Generalized Labeled Multi-Bernoulli Approximation of Multi-Object Densities
Francesco Papi, Ba-Ngu Vo, Ba-Tuong Vo, Claudio Fantacci, Michael Beard
aa r X i v : . [ s t a t . O T ] J u l Generalized Labeled Multi-BernoulliApproximation of Multi-Object Densities
Francesco Papi, Ba-Ngu Vo, Ba-Tuong Vo, Claudio Fantacci, and Michael Beard
Abstract —In multi-object inference, the multi-object prob-ability density captures the uncertainty in the number andthe states of the objects as well as the statistical dependencebetween the objects. Exact computation of the multi-objectdensity is generally intractable and tractable implementationsusually require statistical independence assumptions betweenobjects. In this paper we propose a tractable multi-object densityapproximation that can capture statistical dependence betweenobjects. In particular, we derive a tractable Generalized LabeledMulti-Bernoulli (GLMB) density that matches the cardinalitydistribution and the first moment of the labeled multi-objectdistribution of interest. It is also shown that the proposed approx-imation minimizes the Kullback-Leibler divergence over a specialtractable class of GLMB densities. Based on the proposed GLMBapproximation we further demonstrate a tractable multi-objecttracking algorithm for generic measurement models. Simulationresults for a multi-object Track-Before-Detect example usingradar measurements in low signal-to-noise ratio (SNR) scenariosverify the applicability of the proposed approach.
Index Terms —RFS, FISST, Multi-Object Tracking, PHD.
I. I
NTRODUCTION I N multi-object inference the objective is the estimation ofan unknown number of objects and their individual statesfrom noisy observations. Multi-object estimation is a coreproblem in spatial statistics [1], [2], and multi-target tracking[3], [4], spanning a diverse range of applications. Importantapplications of spatial statistics include agriculture/forestry[5]–[7], epidemiology/public health [1], [2], [8], communi-cations networks [9]–[11], while applications of multi-targettracking include radar/sonar [12]–[14], computer vision [15]–[18], autonomous vehicles [19]–[22], automotive safety [23],[24] and sensor networks [25]–[28]. The multi-object probabil-ity density is fundamental in multi-object estimation becauseit captures the uncertainty in the number and the states ofthe objects as well as the statistical dependence between theobjects. Statistical dependence between objects transpires viathe data when we consider the multi-object posterior density,or from the interactions between objects as in Markov pointprocesses [29], [30] or determinantal point processes [31]–[33].
Acknowledgement: This work is supported by the Australian ResearchCouncil under schemes DP130104404 and DE120102388.Francesco Papi, Ba-Ngu Vo, and Ba-Tuong Vo are with the Department ofElectrical and Computer Engineering, Curtin University, Bentley, WA 6102,Australia. E-mail: {francesco.papi, ba-tuong, ba-ngu.vo}@curtin.edu.auClaudio Fantacci is with the Dipartimento di Ingegneriadell’Informazione,Università di Firenze, Florence 50139, Italy. E-mail:claudio.fantacci@unifi.itMichael Beard is with Maritime Division, Defence Science andTechnology Organisation, Rockingham, WA 6958, Australia. E-mail:[email protected]
Computing the multi-object density is generally intractableand approximations are necessary. Tractable multi-object den-sities usually assume statistical independence between theobjects. For example, the Probability Hypothesis Density(PHD) [34], Cardinalized PHD (CPHD) [35], and multi-Bernoulli filters [36], are derived from multi-object densities inwhich objects are statistically independent. On the other hand,multi-object tracking approaches such as Multiple HypothesesTracking (MHT) [13], [37], [38] and Joint Probabilistic DataAssociation (JPDA) [14] are capable of modeling the statisticaldependence between objects. However, MHT does not havethe notion of multi-object density while JPDA only has thenotion of multi-object density for a known number of objects.A tractable family of multi-object densities that can capture thestatistical dependence between the objects is the recently pro-posed Generalized Labeled Multi-Bernoulli (GLMB) family,which is conjugate with respect to the standard measurementlikelihood function [39], [40].The bulk of multi-object estimation algorithms in the lit-erature, including those discussed above, are designed forthe so-called standard measurement model, where data hasbeen preprocessed into point measurements or detections [12]–[14], [35], [36]. For a generic measurement model the GLMBdensity is not necessarily a conjugate prior, i.e. the multi-objectposterior density is not a GLMB. This is the case in Track-Before-Detect (TBD) [41]–[46], tracking with superpositionalmeasurements [47], [48], merged measurements [49], andvideo measurements [50], [17]. In general, the multi-objectdensity is numerically intractable in applications involvingnon-standard measurement models. A simple strategy thatdrastically reduces the numerical complexity is to approximatethe measurement likelihood by a separable likelihood [50] forwhich Poisson, independently and identically distributed (IID)cluster, multi-Bernoulli and GLMB densities are conjugate.While this approximation can facilitate a trade off betweentractability and performance, biased estimates typically arisewhen the separable assumption is violated.Inspired by Mahler’s IID cluster approximation in theCPHD filter [35], in this paper we consider the approximationof a general labeled RFS density using a special tractableclass of GLMBs. In particular, we derive from this class ofGLMBs, an approximation to any labeled RFS density whichpreserves the cardinality distribution and the first moment.It is also established that our approximation minimizes theKullback-Leibler divergence (KLD) over this class of GLMBdensities. This approximation is then applied to develop anefficient multi-object tracking filter for a generic measurementmodel. As an example application, we consider a radar multi- object TBD problem with low signal-to-noise ratio (SNR)and closely spaced targets. Simulation results verify that theproposed approximation yields effective tracking performancein challenging scenarios.The paper is structured as follows: in Section II we recallsome definitions and results for Labeled random finite sets(RFSs) and GLMB densities. In Section III we propose theGLMB approximation to multi-object distributions via cardi-nality, first moment matching and KLD minimization. In Sec-tion IV we describe the application of our result to multi-objecttracking problems with non-standard measurement models.Simulation results for challenging, low SNR, multi-target TBDin radar scenarios are shown in Section V. Conclusions andfuture research directions are reported in Section VI.II. B
ACKGROUND
This section briefly presents background material on multi-object filtering and labeled RFS which form the basis for theformulation of our multi-object estimation problem.
