Qiang GUO,Long TENG?,,Xinliang WU,Wenming SONG,Dayu HUANG

1College of Information and Communication Engineering,Harbin Engineering University,Harbin 150001,China

2China National Aeronautical Radio Electronics Research Institute,Shanghai 200233,China

Abstract:A novel algorithm that combines the generalized labeled multi-Bernoulli(GLMB)filter with signal features of the unknown emitter is proposed in this paper.In complex electromagnetic environments,emitter features(EFs)are often unknown and time-varying.Aiming at the unknown feature problem,we propose a method for identifying EFs based on dynamic clustering of data fields.Because EFs are time-varying and the probability distribution is unknown,an improved fuzzy C-means algorithm is proposed to calculate the correlation coefficients between the target and measurements,to approximate the EF likelihood function.On this basis,the EF likelihood function is integrated into the recursive GLMB filter process to obtain the new prediction and update equations.Simulation results show that the proposed method can improve the tracking performance of multiple targets,especially in heavy clutter environments.

Key words:Multi-target tracking;Generalized labeled multi-Bernoulli;Signal features of emitter;Fuzzy C-means;Dynamic clustering

1 Introduction

The objective of multi-target tracking is to transform uncertain measurement information into deterministic target state information.At present,multi-target tracking methods include two categories:(1)data association algorithms(Guo YF et al.,2015),such as probabilistic data association(PDA)(Guo YF et al.,2016),joint PDA(JPDA)(Guo YF et al.,2020a,2020b;Zhu Y et al.,2021),and multiple hypotheses tracking(MHT)(Chen et al.,2004,2008);(2)random finite set(RFS)methods(Mahler RPS,2007;Da et al.,2021).The former requires complex data association operations.The latter models the state and measurement information of the target as an RFS,which can effectively avoid the complex data correlation process and is a popular multi-target tracking method nowadays.The core of the RFS is the Bayesian multi-objective filter,which propagates the posterior density of multi-objective states recursively in time.

Because Bayesian multi-objective filters do not have closed solutions,approximation methods such as probability hypothesis density(PHD),cardinality PHD(CPHD),and multi-Bernoulli(MB)filters have been proposed one after another(Mahler RPS,2003;Mahler R,2007;Ristic et al.,2013).It is assumed that the multi-object probability distribution in PHD and CPHD filters is a Poisson process and an independent and identically distributed process,respectively.PHD and CPHD filters recursivelypropagate the combination of the statistical moments and cardinality distribution of the posterior distribution.It is assumed that the multi-target probability distribution in the MB filter is an MB process,and the MB filter directly approximates the multi-objective probability distribution and recurses the parameters of the MB distribution.Implementations of these filters include Gaussian mixture(GM)and sequential Monte Carlo(SMC)(Vo and Ma,2006;Li et al.,2016),as well as multiple extended versions(Li et al.,2017,2018;Wang et al.,2021),such as distributed PHD/CPHD filter(Battistelli et al.,2013;Da et al.,2020;Yi et al.,2020;Li and Hlawatsch,2021;Yi and Chai,2021)and distributed Bernoulli filter(Li et al.,2019).However,these filters cannot directly form track information.

To address the inherent drawback of the above filters,Vo et al.(2014)introduced a conjugate prior based on the Chapman–Kolmogorov equation to derive a multi-target tracking algorithm for obtaining target track labels,i.e.,the generalized labeled multi-Bernoulli(GLMB)filter.Vo et al.(2017)further improved the real-time performance of the GLMB filter by combining the prediction and update steps and introducing Gibbs sampling to truncate the density.More extended applications have been developed,such as the multi-model GLMB filter(Yi et al.,2017;Wu et al.,2021)and the distributed GLMB filter(Herrmann et al.,2021).

