predictions involve the state-transition pdfs (94) and (96), respectively.

More specifically, f(X dk |Z1:k−1) is again an LMB pdf, with

the Bernoulli parameters r(l)k|k−1 and s(l)k|k−1(xk) given by

r(l)k|k−1 = r(l)k−1

∫

ps(xk−1)s(l)k−1(xk−1)dxk−1 (102)

s(l)k|k−1(xk) =

∫

f(xk|xk−1)ps(xk−1)s(l)k−1(xk−1)dxk−1

∫

ps(xk−1)s(l)k−1(xk−1)dxk−1

,

(103)

for l∈Lk−1. If at time k−1 the existence of a target is perfectly

known, i.e., r(l)k−1 = 1, and if also ps(xk−1) = 1, and thus

r(l)k|k−1=1 according to (102), then the spatial-pdf predictionin (103) simplifies to the conventional prediction for a singletarget performed by the JPDA filter (see (38)). In the generalcase, (102) and (103) correspond to the vector prediction (76)

(here, r(l)k|k−1s(l)k|k−1(xk) corresponds to αk(xk, 1)).

Furthermore, f(X uk |Z1:k−1) is again a Poisson pdf, whose

state intensity function is given by

λuk|k−1(xk, lk) =1(lk)

∫

f(xk|xk−1)ps(xk−1)

× λuk−1(xk−1, lk−1=0)dxk−1+λb(xk, lk).

This prediction step converting λuk−1(xk−1, lk−1) into

λuk|k−1(xk, lk) is equivalent to the prediction step of the PHD

filter [111]. The labels of the elements of f(X uk |Z1:k−1) are

again lk = 0. It is important to note that the entire predictionstep preserves the LMB–Poisson model assumed for Xk−1.


B. Update Step

In the update step, the predicted posterior pdf f(Xk|Z1:k−1)is converted into the posterior pdf f(Xk|Z1:k) via the RFSversion of Bayes’ rule (cf. (2))

f(Xk|Z1:k) ∝ f(Zk|Xk)f(Xk|Z1:k−1).

Here, the single-sensor multitarget likelihood functionf(Zk|Xk) (see (99)) incorporates the current measurement setZk.

1) Update for the Detected Targets: The posterior pdfdescribing the detected targets is given by [51]

f(X dk |Z1:k) =

∑

ck

p(ck)fLMBck

(X dk ) (104)

=∑

ck

p(ck)∏

l∈Lk

f (l,c(l)k

)(

X (l)k

)

, (105)

where X dk =

{(

x(1)k , l(1)

)

, . . . ,(

x(id

k)k , l(i

dk))}

, ck is a DA

vector with entries c(l)k , l ∈ Lk and pmf p(ck) (see

below), f (l,c(l)k

)(

X (l)k

)

is a Bernoulli pdf, and X (l)k =

{

(x(i)k , l(i)) ∈ X d

k : l(i)= l}

. According to (90), fLMBck

(X dk ) =

∏

l∈Lkf (l,c

(l)k

)(

X (l)k

)

is an LMB pdf. Thus, (104) means that

f(X dk |Z1:k) is a mixture of LMB pdfs, where each LMB pdf

is indexed by ck, weighted by p(ck), and consists of |Lk|Bernoulli components. The label set is updated as

Lk = Lk−1 ∪ Lnewk , (106)

where Lk−1 corresponds to legacy components that are takenover from time k−1 and Lnew

k corresponds to new componentsthat were undetected so far and generate a measurement forthe first time. Note that Lk−1 ∩ Lnew

k = ∅ and thus |Lk| =|Lk−1| + |Lnew

k |. For each measurement m ∈ {1, . . . ,mk}, anew Bernoulli component is created, to which the unique labell! (k,m)∈Lnew

k is assigned. Thus, in particular, |Lnewk |=mk.

The legacy and new Bernoulli components are the set-typecounterparts of, respectively, the legacy and new PTs usedin the vector-type system model described in Section VIII.In the vector-type system model, the number of PTs increasedroughly linearly with time k (see (52)). As evidenced by (106),the same increase is now exhibited by the number of Bernoulli

components f (l,c(l)k

)(

X (l)k

)

in each LMB pdf fLMBck

(X dk ). In

addition, the number of possible association vectors ck and,in turn, the number of different LMB pdfs fLMB

ck(X d

k ) increasesexponentially. These issues will be addressed in SectionXIII-A.

It will be convenient to partition the DA vector ck =[

c(l)k

]

l∈Lkin (104), (105) as ck !

[

aTk bT

k

]T, with the “legacy

component” DA vector ak =[

a(l)k

]

l∈Lk−1and the “new

component” DA vector bk =[

b(l)k

]

l∈Lnewk

. The entries of these

vectors are defined as follows. For l∈Lk−1, c(l)k = a

(l)k with

a(l)k !

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

m∈ {1, . . . ,mk} , if at time k, legacy componentl generates measurement m

0 , if at time k, legacy componentl does not generate a measure-ment.

For l∈Lnewk , c

(l)k = b

(l)k with

b(l)k !

⎧

⎪

⎪

⎨

⎪

⎪

⎩

l′∈ Lk−1 , if at time k, measurement m isgenerated by legacy component l′

0 , if at time k, measurement m isnot generated by a legacy component.