A. Labeled RFS
An RFS on a space X is simply a random variable takingvalues in F ( X ) , the space of all finite subsets of X . Thespace F ( X ) does not inherit the Euclidean notion of integra-tion and density. Nonetheless, Mahler’s Finite Set Statistics(FISST) provides powerful yet practical mathematical toolsfor dealing with RFSs [3], [34], [51] based on a notion ofintegration/density that is consistent with measure theory [52].A labeled RFS is an RFS whose elements are assignedunique distinct labels [39]. In this model, the single-objectstate space X is the Cartesian product X × L , where X is thekinematic/feature space and L is the (discrete) label space.Let L : X × L → L be the projection L (( x, ℓ )) = ℓ . A finitesubset set X of X × L has distinct labels if X and its labels L ( X ) , { ℓ : ( x, ℓ ) ∈ X } have the same cardinality. An RFSon X × L with distinct labels is called a labeled RFS [39].For the rest of the paper, we use the standard inner productnotation h f, g i , ´ f ( x ) g ( x ) dx , and multi-object exponentialnotation h X , Q x ∈ X h ( x ) , where h is a real-valued function,with h ∅ = 1 by convention. We denote a generalization of theKroneker delta and the inclusion function which take arbitraryarguments such as sets, vectors, etc, by δ Y ( X ) , (cid:26) , if X = Y , otherwise Y ( X ) , (cid:26) , if X ⊆ Y , otherwiseWe also write Y ( x ) in place of Y ( { x } ) when X = { x } .Single-object states are represented by lowercase letters, e.g. x , x , while multi-object states are represented by uppercaseletters, e.g. X , X , symbols for labeled states and their distri-butions are bolded to distinguish them from unlabeled ones,e.g. x , X , π , etc, spaces are represented by blackboard bolde.g. X , Z , L , etc. The integral of a function f on X × L isgiven by ˆ f ( x ) d x = X ℓ ∈ L ˆ f ( x, ℓ ) dx. Two important statistics of an RFS relevant to this paper arethe cardinality distribution ρ ( · ) and the PHD v ( · ) [3]: ρ ( n ) = 1 n ! ˆ π ( { x , ..., x n } ) d ( x , ..., x n ) (1) v ( x, ℓ ) = ˆ π ( { ( x, ℓ ) } ∪ X ) δ X (2)where the integral is a set integral defined for any function f on F ( X ) by ˆ f ( X ) δ X = ∞ X i =0 i ! ˆ f ( { x , ..., x i } ) d ( x , ..., x i ) . The PHD in (2) and the unlabeled PHD in [39], i.e. the PHDof the unlabeled version, are related by v ( x ) = P ℓ ∈ L v ( x, ℓ ) .Hence, v ( · , ℓ ) can be interpreted as the contribution from label ℓ to the unlabeled PHD. B. Generalized Labeled Multi-Bernoulli
An important class of labeled RFS is the generalized labeledmulti-Bernoulli (GLMB) family [39], which forms the basisof an analytic solution to the Bayes multi-object filter [40].Under the standard multi-object likelihood, the GLMB is aconjugate prior, which is also closed under the Chapman-Kolmogorov equation [39]. Thus if initial prior is a GLMBdensity, then the multi-object prediction and posterior densitiesat all subsequent times are also GLMB densities.A GLMB is an RFS of X × L distributed according to π ( X ) = ∆( X ) X c ∈ C w ( c ) ( L ( X )) h p ( c ) i X (3)where ∆( X ) , δ | X | ( |L ( X ) | ) denotes the distinct label indica-tor , C is a discrete index set, and w ( c ) , p ( c ) satisfy: X L ⊆ L X c ∈ C w ( c ) ( L ) = 1 , (4) ˆ p ( c ) ( x, ℓ ) dx = 1 . (5)The GLMB density (3) can be interpreted as a mixture ofmulti-object exponentials. Each term in (3) consists of a weight w ( c ) ( L ( X )) that depends only on the labels of X , and amulti-object exponential (cid:2) p ( c ) (cid:3) X that depends on the labelsand kinematics/features of X .The cardinality distribution and PHD of a GLMB are,respectively, given by [39] ρ ( n ) = X c ∈ C X L ⊆ L δ n ( | L | ) w ( c ) ( L ) , (6) v ( x, ℓ ) = X c ∈ C p ( c ) ( x, ℓ ) X L ⊆ L L ( ℓ ) w ( c ) ( L ) . (7)A Labeled Multi-Bernoulli (LMB) density is a special caseof the GLMB density with one term (in which case thesuperscript ( c ) is not needed) and a specific form for the onlyweight w ( · ) [39], [53]: w ( L ) = Y ℓ ∈ M (cid:16) − r ( ℓ ) (cid:17) Y ℓ ∈ L M ( ℓ ) r ( ℓ ) − r ( ℓ ) , (8) where r ( ℓ ) for ℓ ∈ M ⊆ L represents the existence probabilityof track ℓ , and p ( · , ℓ ) is the probability density of the kinematicstate of track ℓ conditional upon existence [39]. Note thatthe LMB density can always be factored into a product ofterms over the elements of X . The LMB density can thusbe interpreted as comprising multiple independent tracks. TheLMB density is in fact the basis of the LMB filter, a principledand efficient approximation of the Bayes multi-object trackingfilter, which is highly parallelizable and capable of trackinglarge numbers of targets [53], [54].III. M ULTI -O BJECT E STIMATION WITH
GLMB S In this section we discuss the multi-object estimation prob-lem with GLMBs. In particular, in subsection III-A we presenta simple approximation through a separable likelihood func-tion which exploits the conjugacy of the GLMB distributions,while in subsection III-B we propose a more principledapproach for approximating a general labeled RFS densitywith a special form GLMB that matches both the PHD andcardinality distribution.