However,as the number of clutters in the target tracking scene increases,the differentiation between the target measurement and the clutter gradually decreases.The tracking performance of the above RFS filters will be degraded to different degrees.To improve the anti-clutter performance of RFS filters,some algorithms integrate multi-dimensional independent information into the RFS filters.Bar-Shalom et al.(2005)proposed a target tracking algorithm with classification information by integrating target classification information into the data association process.In the target tracking scenarios,the amplitude of the target echo is stronger than those coming from clutter.Clark et al.(2010)proved that the amplitude information of the target echo can improve the multi-target state estimation accuracy,and applied it to PHD and CPHD filters under Gaussian conditions.Similarly,amplitude information was integrated into MB and GLMB filters to improve multi-target tracking performance in the literature(Liu C et al.,2018;Peng et al.,2018;Sun et al.,2020).Doppler information was also widely employed in multi-target tracking(Peng et al.,2018;Jin et al.,2019).

In a multi-heterogeneous sensor tracking system,we can not only obtain the target’s kinetic information(Cao and Zhao,2022),but also intercept the emitter features(EFs)which are called the pulse description words(PDWs),such as radio frequency(RF),pulse width(PW),and pulse repetition frequency(PRF).Each feature reflects the electromagnetic characteristics of the emitter in different dimensions,and plays a vital role in the classification and identification of the emitter.RF is the frequency at which electromagnetic waves are emitted,and is closely related to the working state and modulation mode of the emitter.PW is the duration of the transmitted pulse to the maximum value.PRF determines the maximum unambiguity range and radial velocity of the radar.To our knowledge,there are few studies of RFS filters with EFs;Zhou and Zhu(2015)and Zhu YQ(2015)are the only ones in which EFs were integrated in a PHD filter with the prerequisite that the EFs are known and non-timevarying.The filter does not form track information.

In a real target tracking scenario,the target track is necessary and the EFs are usually unknown and time-varying.In this paper,the state and measurement of the target are extended,and the EF identification method based on dynamic clustering of the data field is proposed to solve the problem of unknown EF.On this basis,an improved fuzzy C-means(FCM)algorithm is proposed,which can approximately calculate the time-varying EF likelihood and solve the problems that EFs are timevarying and the probability distribution is unknown.Then the EFs are integrated into the GLMB filter to solve the problem that the track cannot be generated directly.The filter proposed in this paper can improve multi-target tracking performance,especially in heavy clutter environments.

2 Background

We briefly review the GLMB filter,including labeled RFS,multi-object state transition model,multi-object observation model,and GLMB.Readers can refer to Vo et al.(2014)for details.

2.1 Labeled RFS

Denote X as the single target state space and L as the discrete label space.L:X×L→L is the projection defined byL((x,?))=?.ThenL(X)is called the label of pointx∈X×L.A finite subsetXof X×L is said to have distinct labels if and only ifXand its labelsL(X)={L(x):x∈X}have the same cardinality.The available label indicator function is defined as

where|X|is the cardinality distribution.

2.2 Multi-object state transition model

Given the multi-object stateX,each(x,?)∈Xeither survives with probabilityPs(x,?)and propagates to get a new state(x+,?+),or dies with probability 1?Ps(x,?).For notational compactness,the subscriptkfor the time index is omitted and the subscript“+”is used to denote the next time.The new state includes the objects of survival and new birth.The setB+of the new birth objects is distributed according to the labeled multi-Bernoulli(LMB)density:

whererB,+(?)is the probability of the birth object with label?,pB,+(·,?)is the distribution of the corresponding kinetic states,and B is the label space for the birth objects.Assuming that the target’s move,birth,and death are independent of each other,the multi-object transition density is(Vo et al.,2014)

wherefs,+(·)andfb,+(·)represent the density functions of the surviving target and the birth target,respectively.X+is the multi-target state at the next time,and

where 1L(X)(L(W))is the generalization of the indicator function.WhenL(X)?L(W),1L(X)(L(W))=1;otherwise,1L(X)(L(W))=0.Δ(·)is the distinct label indicator function.Ps(x,?)is the target survival probability.When?+=?,we haveδ?[?+]=1.f+(x+|x,?)represents the state transition function.