(We recall that l= (k,m) for l ∈ Lnewk .) These definitions of

a(l)k and b

(l)k are similar to (10) and (20), respectively. Note

that ck ∈ {0, . . . ,mk}|Lk−1| × ({0}∪Lk−1)mk. To guarantee

that p(ck)= 0 for all ck=[

aTk bT

k

]Tthat violate A4, similarly

as in (21) and (22), we introduce the indicator function

ψ(ak , bk) =∏

l∈Lk−1

mk∏

m=1

Ψl,m

(

a(l)k , b(k,m)k

)

,

with

Ψl,m

(

a(l)k , b(k,m)k

)

!

⎧

⎪

⎨

⎪

⎩

0 , a(l)k =m, b(k,m)k = l

or b(k,m)k = l, a(l)k =m

1 , otherwise.

The DA pmf p(ck) is now given by

p(ck) = p(ak , bk)

∝ ψ(ak , bk)

(

∏

l∈Lk−1

β(l)k

(

a(l)k

)

)

∏

l′∈Lnewk

ξ(l′)

k

(

b(l′)

k

)

,

(107)

where the association weights β(l)k (·) and ξ(l)k

(

·)

are calculated

as follows. For l∈Lk−1, we have for a(l)k =m∈ {1, . . . ,mk}

β(l)k (m) = r(l)k|k−1C

(l)k

(

z(m)k

)

,

with C(l)k

(

z(m)k

)

!∫

f(

z(m)k

∣

∣xk

)

pd(xk)s(l)k|k−1(xk)dxk, and

for a(l)k = 0β(l)k (0) = 1− r(l)k|k−1D

(l)k ,

with D(l)k !

∫

pd(xk)s(l)k|k−1(xk)dxk. For l = (k,m) ∈ Lnew

k ,

we have ξ(l)k

(

b(l)k

)

= 1 if b(l)k = 0, and

ξ(l=(k,m))k (0) = Cu

k

(

z(m)k

)

+ λc

(

z(m)k

)

if b(l)k = 0, with Cuk

(

z(m)k

)

!∫

f(

z(m)k

∣

∣xk

)

pd(xk)λuk|k−1(xk,

lk = 0)dxk and λc

(

z(m)k

)

= µcfc

(

z(m)k

)

. It can be shownthat for pd(xk)λuk|k−1(xk, lk = 0) = µnfn(xk), the joint

marginal DA pmf p(ck) = p(ak , bk) in (107) equals the beliefp(ak , bk) in (80), i.e., the belief calculated by the total-SPAmethod for an unknown number of targets.

Each Bernoulli component f (l,c(l)k

)(

X (l)k

)

in (105) is pa-

rameterized by an existence probability r(l,c(l)

k)

k and a spatial

pdf s(l,c(l)

k)

k (xk). These parameters are calculated as follows.Let us first consider the legacy components, i.e., l ∈ Lk−1,

where c(l)k = a(l)k . Here, the existence probability is given for

a(l)k ∈ {1, . . . ,mk} by r(l,a(l)

k)

k = 1 (this means that legacycomponent l corresponds to an existing target that generated


measurement m= a(l)k ) and for a(l)k = 0 by

r(l,0)k =r(l)k|k−1

(

1−D(l)k

)

1− r(l)k|k−1D(l)k

. (108)

The spatial pdf is given for all a(l)k ∈ {0, . . . ,mk} by

s(l,a(l)

k)

k (xk) =g(

xk, a(l)k ; zk

)

s(l)k|k−1(xk)∫

g(

x′k, a

(l)k ; zk

)

s(l)k|k−1(x′k)dx

′k

, (109)

where g(

xk, a(l)k ; zk

)

was defined in (14). Next, we consider

the new components, i.e., l= (k,m) ∈Lnewk , where c(l)k = b(l)k .

Here, the existence probability is given for b(l)k =0 by

r(l=(k,m),0)k =

Cuk

(

z(m)k

)

Cuk

(

z(m)k

)

+ λc

(

z(m)k

)

(110)

and for b(l)k = 0 by r(l,b

(l)k

)k = 0 (this means that measurement

m was generated by a legacy component). The spatial pdf is

given for b(l)k = 0 by

s(l=(k,m),0)k (xk) =

f(

z(m)k

∣

∣xk

)

pd(xk)λuk|k−1(xk, lk= 0)

Cuk

(

z(m)k

)

,

(111)

whereas for b(l)k = 0, s(l,b

(l)k

)k (xk) does not need to be defined

(since r(l,b(l)

k)

k =0).2) Update for the Undetected Targets: We recall that for the

targets that remain undetected, the posterior pdf f(X uk |Z1:k)=

f(X uk ) does not involve Z1:k (although it involves pd(xk)).

The update step for these targets yields again a Poisson pdff(X u

k ), whose intensity function is calculated as

λuk(xk, lk) = (1−pd(xk))λ

uk|k−1(xk, lk).

This is equivalent to the update step of the PHD filter [111] forthe case where the sensor does not produce any measurements.The labels of all elements of X u

k are again zero.

XIII. SPA-BASED SET-TYPE MTT METHODS

Building on the set-type system model from Section XI andthe set-type multitarget state propagation rules from SectionXII, we will now develop a hybrid labeled/unlabeled variant ofthe TOMB/P filter proposed in [51], [113] (cf. Section II-B2).This labeled/unlabeled TOMB/P filter represents the joint stateof the detected targets by a labeled RFS—more specifically,an LMB RFS—and is thus able to maintain track continuity,i.e., to estimate entire target trajectories.