A. Conjugacy with respect to Separable Likelihoods
A separable multi-object likelihood of the state X given themeasurement z is one of the form [50]: g ( z | X ) ∝ γ X z = Y x ∈ X γ z ( x ) (9)where γ z ( · ) is a non-negative function defined on X .It was shown in [50] that Poisson, IID cluster and multi-Bernoulli densities are conjugate with respect to separablemulti-object likelihood functions. Moreover, this conjugacy iseasily extented to the family of GLMBs. Proposition 1.
If the multi-object prior density π is a GLMBof the form (3) and the multi-object likelihood is separableof the form (9), then the multi-object posterior density is aGLMB of the form: π ( X | z ) ∝ ∆( X ) X c ∈ C w ( c ) z ( L ( X )) h p ( c ) ( ·| z ) i X (10) where w ( c ) z ( L ) = [ η z ] L w ( c ) ( L ) (11) p ( c ) ( x, ℓ | z ) = p ( c ) ( x, ℓ ) γ z ( x, ℓ ) /η z ( ℓ ) (12) η z ( ℓ ) = D p ( c ) ( · , ℓ ) , γ z ( · , ℓ ) E (13) Proof: π ( X | z ) ∝ γ X z π ( X )= ∆( X ) X c ∈ C w ( c ) ( L ( X )) γ X z [ p ( c ) ] X = ∆( X ) X c ∈ C w ( c ) ( L ( X )) [ η z ] L ( X ) (cid:2) γ z p ( c ) (cid:3) X [ η z ] L ( X ) = ∆( X ) X c ∈ C w ( c ) z ( L ( X )) h p ( c ) ( ·| z ) i X . (cid:4) In general, the true multi-object likelihood is not separa-ble, however the separable likelihood assumption can be areasonable approximation if the objects do not overlap in themeasurement space [50].
B. Labeled RFS Density Approximation
In this subsection we propose a tractable GLMB densityapproximation to an arbitrary labeled multi-object density π .Tractable GLMB densities are numerically evaluated via theso-called δ -GLMB form which involves explicit enumerationof the label sets (for more details see [39], [40]). Since thereis no general information on the form of π , a natural choiceis the class of δ -GLMBs of the form ¯ π ( X ) = ∆( X ) X L ∈F ( L ) ¯ w ( L ) δ L ( L ( X )) h ¯ p ( L ) i X (14)where each ¯ p ( L ) ( · , ℓ ) is a density on X , and each weight ¯ w ( L ) is non-negative such that P L ⊆ L w ( L ) = 1 . It follows from (6)and (7) that the cardinality distribution and PHD of (14) aregiven, respectively, by ¯ ρ ( n ) = X L ⊆ L δ n ( | L | ) ¯ w ( L ) , (15) ¯ v ( x, ℓ ) = X L ⊆ L L ( ℓ ) ¯ w ( L ) ¯ p ( L ) ( x, ℓ ) . (16)Note that such δ -GLMB is completely characterised by theparameter set { ( ¯ w ( L ) , ¯ p ( L ) ) } L ∈F ( L ) . Our objective is to seeka density, via its parameter set, from this class of δ -GLMBs,which matches the PHD and cardinality distribution of π .The strategy of matching the PHD and cardinality distribu-tion is inspired by Mahler’s IID cluster approximation in theCPHD filter [35], which has proven to be very effective inpractice [4], [55], [56]. While our result is used to developa multi-object tracking algorithm in the next section, it is notnecessarily restricted to tracking applications, and can be usedin more general multi-object estimation problems.Our result follows from the following representation forlabeled RFS. Definition 1.
Given a labeled multi-object density π on F ( X × L ) , and any positive integer n , we define the jointexistence probability of the label set { ℓ , ..., ℓ n } by w ( { ℓ , ..., ℓ n } ) , ˆ π ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n ) (17)and the joint probability density (on X n ) of x , ..., x n , condi-tional on their corresponding labels ℓ , ..., ℓ n , by p ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) , π ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) w ( { ℓ , ..., ℓ n } ) (18)For n = 0 , we define w ( ∅ ) , π ( ∅ ) and p ( ∅ ) , . It isimplicit that p ( X ) is defined to be zero whenever w ( L ( X )) is zero. Consequently, the labeled multi-object density can beexpressed as π ( X ) = w ( L ( X )) p ( X ) (19) Remark . Note that P L ∈F ( L ) w ( L ) = 1 , and since π issymmetric in its arguments it follows from Lemma 1 that w ( · ) is also symmetric in ℓ , ..., ℓ n . Hence w ( · ) is indeed aprobability distribution on F ( L ) . Lemma 1.