2.3 Multi-object observation model

Given the multi-object stateX,each(x,?)∈Xis either detected with probabilityPD(x,?)or missed with probability 1?PD(x,?).The multi-target observation set consists of detected objects and Poisson clutter(densityK).Assuming that the detections are independent of each other and of clutter,the multi-object likelihood function is given by(Vo et al.,2014)

whereΘis the set of positive 1?1 mapsθ:L→{0:|Z|},and

Here,mapθrepresents that?generates detectionzθ(?)∈Zwithθ(?)=0 if?is undetected,and any measurement is assigned to at most one object.

2.4 GLMB

An effective implementation of GLMB isδ-GLMB,which is a special GLMB.In this study,δ-GLMB is abbreviated as GLMB and its density is defined by

whereω(I,ξ)andp(ξ)(·,?)are the weights and probability density functions respectively,and Σω(I,ξ)=1.ξis the history of association maps,andIis the set of object labels.

The cardinality distribution of the GLMB is given by

while the existence probability and probability density of the track with label?are

3 GLMB filter with EFs

This section describes the implementation of the GLMB filter with EFs.Section 3.1 gives the likelihood approximation calculations for the EFs.The recursive process of the GLMB filter with EFs is shown in Section 3.2.

3.1 Likelihood approximation of EFs

Due to the complexity of the real electromagnetic environment,EFs are usually unknown and time-varying,and their distribution is also unknown.Therefore,the likelihood of the emitters is not directly available.To address these issues,dynamic clustering based on data fields is first used to estimate the EFs.Then,inspired by PDA(it considers that all measurements can be derived from the target with different probabilities),an improved FCM algorithm is proposed to approximate the correlation weights of measurements with respect to the target and clutter,i.e.,the EF likelihood.

3.1.1 EF estimation

The core idea of dynamic clustering based on data fields is to treat the EF samples as particles with mass.Dynamic clustering can radiate energy into the feature space,thus creating a data field.According to the data field principle,the samples in the feature space interact with each other.The dynamic clustering is accomplished by moving to different clustering centers under the action of field forces(Guo Q et al.,2016).

Given a set{v1,v2,···,vN}of EF samples,wherevi(i=1,2,···,N)is the position vector including the RF,PW,and PRF information,according to the nuclear radiation field model,the potential function of any position vectorvis

wheremiis the mass of featurei,andmi=1.“‖·‖”represents the Euclidean distance.σis the impact factor,controlling the range of effect on the data sample.Based on Eq.(12),the field intensity of targetviat timetis

whererij(t)is the distance vector between targetsviandvj.Each target moves in the direction of the larger potential value under the action of the field force.After many iterations,we will obtain the cluster centers of the EFs.Fig.1 shows an example of dynamic clustering.Each of the equipotential lines has the same potential value.The local maximum potential value surrounded by equipotential lines is called the potential center.Usually,the potentialcenter is located at the center of the data sample,so the potential center is often regarded as the cluster center of the data sample.

Fig.1 An example of dynamic clustering:(a)potential field formed by the data sample in a two-dimensional space;(b)potential value of the data sample

3.1.2 Likelihood calculation of the EFs

The EFs obtained in Section 3.1.1 are usually time-varying and their density distribution is unknown.Inspired by PDA(all measurements in the correlation gate can originate from the target with different probabilities;i.e.,each measurement can be associated with both target and clutter),an improved FCM algorithm is proposed.It can approximate the likelihood of EFs by calculating the correlation coefficient of the measurement with respect to the target and clutter.Because the FCM algorithm is a data clustering method based on the optimization of the objective function,the clustering result is the degree of correlation of each sample to the cluster center.The concept of correlation degree is the same as the concept of likelihood in this study.They are all correlation coefficients between the measurement and the target.The objective function of the FCM clustering algorithm can be given by

wherensandncrepresent the numbers of samples and clusters respectively,mis the weighted index,μij∈[0,1]denotes the correlation coefficient between sampleiand clusterj,anddijrepresents the distance between sampleiand clusterj.To optimize the objective functionJ,the correlation coefficient is given by a Lagrange multiplier algorithm:

In the GM implementation,each Gaussian component of the predicted intensity represents a candidate target,so the feature parameter corresponding to each Gaussian component can be treated as a center of the cluster.The correlation coefficient of each measurement with respect to the target and clutter can be calculated based on the center of the cluster.