Whereas the prediction step presented in Section XII-Aand the update step for the undetected targets presented inSection XII-B2 preserve the LMB–Poisson model used forXk in Section XII, the update step for the detected targetspresented in Section XII-B1 converts the LMB RFS describingthe detected targets into a mixture of LMB RFSs. In orderto reobtain an LMB pdf also for the detected targets, wewill use an approximation that is similar to the one usedin the derivation of the JPDA filter. The marginal posteriorDA pmfs are then calculated by SPADA. As in the vector-type MTT methods discussed in Section VII, this is key to

obtaining scalability of the resulting filter. We first considerthe single-sensor case (ns = 1) and later extend to multiplesensors. A detailed derivation of our results and a closed-formimplementation for linear-Gaussian models can be found in[51], [152], and a particle-based implementation for nonlinearand non-Gaussian models is presented in [107].

A. Approximate Update Step for the Detected Targets

According to (105) and (106), the update step for thedetected targets produces a mixture of LMB pdfs, and itscomputational complexity scales exponentially with the num-ber |Lk| of Bernoulli components and the number mk ofmeasurements. To address these issues, we approximate thejoint DA pmf p(ck) by the product of its marginals, i.e.,

p(ck) ≈∏

l∈Lk

p(

c(l)k

)

, with p(

c(l)k

)

=∑

∼c(l)k

p(ck). (112)

Note that a similar approximation was used in the derivationof the JPDA filter, cf. (42). Inserting (112) in (105) yields

f(X dk |Z1:k) ≈

∏

l∈Lk

∑

c(l)k

p(

c(l)k

)

f (l,c(l)k

)(

X (l)k

)

=∏

l∈Lk

f (l)(

X (l)k

)

. (113)

Here, f (l)(

X (l)k

)

!∑

c(l)k

p(

c(l)k

)

f (l,c(l)k

)(

X (l)k

)

is a weighted

sum of Bernoulli pdfs, where the weights p(

c(l)k

)

are normal-

ized. Thus (cf. Section X-B), f (l)(

X (l)k

)

is again a Bernoullipdf. This means that f(X d

k |Z1:k) is approximated in (113) by

the product of Bernoulli pdfs f (l)(

X (l)k

)

, each associated withone of the components l∈Lk. We conclude that the obtainedapproximation of f(X d

k |Z1:k) is again an LMB pdf, i.e., theapproximate update step preserves the LMB structure.

The existence probability r(l)k and spatial pdf s(l)k (xk)

parameterizing the Bernoulli pdf f (l)(

X (l)k

)

are obtained asfollows. For the legacy components, i.e., l ∈ Lk−1, where

c(l)k = a(l)k ,

r(l)k =mk∑

a(l)k

=0

p(

a(l)k

)

r(l,a(l)

k)

k (114)

s(l)k (xk) =1

r(l)k

mk∑

a(l)k

=0

p(

a(l)k

)

r(l,a

(l)k

)k s

(l,a(l)k

)k (xk), (115)

and for new components, i.e., l∈Lnewk , where c(l)k = b(l)k ,

r(l)k = p(

b(l)k = 0)

r(l,0)k , s(l)k (xk) = s(l,0)k (xk). (116)

Here, expressions of r(l,c(l)

k)

k and s(l,c(l)

k)

k (xk) were providedin Section XII-B1. In analogy to the JPDA filter, where theapproximation (42) ensured that the approximate posterior pdff(xk|z1:k) of the multitarget state factorizes into the approx-

imate posterior pdfs f(

x(i)k

∣

∣z1:k)

of the single-target states,the similar approximation (112) ensures that f(X d

k |Z1:k) is anLMB pdf.

It remains to calculate the marginal DA pmfs p(

a(l)k

)

,

l ∈ Lk−1 and p(

b(l)k

)

, l ∈ Lnewk occurring in (114)–(116).


The factorization in (107) is analogous to that in (26) with{1, . . . , nt} and {1, . . . ,mk} replaced by Lk−1 and Lnew

k ,respectively. Hence, we can use SPADA with association

weights β(l)k (a(l)k ), l ∈ Lk−1 and ξ(l)k

(

b(l)k

)

, l ∈ Lnewk —to

calculate approximations of p(

a(l)k

)

and p(

b(l)k

)

.

After (114)–(116) have been evaluated, target declarationand state estimation can be performed. LMB component l∈Lk

is declared to be an existing target at time k if r(l)k is largerthan a threshold Pth. Furthermore, for a target l that is declared

to exist, an estimate x(l)k of the state x

(l)k is calculated as in

(3) with f(

xk|z1:k)

replaced by s(l)k (xk).Whereas the approximate update step discussed above

avoids the exponential scaling of complexity with |Lk| andmk, it does not avoid the roughly linear increase of |Lk|with time k (cf. (106)). Therefore, a pruning step is typicallyperformed, which removes Bernoulli components l whose

existence probability r(l)k is below a predefined threshold Ppr.

It is interesting to note that the expression (109) of the

spatial pdf s(l,a(l)

k)

k (xk) is analogous to the expression (41) of

the approximate posterior pdf f(

x(i)k

∣

∣a(i)k , z1:k)

in the JPDAfilter. Indeed, if the predicted existence probability of all

legacy components l ∈ Lk−1 is r(l)k|k−1 = 1, and thus (cf.