Let f : ( X × Y ) n → R be symmetric. Then g : X n → R given by g ( x , ..., x n ) = ˆ f (( x , y ) , ..., ( x n , y n )) d ( y , ..., y n ) is also symmetric on X n .Proof: Let σ be a permutation of { , ..., n } , then g ( x σ (1) , ..., x σ ( n ) )= ˆ f (( x σ (1) , y σ (1) ) , ..., ( x σ ( n ) , y σ ( n ) )) d ( y σ (1) , ..., y σ ( n ) )= ˆ f (( x , y ) , ..., ( x n , y n )) d ( y σ (1) , ..., y σ ( n ) )= ˆ f (( x , y ) , ..., ( x n , y n )) d ( y , ..., y n ) where the last step follows from the fact that the order ofintegration is interchangeable. Proposition 2.
Given any labeled multi-object density π , the δ -GLMB density in the class defined by (14) which preservesthe cardinality distribution and PHD of π , and minimizes theKullback-Leibler divergence from π , is given by ˆ π ( X ) = ∆( X ) X I ∈F ( L ) ˆ w ( I ) δ I ( L ( X )) h ˆ p ( I ) i X (20) where ˆ w ( I ) = w ( I ) , (21) ˆ p ( I ) ( x, ℓ ) = 1 I ( ℓ ) p I −{ ℓ } ( x, ℓ ) , (22) p { ℓ ,...,ℓ n } ( x, ℓ ) = ˆ p ( { ( x, ℓ ) , ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n ) . (23) Remark . Note from the definition of ˆ p ( I ) ( x, ℓ ) in (22) that ˆ p ( { ℓ,ℓ ,...,ℓ n } ) ( x, ℓ )= ˆ p ( { ( x, ℓ ) , ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n ) (24)Hence, ˆ p ( { ℓ ,...,ℓ n } ) ( · , ℓ i ) , i = 1 , ..., n , defined in (22)are the marginals of the label-conditioned joint density p ( { ( · , ℓ ) , ..., ( · , ℓ n ) } ) of π .Proposition 2 states that replacing the label-conditionedjoint densities, of a labeled multi-object density π , by theproducts of their marginals yields a δ -GLMB of the form (14),which minimises the Kullback-Leibler divergence from π , andmatches its PHD and cardinality distribution. Proof:
Since p { ℓ ,...,ℓ n } ( x, ℓ ) is symmetric in ℓ , ..., ℓ n ,via Lemma 1, ˆ p ( I ) ( x, ℓ ) is indeed a function of the set I .The proof uses the fact (14) can be rewritten as ¯ π ( X ) =¯ w ( L ( X ))¯ p ( X ) where ¯ w ( L ) = ¯ w ( L ) , ¯ p ( X ) = ∆( X ) h ¯ p ( L ( X )) i X . To show that ˆ π preserves the cardinality of π , observe thatthe cardinality distribution of any labeled RFS is completelydetermined by the joint existence probabilities of the labels w ( · ) , i.e. ρ ( n ) = 1 n ! X ( ℓ ,...,ℓ n ) ∈ L n ˆ w ( { ℓ , ..., ℓ n } ) × p ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n )= X L ⊆ L δ n ( | L | ) w ( L ) Since both ˆ π and π have the same joint existence probabilities,i.e. ˆ w ( L ) = ˆ w ( L ) = w ( L ) , their cardinality distributions arethe same.To show that the PHDs of ˆ π and π are the same, note from(16) that the PHD of ˆ π can be expanded as ˆ v ( x, ℓ ) = ∞ X n =0 n ! X ( ℓ ,...,ℓ n ) ∈ L n ˆ w ( { ℓ,ℓ ,...,ℓ n } ) ˆ p ( { ℓ,ℓ ,...,ℓ n } ) ( x, ℓ )= ∞ X n =0 n ! X ( ℓ ,...,ℓ n ) ∈ L n w ( { ℓ, ℓ , ..., ℓ n } ) × ˆ p ( { ( x, ℓ ) , ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n ) where the last step follows by substituting (21) and (24).The right hand side of the above equation is the set integral ´ π ( { ( x, ℓ ) } ∪ X ) δ X . Hence ˆ v ( x, ℓ ) = v ( x, ℓ ) .The Kullback-Leibler divergence from π and any δ -GLMBof the form (14) is given by D KL ( π ; ¯ π )= ˆ log (cid:18) w ( L ( X )) p ( X )¯ w ( L ( X ))¯ p ( X ) (cid:19) w ( L ( X )) p ( X ) δ X = ∞ X n =0 n ! X ( ℓ ,...,ℓ n ) ∈ L n log (cid:18) w ( { ℓ , ..., ℓ n } )¯ w ( { ℓ , ..., ℓ n } ) (cid:19) × w ( { ℓ , ..., ℓ n } ) ˆ p ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n )+ ∞ X n =0 n ! X ( ℓ ,...,ℓ n ) ∈ L n ˆ log (cid:18) p ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) Q ni =1 ¯ p ( { ℓ ,...,ℓ n } ) ( x i , ℓ i ) (cid:19) × w ( { ℓ , ..., ℓ n } ) p ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n ) Noting that p ( { ( · , ℓ ) , ..., ( · , ℓ n ) } ) integrates to 1, we have D KL ( π ; ¯ π ) = D KL ( w ; ¯ w ) + ∞ X n =0 n ! X ( ℓ ,...,ℓ n ) ∈ L n w ( { ℓ , ..., ℓ n } ) × D KL p ( { ( · , ℓ ) , ..., ( · , ℓ n ) } ); n Y i =1 ¯ p ( { ℓ ,...,ℓ n } ) ( · , ℓ i ) ! Setting ¯ π = ˆ π we have D KL ( w ; ˆ w ) = 0 since ˆ w ( I ) = w ( I ) .Moreover, for each n and each { ℓ , ..., ℓ n } , ˆ p ( { ℓ ,...,ℓ n } ) ( · , ℓ i ) , i = 1 , ..., n , are the marginals of p ( { ( · , ℓ ) , ..., ( · , ℓ n ) } ) .Hence, it follows from [57] that each Kullback-Leibler diver-gence in the above sum is minimized. Therefore, D KL ( π ; ˆ π ) is minimized over the class of δ -GLMB of the form (14). The cardinality and PHD matching strategy in the aboveProposition can be readily extended to the approximation ofany labeled multi-object density of the form π ( X ) = ∆( X ) X c ∈ C w ( c ) ( L ( X )) p ( c ) ( X ) (25)where the weights w ( c ) ( · ) satisfy (4) and ˆ p ( c ) ( { ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n ) = 1 (26)by approximating each p ( c ) ( { ( · , ℓ ) , ..., ( · , ℓ n ) } ) by the productof its marginals. This is a better approximation than directlyapplying Proposition 2 to (25), which only approximates thelabel-conditioned joint densities of (25). However, it is difficultto establish any results on the Kullback-Leibler divergence forthis more general class. Proposition 3.