Assume that the feature measurement set at timekiswhereNekis the number of measurement points.Because the three-dimensional features of RF,PRF,and PW can be processed independently,we take RF as an example to illustrate the calculation process of the correlation coefficient,and then fuse the correlation coefficient of the three-dimensional features.

However,the time-varying features lead to the time-varying center of the cluster.We need to calculate the optimal center of the cluster at each time.First,a set of cluster center candidates is established which contains all possible values of RF features.It has been obtained in Section 3.1.1:

where rfcenteris the cluster center of the RF features,andJkis the number of candidate targets at timek.Then,we take a sample as an example to calculate the correlation coefficient of each cluster center inEcenter:

According to the principle of the maximum correlation coefficient,the optimal cluster center rfcenter={rfk,1,rfk,2,···,rfk,Jk}of each candidate target is obtained.The corresponding distance matrix is given by

where

Here,Δrfis the resolution about RF.Finally,the correlation coefficient matrixUrfof the RF feature is

InUrf,represents the correlation coefficient of measurementiwith respect to the clutter.Similarly,we can obtain the correlation coefficient matrices of PRI and PW asUprfandUpw,respectively.Therefore,for arbitrary feature measurements,the likelihoods of EFs with respect to the candidate target and clutter can be approximately calculated by fusing different features(RF,PRF,and PW):

3.2 Recursive process of the GLMB filter with EFs

The signal features of the radar emitter are usually highly related to the actual application of the radar,and are not necessarily related to whether the platform is moving.Therefore,we consider the EFs and the target kinetic information to be independent.

The state and measurement of the target are augmented.The augmented state?xand measurement?zinclude not only kinetic information but also EF information:

wherexandzrepresent the target kinetic state and measurement respectively,andxeandzeindicate the EF state and measurement information respectively.

Given the GLMB filtering density(Eq.(8))at timek,the likelihood of the EFs in Section 3.1 is integrated into the GLMB recursion(Vo et al.,2017).The GLMB filtering density at timek+1 is given by

whereI∈F(L),ξ∈Ξ,I+∈F(L+),θ+∈Θ+,and

Here,〈·〉denotes the inner product,andandrepresent the measurement likelihood and the clutter density of the kinetic state,respectively.denote the measurement and clutter likelihood functions of the EF,respectively.λis the average clutter intensity.

The GLMB filter recursive process with EFs follows the efficient implementation in Vo et al.(2017).Ranked assignment and Gibbs sampling are used to efficiently generate GLMB components with high filtering weights,while maintaining diversity across the generated samples.For the state estimation of the multiple targets,we use a suboptimal marginal multi-objective estimator(Vo et al.,2014).The maximum a posteriori(MAP)cardinality estimate is first found from the cardinality distribution.Then,we extract the labels and average estimates of the multitarget states from the highest weighted component that has the same cardinality as the MAP cardinality estimate.

4 Simulations

In this section,we compare the tracking performances of the proposed EFs aided by GLMB and GLMB filters(Vo et al.,2017)under linear Gaussian conditions.

The size of the target surveillance region isV=[?1000,1000]m×[?1000,0]m.Each target can be described by its state vectorx=(x,˙x,y,˙y)T,which includes the positions and speeds of thex-axis andyaxis.Assume that the velocity of the target is nearlyconstant,and its state equation can be given by

whereTs=1 s is the sampling period.μk?1is state process noise and its covariance matrix is

whereσ=2 m/s2.In the simulations,the EFs areek=(rfk,prfk,pwk)T.Their specific parameter settings,which are unknown in our algorithm,are shown in Table 1.Take RF as an example to illustrate the meaning of stagger,agility,and jitter in Table 1.

Stagger uses two or more RF features to form a set{rf1,rf2,···,rfme},wheremeis the number of elements.The RF features are repeatedly generated by

RF jitter is given by rfk=rf0+ε,where rf0is the mean of the feature andε=[??,?]follows a Gaussian distributed random variable.?is the maximum jitter for the feature,set to 5%.