(108) and the discussion preceding it) r(l,a(l)

k)

k = 1 for all

l ∈ Lk−1 and all a(l)k ∈ {0, . . . ,mk}, and if there are nopredicted undetected targets, i.e., λu

k|k−1(xk, lk) = 0, then

r(l=(k,m),0)k = 0 in (110). This means that the existence

probability of all new components is zero. As a consequence,(105) is equivalent to (39), and thus the update step of theTOMB/P filter reduces to the update step of the JPDA filter.Furthermore, it can be shown that if the intensity functionλn

(

xk

)

= µnfn

(

xk

)

for newly detected targets in the vector-type total-SPA method from Section IX-A is set to λn

(

xk

)

=pd(xk)λu

k|k−1(xk, lk = 0), then the beliefs resulting fromthat method are equivalent to the Bernoulli component pdfs

f (l)(

X (l)k

)

. More specifically, let f(

x(j)k , r(j)k

)

be the beliefproduced by the vector-type total-SPA method for the PT thatwas detected for the first time due to measurement m at timek, i.e., j = jk−1 +m. Furthermore, let f (l)

(

X (l)k

)

be the pdfof the Bernoulli component with label l = (m, k) obtainedby the hybrid labeled/unlabeled variant of the TOMB/P filter.

Then the existence probability r(l)k and spatial pdf s(l)k

(

x(l)k

)

of

that Bernoulli component are related to f(

x(j)k , r(j)k

)

according

to r(l)k s(l)k

(

x(l)k

)

= f(

x(j)k , 1

)

. While in the vector-type total-SPA method the time step k and the measurement index mrelated to the first measurement of a PT are implicitly encoded

by the order of the subvectors y(j)k , j = 1, . . . jk in the

joint PT vector yk, in the hybrid labeled/unlabeled variant ofthe TOMB/P filter this information is explicitly given by theBernoulli component label l.

B. Multisensor Extensions

In the case of multiple sensors (ns≥ 2), the prediction stepdiscussed in Section XII-A is unchanged as it does not involvethe sensor measurements. Regarding the update step, using

Bayes’ rule, A11, and (100), we obtain

f(Xk|Z1:k) ∝ f(Zk|Xk)f(Xk|Z1:k−1)

=

(

ns∏

s=1

f(Zk,s|Xk)

)

f(Xk|Z1:k−1).

Based on this expression, the update step can be performedsensor-sequentially within an iterated-corrector method. Iter-ated marginal posterior pdfs f(Xk|Zs

1:k) are calculated foreach sensor s=1, . . . , ns. Here, f(Xk|Zs

1:k) denotes the pdfof Xk conditioned on Z1:k−1 and on Zk,s′ for s′ =1, . . . , s.The sth update step converts f(Xk|Z

s−11:k ) into f(Xk|Zs

1:k),thereby incorporating the measurement Zk,s of sensor s. Thisrecursion is initialized by f(Xk|Z0

1:k) = f(Xk|Z1:k−1). Thesth update step is equal to the single-sensor update stepdiscussed in Section XIII-A except for the following differ-ences. The input f(Xk|Z1:k−1) of the single-sensor update

step is replaced by f(Xk|Zs−11:k ), and the measurements z

(m)k ,

m ∈ {1, . . . ,mk} are replaced by z(m)k,s , m ∈ {1, . . . ,mk,s}.

Furthermore, for the labels of the newly detected targets to beunique, they are now given by the triple (k, s,m).

This sequential multisensor MTT method exhibits excellentperformance in many scenarios; an example will be consideredin Section XIV. Due to the approximation performed in eachupdate step, the final result of the method depends on theorder in which the sensors are updated. In certain set-typefilters, such as the PHD, CPHD, and MeMBer filters, a sensor-sequential update can lead to a poorer performance than in thesingle-sensor case [112], [154]. In the method discussed above,because of the different approximation employed, the influenceof the sensor order on performance is significantly less strong.We note that “parallel” set-type multisensor MTT methodsthat perform the update steps for all sensors simultaneouslyhave been proposed recently [104], [105]. However, thesemethods do not exploit the independence of the measurementsof different sensors (cf. A10) and hence do not scale well withthe number of sensors.

XIV. PERFORMANCE EVALUATION

In this section, we present simulation and real-data experi-ments in order to demonstrate the performance of SPA-basedMTT methods and compare it with that of other MTT methods.

A. Simulation Results

We consider a region of interest (ROI) given by[−3000m, 3000m] × [−3000m, 3000m], with up to five tar-gets. The target states comprise 2-D position and velocity, i.e.,

x(i)k =

[

x(i)1,k x(i)

2,k x(i)1,k x(i)

2,k

]T, and they evolve according to

the near constant-velocity motion model [155, Sec. 6.3.2] suchthat the target trajectories tend to intersect at the ROI center.The sensors are placed uniformly on a circle of radius 3000mabout the ROI center. The ROI with the sensor positions (forns = 3 sensors) and an example realization of the targettrajectories is shown in [53, Fig. 4]. The sensors perform rangeand bearing measurements within a measurement range of

6000m. The mean number of clutter measurements is µ(s)c =2.

The clutter pdf f (s)c

(

z(s)n,m)

is uniform on [0m, 6000m] with


respect to the range component and uniform on [0◦, 360◦) withrespect to the angle component.