Given any labeled multi-object density ofthe form (25) a δ -GLMB which preserves the cardinalitydistribution and the PHD of π is given by ˆ π ( X ) = ∆( X ) X ( c,I ) ∈ C ×F ( L ) δ I ( L ( X )) ˆ w ( c,I ) h ˆ p ( c,I ) i X (27) where ˆ w ( c,I ) = w ( c ) ( I ) , (28) ˆ p ( c,I ) ( x, ℓ ) = 1 I ( ℓ ) p ( c ) I −{ ℓ } ( x, ℓ ) , (29) p ( c ) { ℓ ,...,ℓ n } ( x, ℓ ) = ˆ p ( c ) ( { ( x, ℓ ) , ( x , ℓ ) , ..., ( x n , ℓ n ) } ) d ( x , ..., x n ) . (30)The proof follows along the same lines as Proposition 2. Remark . Note that in [49, Sec. V] a δ -GLMB was proposedto approximate a particular family of labeled RFS densitiesthat arises from multi-target filtering with merged measure-ments. Our results show that the approximation used in [49,Sec. V] preserves the cardinality distribution and the PHD.In multi-object tracking, the matching of the cardinalitydistribution and PHD in Proposition 2 is a stronger result thansimply matching the PHD alone. Notice that this property doesnot hold for the LMB filter, as shown in [53] (Section III),due to the imposed multi-Bernoulli parameterization of thecardinality distribution.IV. A PPLICATION TO MULTI - TARGET T RACKING
In this section we propose a multi-target tracking filterfor generic measurement models by applying the GLMBapproximation result of Proposition 2. Specifically, we presentthe prediction and update of the Bayes multi-target filter (32)-(33) for the standard multi-target dynamic model as well as ageneric measurement model.
A. Multi-target Filtering
Following [39], [40], to ensure distinct labels we assigneach target an ordered pair of integers ℓ = ( k, i ) , where k isthe time of birth and i is a unique index to distinguish targetsborn at the same time. The label space for targets born at time k is denoted by L k , and the label space for targets at time k (including those born prior to k ) is denoted as L k . Note that L k and L k − are disjoint and L k = L k − ∪ L k .A multi-target state X k at time k , is a finite subset of X = X × L k . Similar to the standard state space model, themulti-target system model can be specified, for each time step k , via the multi-target transition density f k | k − ( ·|· ) and the multi-target likelihood function g k ( ·|· ) , using the FISST notionof integration/density. The multi-target posterior density (orsimply multi-target posterior) contains all information onthe multi-target states given the measurement history. Themulti-target posterior recursion generalizes directly from theposterior recursion for vector-valued states [58], i.e. for k ≥ π k ( X k | z k ) ∝ g k ( z k | X k ) f k | k − ( X k | X k − ) π k − ( X k − | z k − ) , (31)where X k = ( X , ..., X k ) is the multi-target state history,and z k = ( z , ..., z k ) is the measurement history with z k denoting the measurement at time k . Target trajectories ortracks are accommodated in this formulation through theinclusion of a distinct label in the target’s state vector [3],[39], [51], [59]. The multi-target posterior (31) then containsall information on the random finite set of tracks, given themeasurement history.In this work we are interested in the multi-target filteringdensity π k , a marginal of the multi-target posterior, which canbe propagated forward recursively by the multi-target Bayesfilter [3], [34] π k ( X k | z k ) = g k ( z k | X k ) π k | k − ( X k ) ´ g k ( z k | X ) π k | k − ( X ) δ X , (32) π k +1 | k ( X k +1 ) = ˆ f k +1 | k ( X k +1 | X ) π k ( X | z k ) δ X , (33)where π k +1 | k is the multi-target prediction density to time k +1 (the dependence on the data is omitted for compactness).An analytic solution to the multi-target Bayes filter for labeledstates and track estimation from the multi-target filteringdensity is given in [39]. Note that a large volume of workin multi-target tracking is based on filtering, and often theterm "multi-target posterior" is used in place of "multi-targetfiltering density". In this work we shall not distinguish betweenthe filtering density and the posterior density. B. Update