Agility means that the carrier frequency of adjacent pulses changes rapidly and randomly within a certain frequency band,and its model is

whereBsis the slip bandwidth,fis the agile frequency,Tris the arrival time,and?0is the initial phase.

The measurement equations of the target are

wheree′kis the EF measurement,ekis the EF,andwkandwekare the zero mean Gaussian noises with covariance matrices

Hereσx=σy=10 m,σrf=30 MHz,σprf=5 kHz,andσpw=10 μs.

The resolution of the feature is the variance of the measurement noise.The probability of target detection ispD,k=0.98.The simulation time isT=100 s.Clutter can be generated according to a Poisson point-process withKk(z)=λV c(z),whereλis the average number of clutters per scan,Vis the surveillance region,andc(z)is the spatial distribution of clutter,which is assumed to be uniform in the surveillance region.RF,PRF,and PW of the clutter are uniformly distributed in[0,5000]MHz,[0,1000]kHz,and[0,1000]μs,respectively.The target survival probability isps,k=0.99,and the birth model is an LMB RFS withπb=,whererb=0.03 and

As for the tracking performance evaluation,we use the optimal subpattern assignment(OSPA)metric(Schuhmacher et al.,2008),whose cut-offiscOSPA=100 and norm order ispOSPA=100.For each case,we perform 100 Monte–Carlo(MC)simulations.

The cluster center features after dynamic clustering based on the data field are shown in Fig.2.It can be seen that there are seven cluster centers after clustering,and the EFs of each cluster center are shown in Table 2.

Table 2 Features of each cluster center

Fig.2 Dynamic clustering based on the data field

The trajectory of the target is shown in Fig.3.These trajectories with cluttered measurements and position estimates from a single simulation are shown in Fig.4(150 clutter returns per scan over the region).The born time of targets 1,2,and 3 is the same,k=1 s.Target 4 is born atk=20 s.Targets 1 and 3 die at timek=70 s,and targets 2 and 4 die at timek=100 s.It can be seen from Fig.4 that the proposed method has satisfactory tracking performance in the cluttered environment.

Fig.3 Target trajectories

Fig.4 True target positions and position estimates on x(a)and y(b)coordinates

To verify the advantages of the proposed method,it is compared with the GLMB filter(Vo et al.,2017)with the same clutter density(150 clutter returns per scan over the region).Figs.5 and 6 show the cardinality estimation and OSPA distance of the proposed method and GLMB filter over the time,respectively.It can be seen that the proposed method has significant advantages.

To further verify the anti-clutter performance of the proposed method,the average OSPA distances of the proposed method and GLMB filter are compared under different clutter intensities.As shown in Fig.7,the average OSPA distance of the GLMB filter increases rapidly with increasing clutter intensity,while the average OSPA distance of the proposed method increases slowly.This indicates that the performance of the GLMB filter is susceptibleto the clutter,while the proposed method is less susceptible to the clutter(i.e.,it has stronger resistance to the clutter).

Fig.5 Cardinality estimation of the GLMB and the GLMB with the EF filter

Fig.7 Average OSPA distance vs.the number of clutters per scan

Fig.6 OSPA distance of the GLMB and GLMB with the EF filter

In summary,the proposed method has not only satisfactory multi-target tracking performance,but also satisfactory anti-clutter performance.

5 Conclusions

This paper proposes an improved GLMB filter that integrates unknown and time-varying features of the emitter signals.It employs the feature information of the emitter to enhance the discrimination between the target and the clutter.It can be seen from the simulation results that the proposed method has significant performance advantages compared with the GLMB filter in heavy clutter scenarios.

Contributors

Qiang GUO and Long TENG designed the research and addressed the problems.Long TENG processed the data and drafted the paper.Xinliang WU and Dayu HUANG helped with the technical information.Wenming SONG supervised the study and helped organize the paper.Qiang GUO and Long TENG revised and finalized the paper.

Compliance with ethics guidelines

Qiang GUO,Long TENG,Xinliang WU,Wenming SONG,and Dayu HUANG declare that they have no conflict of interest.