We present simulation results for the following SPA-basedmultisensor MTT methods: the sequential and parallel multi-sensor JPDA methods with SPA-based DA described in Sec-tion VII-C (abbreviated9 “JPDA-SPA”), the sequential and par-allel multisensor total-SPA methods described in Section IX-B(abbreviated “Total-SPA-S” and “Total-SPA-P,” respectively),and the sequential multisensor TOMB/P filter with SPA-based DA described in Section XIII-B (abbreviated “TOMB/P-SPA”). These SPA-based methods are compared with theiterated-corrector PHD filter [3], [101], [112] (“IC-PHD”), theiterated-corrector CPHD filter [3], [102], [112] (“IC-CPHD”),the partition-based multisensor PHD filter [104] (“MS-PHD”),and the partition-based multisensor CPHD filter [104] (“MS-CPHD”). Particle implementations are used for all filters.

The simulation parameters are as follows. For methodsthat model the number of targets as unknown, the birthpdf fb(xk) is uniform on the ROI, the number of newborntargets is Poisson distributed with mean µb = 10−2, and thesurvival probability is ps(xk) = ps = 0.999. The thresholdfor target declaration is Pth = 0.5. The SPA-based methodsperform nit = 20 message passing iterations. In Total-SPA-S,we choose the intensity function for newly detected targets

as λ(s)n (xk) = p(s)d (xk)µbfb(xk) for all sensors s. (Note

that µ(s)n =

∫

λ(s)n (xk)dxk and fn(xk) = λ(s)n (xk)/µ(s)n .)

In Total-SPA-P, we set λ(1)n (xk) = p(1)d (xk)µbfb(xk) and

λ(s)n (xk) = 0 for s = 2, . . . , ns. In TOMB/P-SPA, the meannumber of targets at time k=0 is µp=0 (cf. S9). In TOMB/P-SPA, Total-SPA-P, and Total-SPA-S, the pruning threshold isPpr = 10−5. For track management in JPDA-SPA, we performgating with a gate threshold of 18.4, we use the m-of-nheuristic [1] with m = 4 and n = 6 across time and sensorsfor track initialization, and we terminate a track if for sixconsecutive update steps no measurement falls into the gateof the respective target. The maximum numbers of subsets andpartitions used by MS-PHD and MS-CPHD are 120 and 720,respectively, similarly to [104]. We performed 400 simulationruns, each with 150 time steps k. A new target trajectorywas generated in each simulation run. Further details of thesimulation setup and parameters (e.g., the number of particlesused by the various filters) are provided in [53].

We measure the performance of the various MTT methodsby the Euclidean distance based optimal sub-pattern assign-ment (OSPA) metric with cutoff parameter 200 [156]. (Werecall that complementary performance results assessing DAaccuracy were presented in Section VI-C.) For ns =3 sensors

and a detection probability of p(s)d

(

x(i)k

)

= pd = 0.8, Fig. 5shows the mean OSPA (MOSPA) error—averaged over 400simulation runs—of all methods versus time k. One can seethat for all methods, the error peaks at times k=5, 10, 15, 20,and 25, i.e., when there are target births. However, very soonafter each target birth, all methods except IC-PHD reliablyestimate the number of targets. The performance of TOMB/P-SPA, Total-SPA-S, and Total-SPA-P is seen to be almost

9We do not distinguish between the sequential and parallel multisensorJPDA methods because they exhibited identical performance.

25 50 75 100 125 1500

40

80

120

160

200

Time k

MO

SPA

IC-PHD

MS-PHD

IC-CPHD

MS-CPHD

JPDA-SPA

Total-SPA-P

Total-SPA-S

TOMB/P-SPA

Fig. 5. MOSPA error versus time k for ns = 3 and pd = 0.8. (The curvesfor Total-SPA-S and TOMB/P-SPA coincide.)

identical. The performance of JPDA-SPA is inferior to that ofthe other SPA-based methods immediately after target births;this can be explained by the fact that JPDA-SPA uses a heuris-tic to initialize new targets. Otherwise, JPDA-SPA performslike the other SPA-based methods. The SPA-based methodsare seen to outperform all the other simulated methods. Inparticular, IC-CPHD, MS-PHD, and MS-CPHD perform worsethan the SPA-based methods because particle implementationsof (C)PHD filters involve a potentially unreliable clusteringstep. This clustering step is especially unreliable when targetsare closely spaced, which occurs in our simulation around timek=100, i.e., when the five target trajectories intersect in theROI center. In fact, the MOSPA error of IC-CPHD, MS-PHD,and MS-CPHD is seen to be higher around that time. Finally,IC-PHD performs significantly worse than all the other filtersbecause it is unable to reliably estimate the number of targetsin the simulated scenario. We note that simulation results fora significantly larger scenario with closely spaced targets arepresented in [53].

The average computation time per time step k for a MAT-LAB implementation on a single core of an Intel Xeon E5-2640 v3 CPU was measured as 0.03s for JPDA-SPA; 0.08s forTotal-SPA-S, Total-SPA-P, and TOMB/P-SPA; 0.11s for IC-PHD; 0.14s for IC-CPHD; 13.21s for MS-PHD; and 13.82sfor MS-CPHD. We note that JPDA-SPA used gating whereasthe other filters did not. The high computation times of theMS-PHD and MS-CPHD filters are due to the fact that thetrellis algorithm used for partition extraction is tailored toa Gaussian mixture implementation and becomes computa-tionally intensive in a particle-based implementation. Furtherresults demonstrating the excellent scalability of the SPA-based methods are reported in [52], [53], [148].