In this section we apply the proposed δ -GLMB approx-imation to multi-target tracking with a generic measurementmodel. We do not assume any particular structure for the multi-target likelihood function g ( ·|· ) and hence the approach inthis section is applicable to any measurement model includingpoint detections, superpositional sensors and imprecise mea-surements [3], [60]. If the multi-target prediction density is a δ -GLMB of the form π k | k − ( X ) = ∆( X ) X I ∈F ( L k ) δ I ( L ( X )) w ( I ) k | k − h p ( I ) k | k − i X , (34) then the multi-target posterior density (32) becomes π k ( X | z k ) = ∆( X ) X I ∈F ( L k ) δ I ( L ( X )) w ( I ) k ( z k ) p ( I ) k ( X | z k ) , (35)where w ( I ) k ( z k ) ∝ w ( I ) k | k − η z k ( I ) , (36) p ( I ) k ( X | z ) = g ( z k | X )[ p ( I ) k | k − ] X /η z k ( I ) , (37) η z k ( { ℓ , ..., ℓ n } ) = ˆ g ( z k |{ ( x , ℓ ) , ..., ( x n , ℓ n ) } ) × n Y i =1 p ( { ℓ ,...,ℓ n } ) k | k − ( x i , ℓ i ) d ( x , ..., x n ) . (38)Note from (37) that after the update each multi-objectexponential [ p ( I ) k | k − ] X from the prior δ -GLMB becomes p ( I ) k ( X | z k ) , which is not necessarily a multi-object exponen-tial. Hence, in general, (35) is not a GLMB density.
1) Separable Likelihood:
If targets are well separated in themeasurement space, we can approximate the likelihood by aseparable one, i.e. g ( z k | X ) ≈ γ X z k , and obtain an approximateGLMB posterior from Proposition 1: ˆ π k ( X | z k ) = ∆( X ) X I ∈F ( L k ) δ I ( L ( X )) ˆ w ( I ) k ( z k ) h ˆ p ( I ) k ( ·| z k ) i X , (39)where ˆ w ( I ) k ( z k ) ∝ w ( I ) k | k − [ η z k ] I , (40) ˆ p ( I ) k ( x, ℓ | z k ) = p ( I ) k | k − ( x, ℓ ) γ z k ( x, ℓ ) /η z k ( ℓ ) , (41) η z k ( ℓ ) = D p ( I ) k | k − ( · , ℓ ) , γ z k ( · , ℓ ) E . (42)
2) General Case:
If instead targets are closely spaced, theseparable likelihood assumption is violated, then it becomesnecessary to directly approximate the multi-target posterior in(35) which can be rewritten as: π k ( X | z k ) = w ( L ( X )) k ( z k )∆( X ) p ( L ( X )) k ( X | z k ) (43)It follows from Proposition 2 that an approximate δ -GLMBof the form (14), which matches the cardinality and PHD ofthe above multi-target posterior, as well as minimizing theKullback-Leibler divergence from it, is given by ˆ π k ( X | z k ) = ∆( X ) X I ∈F ( L k ) δ I ( L ( X )) w ( I ) k ( z k ) h ˆ p ( I ) k ( ·| z k ) i X , (44)where for each label set I = { ℓ , ..., ℓ n } , the densi-ties ˆ p ( { ℓ ,...,ℓ n } ) k ( · , ℓ i | z k ) , i = 1 , ..., n are the marginals of p ( { ℓ ,...,ℓ n } ) k { ( · , ℓ ) , ..., ( · , ℓ n ) }| z k ) . Notice that we retained theweights w ( I ) k ( z k ) , given by (36), from the true posterior (35). C. Prediction
The standard multi-target dynamic model is described asfollows. Given the current multi-target state X ′ , each state ( x ′ , ℓ ′ ) ∈ X ′ either continues to exist at the next time stepwith probability p S ( x ′ , ℓ ′ ) and evolves to a new state ( x, ℓ ) with probability density f k +1 | k ( x | x ′ , ℓ ′ ) δ ℓ ( ℓ ′ ) , or dies withprobability − p S ( x ′ , ℓ ′ ) . The multi-target state at the next time is the superposition of surviving and new born targets.The set of new targets born at the next time step is distributedaccording to a birth density f B on F ( X × L k +1 ) , given by f B ( Y ) = ∆( Y ) w B ( L ( Y )) [ p B ] Y (45)This birth model covers labeled Poisson, labeled IID clusterand LMB. We use an LMB birth model with w B ( L ) = Y i ∈ L k (cid:16) − r ( i ) B (cid:17) Y ℓ ∈ L L k ( ℓ ) r ( ℓ ) B − r ( ℓ ) B , (46) p B ( x, ℓ ) = p ( ℓ ) B ( x ) . (47)Following [39], if the current multi-target posterior has thefollowing δ -GLMB form π k ( X ) = ∆( X ) X I ∈F ( L k ) δ I ( L ( X )) w ( I ) k h p ( I ) k i X , (48)then the multi-target prediction (33) is also a δ -GLMB: π k +1 | k ( X ) = ∆( X ) X I ∈F ( L k +1 ) δ I ( L ( X )) w ( I ) k +1 | k h p ( I ) k +1 | k i X (49)where w ( I ) k +1 | k = w ( I ) S ( I ∩ L k ) w B ( I ∩ L k +1 ) ,w ( I ) S ( L ) = [ η ( I ) S ] L X J ⊆ L k J ( L )[1 − η ( I ) S ] J − L w ( I ) k ( J ) ,p ( I ) k +1 | k ( x, ℓ ) = 1 L k ( ℓ ) p ( I ) S ( x, ℓ ) + (1 − L k ( ℓ )) p B ( x, ℓ ) ,p ( I ) S ( x, ℓ ) = D p S ( · , ℓ ) f k +1 | k ( x |· , ℓ ) , p ( I ) k ( · , ℓ ) E η ( I ) S ( ℓ ) ,η ( I ) S ( ℓ ) = D p S ( · , ℓ ) , p ( I ) k ( · , ℓ ) E . The above Eqs. explicitly describe the calculation of theparameters of the predicted multi-target density from theparameters of the previous multi-target density [40].V. N
UMERICAL R ESULTS
In this section we verify the proposed GLMB approximationtechnique via an application to recursive multi-target trackingwith radar power measurements. Target tracking is usuallyperformed on data that have been preprocessed into pointmeasurements or detections. The bulk of multi-target trackingalgorithms in the literature are designed for this type of data[3], [12], [61], [62]. Compressing information from the rawmeasurement into a finite set of points is very effective for awide range of applications. However, for applications with lowSNR, this approach may not be adequate as the informationloss incurred in the compression becomes significant. Conse-quently, it becomes necessary to make use of all informationcontained in the pre-detection measurements, which in turnrequires more advanced sensor models and algorithms.We first describe the single-target dynamic model andmulti-target measurement equation used to simulate the radarpower measurements. We then report numerical results for theseparable likelihood approximation and GLMB posterior ap-proximation. Throughout this section our recursive multi-targettracker is implemented with a particle filter approximation[58], [63] of the GLMB density given in [40].
A. Dynamic Model
The kinematic part of the single-target state x k = ( x k , ℓ k ) at time k comprises the planar position, velocity vectorsin 2D Cartesian coordinates, and the unknown modulus ofthe target complex amplitude ζ k , respectively, i.e. x k =[ p x,k , ˙ p x,k , p y,k , ˙ p y,k , ζ k ] T . A Nearly Constant Velocity (NCV)model is used to describe the target dynamics, while a zero-mean Gaussian random walk is used to model the fluctuationsof the target complex amplitude, i.e. x k +1 = F x k + v k , v k ∼ N (0; Q ) where F = diag ( F , F , , Q = diag ( qQ , qQ , a ζ T s ) , F = (cid:20) T s (cid:21) , Q = " T s T s T s T s with T s , q , and a ζ denoting the radar sampling time, thepower spectral density of the process noise, and the amplitudefluctuation in linear domain, respectively. B. TBD Measurement Equation
A target x ∈ X illuminates a set of cells C ( x ) , usually re-ferred to as the target template . A radar positioned at the Carte-sian origin collects a vector measurement z = [ z (1) , ..., z ( m ) ] consisting of the power signal returns z ( i ) = | z ( i ) A | , where z ( i ) A = X x ∈ X C ( x ) ( i ) A ( x ) h ( i ) A ( x ) + w ( i ) is the complex signal in cell i , with: • w ( i ) denoting zero-mean white circularly symmetric com-plex Gaussian noise with variance σ w ; • h ( i ) A ( x ) denoting the point spread function value in cell i from a target with state x h ( i ) A ( x ) = exp (cid:18) − ( r i − r ( x )) R − ( d i − d ( x )) D − ( b i − b ( x )) B (cid:19) where R , D , B are resolutions for range, Doppler, bear-ing; r ( x ) = q p x + p y , d ( x ) = − ( ˙ p x p x + ˙ p y p y ) /r ( x ) , b ( x ) = atan2 ( p y , p x ) are range, Doppler, bearing, giventhe target state x ; and r i , d i , b i are cell centroids; • A ( x ) denoting the complex echo of target x , which fora Swerling model is constant in modulus A ( x ) = ¯ Ae jθ , θ ∼ U [0 , π ) . Let ˆ z ( i ) = | ˆ z ( i ) A | be the noiseless power return in cell i , where ˆ z ( i ) A = X x ∈ X C ( x ) ( i ) ¯ Ah ( i ) A ( x ) . The measurement z ( i ) in each cell follows a non-central chi-squared distribution with degrees of freedom and non-centrality parameter ˆ z ( i ) A , and simplifies to a central chi-squared distribution with degrees of freedom when ˆ z ( i ) A = 0 .Consequently, the likelihood ratio for cell ( i ) is given by: ℓ ( z ( i ) | X ) = exp (cid:16) − . z ( i ) (cid:17) I (cid:16)p z ( i ) ˆ z ( i ) (cid:17) (50)where I ( · ) is the modified Bessel function, which can beevaluated using the approximation given in [64]. Given a vector measurement z the likelihood function ofthe multi-target state X takes the form g ( z | X ) ∝ Y i ∈ ∪ x ∈ X C ( x ) ℓ ( z ( i ) | X ) , (51)Notice that eqs. (50)-(51) capture the superpositional natureof the power returns for each measurement bin due to thepossibility of closely spaced targets target, i.e. overlappingtarget templates. The separable likelihood assumption is ob-tained from eqs. (50)-(51) by assuming that at most one targetcontributes to the power return from each cell ( i ) , ˆ z ( i ) = | ˆ z ( i ) A | = ( | ¯ Ah ( i ) A ( x ) | , ∃ x ∈ X : i ∈ C ( x )0 , otherwiseIn the numerical examples we use
10 log (cid:0) ¯ A / (2 σ w ) (cid:1) as thesignal-to-noise ratio (SNR) definition, and choosing σ w = 1 implies ¯ A = √ · SNR/ . Table IC
OMMON P ARAMETERS USED IN S IMULATIONS
Parameter Symbol ValueSignal-to-Noise Ratio SNR dBPower Spectral Density q m / s Amplitude Fluctuation a ρ st Birth Point Coordinates x B [1250 , − , , − nd Birth Point Coordinates x B [1000 , − , , − rd Birth Point Coordinates x B [1250 , − , , − Birth Probability P B . Survival Probability P S . n ◦ of particles per target N p Table IIS
EPARABLE L IKELIHOOD P ARAMETERS
Parameter Symbol ValueRange Resolution R mAzimuth Resolution B ◦ Doppler Resolution D m / sSampling Time T s sBirth Covariance Q B diag ([25 , , , C. Separable Likelihood Results