Finally, for detection probability pd = 0.6, Fig. 6 showsthe time-averaged MOSPA error—i.e., averaged over all 150simulated time steps—versus the number of sensors ns. JPDA-SPA performs worse than TOMB/P-SPA, Total-SPA-S, andTotal-SPA-P, due to its heuristic track initialization. Total-SPA-P performs worse than Total-SPA-S and TOMB/P-SPA,because it does not implement sensor fusion for new PTs.The increase of the time-averaged MOSPA error of MS-PHD


1 2 3 4 5 6 7 80

20

40

60

80

100

Number of Sensors ns

MO

SPA

IC-PHD MS-PHD

IC-CPHD MS-CPHD

JPDA-SPA Total-SPA-P

Total-SPA-S TOMB/P-SPA

Fig. 6. Time-averaged MOSPA error versus number of sensors ns for pd =0.6. (The curves for Total-SPA-S and TOMB/P-SPA coincide.)

and MS-CPHD for ns > 5 is due to the fact that the chosenmaximum numbers of subsets (120) and partitions (720) aretoo small for that case; however, larger maximum numberswould lead to excessive simulation times.

B. Results for a Radar Tracking Experiment

For further validation of the SPA-based MTT methods,we use real measurements that were acquired by two high-frequency surface-wave (HFSW) radars for maritime surveil-lance [157], [158]. HFSW radars feature over-the-horizon cov-erage and a continuous-time mode of operation. On the otherhand, they suffer from poor range and azimuth resolution,high nonlinearity, and strong clutter; these effects make MTTa challenging task [157], [158]. The two radar stations arelocated close to the cities of Pisa and La Spezia in Italy.The ROI is the intersection of the fields-of-view of the tworadar stations. All vessels in the ROI are considered as targets.Ground truth information about the target positions is reportedby the Automatic Identification System (AIS). However, thisinformation is incomplete since AIS reports do not includevessels below a certain gross tonnage and military vessels. Theoverall tracking scenario is shown in Fig. 7. At each of thetwo radar sensors, measurements are available every 33.28s.

We processed the real measurements provided by the twosensors by the same MTT methods as in the previous sub-section. Target motion is modeled by the near constant-velocity model with driving process variance 0.01m2/s4.We use a range-bearing measurement model involving aGaussian measurement noise vector with covariance ma-trix diag{(150m)2, (1.5◦)2}. The probability of detection is

p(s)d

(

x(i)k

)

= pd = 0.65 for both sensors. The clutter pdf

f (s)c

(

z(m)k,s

)

is chosen uniform on the ROI, and the mean

number of clutter measurements is µ(s)c = 15. For methods

that model the number of targets as unknown, the birth pdffb(xk) is uniform on the ROI, the number of newborn targetsis Poisson distributed with mean µb = 10−1, and the survivalprobability is ps(xk) = ps = 0.999. The threshold for targetdeclaration is Pth =0.5. Track management in JPDA-SPA is asdescribed earlier, except that the gate threshold is 9 and tracks

x1 [km]

x2

[km

]

−100 −75 −50 −25 0−50

−25

0

25

50

Fig. 7. Three hours of measurements acquired by two HFSW radar stations onthe west coast of Italy. The positions of radar stations #1 (near Pisa) and #2(near La Spezia) are indicated by a green and a blue square, respectively.The corresponding measurements are indicated by green and blue dots,respectively. AIS ground truth information is indicated by black lines.

are terminated if for six consecutive update steps less thantwo measurements fall into the gate of the respective target.The (C)PHD-type filters use 45000 particles to represent thePHD of the target states. JPDA-SPA, TOMB/P-SPA, Total-SPA-S, and Total-SPA-P use 3000 particles for each target.In TOMB/P-SPA, the mean number of targets at time k = 0is µp =5, and the spatial pdf fp(x0) is uniform on the ROI.In TOMB/P-SPA, Total-SPA-S, and Total-SPA-P, after eachsingle-sensor update step, targets with existence probabilitysmaller than Ppr =10−3 are removed. All remaining parame-ters are as in Section XIV-A.

Table I shows the time-on-target (TOT) and the false alarmrate (FAR) [1] obtained with the various methods for 24hof measurement data. TOT is the fraction of time that thetarget is successfully detected, which is considered to be trueif the estimated target position is within 500m of the truetarget position. FAR is the number of false trajectories (orfalse detections) generated in the surveillance region per unitof 2-D space and unit of time. Ideally, the TOT would beone and the FAR would be zero. It can be seen in Table Ithat the SPA-based methods achieve an attractive TOT-FARcompromise. In particular, the TOT of Total-SPA-S, Total-SPA-P, and TOMB/P-SPA is very similar, and larger thanthat of the other filters. The smaller TOT of JPDA-SPA-S and JPDA-SPA-P (the sequential and parallel multisensorextensions of JPDA-SPA discussed in Section VII-C) can beexplained by the heuristic that is used to initialize new targets.The relatively high TOT of MS-PHD and MS-CPHD is seento come at the cost of an increased FAR, while Total-SPA-S, Total-SPA-P, and TOMB/P-SPA achieve both a high TOTand a rather low FAR. We caution that the FAR reported inTable I is pessimistic in that the AIS ground truth informationis incomplete and thus measurements that are generated bytargets without ground truth information and detected by atracking algorithm are considered as false alarms. In this light,the relatively low FAR of IC-PHD, JPDA-SPA-S, and JPDA-SPA-P can be partly attributed to the fact that these methods,


TOT [%] FAR [s−1km−2]

IC-PHD 60.8 1.588 · 10−5

MS-PHD 71.9 2.041 · 10−5

IC-CPHD 68.9 1.933 · 10−5

MS-CPHD 73.2 2.120 · 10−5

JPDA-SPA-S 67.7 1.518 · 10−5

JPDA-SPA-P 68.1 1.572 · 10−5

Total-SPA-S 75.5 1.883 · 10−5

Total-SPA-P 75.0 1.663 · 10−5

TOMB/P-SPA 75.6 1.688 · 10−5

TABLE ITIME-ON-TARGET (TOT) AND FALSE ALARM RATE (FAR).

as evidenced by their low TOT, are unable to track sometargets that do not provide AIS information (and that wouldthus contribute to the FAR if successfully tracked).