In this section we report simulation results for a radar TBDscenario under the separable likelihood assumption, which isvalid when targets do not overlap at any time. This impliesthat the birth density is relatively informative compared tothe targets kinematics. This apparently obvious requirement isnecessary to avoid a bias in the estimated number of targetsdue to new target or birth hypotheses which always violate theseparable likelihood assumption.The considered scenario is depicted in Fig. 1: we havea time varying number of targets due to various births anddeaths with a maximum of targets present mid scenario.The parameters are reported in Tables I and II. Fig. 2 showsthe estimation results for a single trial along the x and y coordinates, and Fig. 3 shows the Monte Carlo results for theestimated number of targets and positional OSPA distance.Notice that the average estimated number of targets slightlydiffers from the true number due to closely spaced targets (seeFig. 1), but the overall performance is satisfactory given thelow SNR of dB.
700 800 900 1000 1100 1200 13005006007008009001000110012001300
Coordinate x (m) C oo r d i n a t e y ( m ) Start PointEnd PointTrajectoriesEstimates
Fig. 1. Separable likelihood scenario. Targets appear from the top rightcorner and move closer to the radar positioned at the Cartesian origin.
Simulation Scan (n) x - c oo r d i n a t e ( m ) TrajectoriesEstimates
Simulation Scan (n) y - c oo r d i n a t e ( m ) Fig. 2. Separable likelihood scenario. Estimated trajectories along the x and y coordinates. D. Non-Separable Likelihood Results
In this section we consider a more difficult radar TBDscenario where the separable likelihood assumption would leadto a bias on the estimated number of targets. Fig. 4 showsa time varying number of targets due to various births anddeaths with a maximum of targets present mid scenario. Fig.5 shows range-azimuth, range-Doppler, and azimuth-Dopplermaps of the received power returns. Notice that for each 2Dmap, the index of the rd coordinate is such that all mapsrefer to the same group of targets. Specifically, the targetreflection around ( m, . ◦ , m/s) is due to two targetsin the same Radar cell. This leads to the so-called unresolvedtarget problem, which usually results in track loss when usinga standard detection based approach or a separable likelihoodassumption. The parameters used in simulation are reported inTables I and III. Simulation Scan (n) O S P A D i s t . ( m ) Simulation Scan (n) n ◦ o f T a r g e t s True NumberEstimated NumberEst. n ◦ ± std. dev. Fig. 3. Separable likelihood scenario. Monte Carlo results for estimatednumber of targets (top) and the OSPA distance (bottom) with cut-off c = 50 mTable IIIN ON -S EPARABLE L IKELIHOOD P ARAMETERS
Parameter Symbol ValueRange Resolution R mAzimuth Resolution B ◦ Doppler Resolution D m / sSampling Time T s sBirth Covariance Q B diag ([400 , , , The estimation results for a single trial along the x and y coordinates are shown in Fig. 6, and the Monte Carlo resultsfor the estimated number of targets and positional OSPAerror is shown in Fig. 7. The results demonstrate that theproposed GLMB approximation exhibits satisfactory trackingperformance.
400 600 800 1000 1200 140040050060070080090010001100120013001400
Coordinate x (m) C oo r d i n a t e y ( m ) Fig. 4. Non-separable likelihood scenario. Targets appear from the top rightcorner and move closer to the radar positioned at the Cartesian origin.
Fig. 5. Non-separable likelihood scenario. Range-Azimuth, Range-Doppler,and Azimuth-Doppler maps at time instant k = 19 . Ideal or noiselessmeasurement ( right column ), and noisy measurement ( left column ). Noticethat for each 2D map, the index of the rd coordinate is such that all mapsrefer to the same group of targets. Specifically, the target reflection around( m, . ◦ , m/s) is due to two targets in the same Radar cell. VI. C
ONCLUSIONS
This paper has proposed a tractable class of GLMB ap-proximations for labeled RFS densities. In particular, wederived from this class of GLMBs an approximation that cancapture the statistical dependence between targets, preservesthe cardinality distribution and the PHD, as well as minimizesthe Kullback-Leibler divergence. The result has particularsignificance in multi-target tracking since it leads to tractablerecursive filter implementations with formal track estimatesfor a wide range of non-standard measurement models. Aradar based TBD example with low SNR and a time varyingnumber of closely spaced targets was presented to verify thetheoretical result. The key result presented in Section III isnot only important to recursive multi-target filtering but isalso generally applicable to statistical estimation problemsinvolving point processes or random finite sets.R
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