XV. CONCLUSION

Multitarget tracking (MTT) is an important contributionto acquiring and maintaining an awareness of the environ-ment. Although measurement origin uncertainty makes MTTa complicated and challenging task, large-scale scenarios andreal-time operation on resource-limited devices call for MTTmethods whose complexity is moderate and scales well withthe number of targets, sensors, and measurements. In thistutorial paper, we showed that the development of high-performance MTT methods with moderate complexity andexcellent scalability can be based on the recently emergedparadigm of factor graphs and message passing using the sum-product algorithm (SPA). We presented SPA-based BayesianMTT methods within both a random vector framework anda random finite set framework. A core component of thesemethods, and a major reason for their scalability, is a highlyeffective and efficient SPA-based algorithm for probabilisticdata association. We discussed the integration of SPA-basedprobabilistic data association into existing MTT methods andshowed that certain existing methods can be reformulatedwithin the SPA framework. We also presented new vector-typeand set-type MTT methods in which the SPA is used either forprobabilistic data association or for the entire MTT problem.

The SPA approach to designing MTT methods has importantadvantages regarding scalability, accuracy, complexity, versa-tility, and intuitiveness. While rooted in the framework ofoptimum Bayesian inference, the SPA-based approach easilyaccomodates different system models and application-relatedaspects. In particular, it is suited to general nonlinear, non-Gaussian, and time-varying scenarios. The factor graph formu-lation of the MTT system model combined with the principleof stretching factor nodes introduces a beneficial intuitivenessand flexibility into the SPA-based design of MTT methods.Finally, SPA-based MTT methods are able to cope withunknown and time-varying hyperparameters, such as detectionprobabilities [54] and motion model parameters [159]. Usingboth simulated and real measurements, we demonstrated theexcellent performance and low complexity of the presentedSPA-based MTT methods.

The beliefs obtained by the loopy SPA can be overconfidentin that their spread underestimates the uncertainty of theestimates [145]. This can be a limitation in certain applications[160]. A variational message passing approach that performsiterative data association across multiple sensors and/or timesteps and avoids overconfident beliefs will be proposed inforthcoming work [145].

Possible directions of future research include an extensionof the network localization and navigation (NLN) paradigm[161]–[165] to noncollaborative objects; a combination ofMTT with resource allocation, sensor selection, and con-trol [166]–[173]; network experimentation [174]–[176] withnoncollaborative objects; network localization systems [177]–[179] for MTT; and distributed SPA-based MTT methods fordecentralized wireless sensor networks [180]–[182].

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Florian Meyer (S’12–M’15) received the Dipl.-Ing.(M.Sc.) and Ph.D. degrees in electrical engineeringfrom TU Wien, Vienna, Austria in 2011 and 2015,respectively. He was a visiting researcher with theDepartment of Signals and Systems, Chalmers Uni-versity of Technology, Gothenburg, Sweden in 2013and with the NATO Centre of Maritime Researchand Experimentation (CMRE), La Spezia, Italy in2014 and 2015. In 2016 he joint CMRE as a re-search scientist. Currently, Dr. Meyer is an ErwinSchrodinger Fellow at the Massachusetts Institute

of Technology (MIT). His research interests include signal processing forwireless sensor networks, localization and tracking, information-seeking con-trol, message passing algorithms, and finite set statistics. Dr. Meyer served asreviewer for several IEEE journals and as a TPC member in IEEE conferences.

Thomas Kropfreiter received the B.Sc. degreein electrical engineering and the Dipl.-Ing degree(M.Sc. equivalent) in telecommunication engineer-ing with distinction from TU Wien, Vienna, Austria,in 2012 and 2014, respectively. From 2013 to 2015,he was a member of the Mobile CommunicationsGroup, TU Wien, where he worked on the topic ofbeamforming in multi-cell scenarios. Since 2015, hehas been a member of the Signal Processing Group,TU Wien, where he is currently pursuing the Ph.D.degree. His research interests include multi-object

tracking, distributed signal processing in wireless sensor networks, messagepassing algorithms, and finite set statistics.

Jason L. Williams (S’01–M’07–SM’16) receiveddegrees of BE(Electronics)/BInfTech from Queens-land University of Technology in 1999, MSEE fromthe United States Air Force Institute of Technologyin 2003, and PhD from Massachusetts Institute ofTechnology in 2007.

He worked for several years as an engineeringofficer in the Royal Australian Air Force, beforejoining Australia’s Defence Science and TechnologyGroup (formerly Defence Science and TechnologyOrganisation) in 2007. He is also an adjunct asso-

ciate professor at Queensland University of Technology. His research interestsinclude target tracking, sensor resource management, Markov random fieldsand convex optimisation.

Roslyn A. Lau (S’14) received the degrees of BE(computer systems)/BMa&CS (statistics) in 2005,and MS (signal processing) in 2009, all from theUniversity of Adelaide, Adelaide, Australia.

She is currently a PhD candidate at the AustralianNational University. She is also a scientist at theDefence Science and Technology Group, Australia.Her research interests include target tracking, prob-abilistic graphical models, and variational inference.

Franz Hlawatsch (S’85–M’88–SM’00–F’12) re-ceived the Diplom-Ingenieur, Dr. techn., and Univ.-Dozent (habilitation) degrees in electrical engineer-ing/signal processing from TU Wien, Vienna, Aus-tria in 1983, 1988, and 1996, respectively. Since1983, he has been with the Institute of Telecommuni-cations, TU Wien, where he is currently an AssociateProfessor. During 1991–1992, as a recipient of anErwin Schrodinger Fellowship, he spent a sabbaticalyear with the Department of Electrical Engineering,University of Rhode Island, Kingston, RI, USA.

In 1999, 2000, and 2001, he held one-month Visiting Professor positionswith INP/ENSEEIHT, Toulouse, France and IRCCyN, Nantes, France. He(co)authored a book, three review papers that appeared in the IEEE SIGNAL

PROCESSING MAGAZINE, about 200 refereed scientific papers and bookchapters, and three patents. He coedited three books. He was TechnicalProgram Co-Chair of EUSIPCO 2004 and served on the technical committeesof numerous IEEE conferences. He was an Associate Editor for the IEEETRANSACTIONS ON SIGNAL PROCESSING from 2003 to 2007 and for theIEEE TRANSACTIONS ON INFORMATION THEORY from 2008 to 2011. From2004 to 2009, he was a member of the IEEE SPCOM Technical Committee.He currently serves as an Associate Editor for the IEEE TRANSACTIONS ON

SIGNAL AND INFORMATION PROCESSING OVER NETWORKS. He coauthoredpapers that won an IEEE Signal Processing Society Young Author Best PaperAward and a Best Student Paper Award at IEEE ICASSP 2011. His researchinterests include statistical and compressive signal processing methods andtheir application to sensor networks and wireless communications.


Paolo Braca (M’14–SM’17) received the Laureadegree (summa cum laude) in electronic engineering,and the Ph.D. degree (highest rank) in informationengineering from the University of Salerno, Italy,in 2006 and 2010, respectively. In 2009, he was aVisiting Scholar with the Department of Electricaland Computer Engineering, University of Connecti-cut, Storrs, CT, USA. In 2010?2011, he was a Post-doctoral Associate with the University of Salerno,Italy. In 2011, Dr. Braca joined the NATO Science& Technology Organization Centre for Maritime

Research and Experimentation (CMRE), and currently is a Senior Scientistwith the Research Department. He conducts research in the general area ofstatistical signal processing with emphasis on detection and estimation theory,wireless sensor network, multi-agent algorithms, target tracking and datafusion, adaptation and learning over graphs, radar (sonar) signal processing.He is coauthor of about one hundred publications in international scientificjournals and conference proceedings. In 2017 Dr. Braca was awarded withNational Scientific Qualification to function as associate professor in ItalianUniversities. Dr. Braca serves as an Associate Editor of IEEE Transactions onSignal Processing, IEEE Transactions on Aerospace and Electronic Systems,ISIF Journal of Advances in Information Fusion, EURASIP Journal onAdvances in Signal Processing. In 2017 he was Lead Guest Editor of thespecial issue “Sonar Multi-Sensor Applications and Techniques” in IETRadar Sonar and Navigation. He served as Associate Editor of IEEE SignalProcessing Magazine (E-Newsletter) from 2014 to 2016. He is in the TechnicalCommittee of the major international conferences in the field of signalprocessing and data fusion. He was the recipient of the Best Student PaperAward (first runner-up) at the 12th International Conference on InformationFusion in 2009.

Moe Z. Win (S’85–M’87–SM’97–F’04) is a pro-fessor at the Massachusetts Institute of Technology(MIT) and the founding director of the WirelessInformation and Network Sciences Laboratory. Priorto joining MIT, he was with AT&T Research Labo-ratories and NASA Jet Propulsion Laboratory.

His research encompasses fundamental theories,algorithm design, and network experimentation fora broad range of real-world problems. His currentresearch topics include network localization andnavigation, network interference exploitation, and

quantum information science. He has served the IEEE CommunicationsSociety as an elected Member-at-Large on the Board of Governors, as electedChair of the Radio Communications Committee, and as an IEEE DistinguishedLecturer. Over the last two decades, he held various Editorial posts for IEEEjournals and organized numerous international conferences. Currently, he isserving on the SIAM Diversity Advisory Committee.

Dr. Win is an elected Fellow of the AAAS, the IEEE, and the IET. He washonored with two IEEE Technical Field Awards: the IEEE Kiyo TomiyasuAward (2011) and the IEEE Eric E. Sumner Award (2006, jointly with R.A. Scholtz). Together with students and colleagues, his papers have receivednumerous awards. Other recognitions include the IEEE Communications So-ciety Edwin H. Armstrong Achievement Award (2016), the International Prizefor Communications Cristoforo Colombo (2013), the Copernicus Fellowship(2011) and the Laurea Honoris Causa (2008) from the University of Ferrara,and the U.S. Presidential Early Career Award for Scientists and Engineers(2004). He is an ISI Highly Cited Researcher.

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