A Particle MultiTarget Tracker for Superpositional Measurements using Labeled Random Finite Sets
Abstract
In this paper we present a general solution for multitarget tracking with superpositional measurements. Measurements that are functions of the sum of the contributions of the targets present in the surveillance area are called superpositional measurements. We base our modelling on Labeled Random Finite Set (RFS) in order to jointly estimate the number of targets and their trajectories. This modelling leads to a labeled version of Mahler’s multitarget Bayes filter. However, a straightforward implementation of this tracker using Sequential Monte Carlo (SMC) methods is not feasible due to the difficulties of sampling in high dimensional spaces. We propose an efficient multitarget sampling strategy based on Superpositional Approximate CPHD (SACPHD) filter and the recently introduced Labeled MultiBernoulli (LMB) and VoVo densities. The applicability of the proposed approach is verified through simulation in a challenging radar application with closely spaced targets and low signaltonoise ratio.
I Introduction
Superpositional sensors are an important class of predetection sensor models which arise in a wide range of joint detection and estimation problems. For example, in problems such as directionofarrival estimation for linear antenna arrays [1], multiuser detection for wireless communication networks [2], acoustic amplitude sensors [3], radio frequency (RF) tomography [4], target tracking with unresolved or merged measurements [5, 6], multitarget TrackBeforeDetect with closely spaced targets [7, 8], the sensor output is a function of the sum of contributions from individual sources. In classical estimation theory, a frequency domain model for the superpositional sensor is generally used to design algorithms for source separation and parameter estimation. Conversely in dynamic state estimation, a detection based model is typically employed to transform the collected data into a set of point measurements, in order to facilitate the development of computationally efficient estimation algorithms. This is specifically the case in multitarget tracking [9, 10, 11], which is an important problem in estimation theory involving the joint estimation of an unknown and time varying number of targets and their trajectories.
Many real life applications in radar, sonar [9, 10, 11, 12], computer vision [13, 14, 15], robotics [16, 17, 18, 19], automotive safety [20, 21], cell biology [22, 23, 24, 25, 26], etc., can be described as multitarget tracking problems. Most of the multitarget tracking algorithms existing in the literature are designed for data that have been preprocessed into point measurements or detections [9, 10, 11, 27]. These algorithms are based on the “detection sensor” model which assumes that each target generates at most one detection, and that each measurement belongs to at most one target [10]. The performed preprocessing of raw measurements into a finite sets of points is efficient in terms of memory and computational requirements, and is usually effective for a wide range of applications. However, the compression might lead to significant information loss in the presence of low signaltonoise ratio (SNR) and/or closely spaced targets. The standard “detection sensor” approach may not be adequate in this case, and making use of all information contained in the predetection measurements becomes necessary. In turn, this requires more advanced sensor models and new algorithms.
In a superpositional sensor model, the measurement at each time step is a superposition of measurements generated by each of the targets present [28]. In [28] Mahler derived a superpositional Cardinalized Probability Hypthesis Density (CPHD) filter as a tractable approximation to the Bayes multitarget filter for superpositional sensor. The approach was implemented in [4] using SMC methods, and successfully applied to a passive acoustics application as well as RF tomography. The technique was also extended to multiBernoulli and a combination of multiBernouli and CPHD [29, 30]. These filters, however, are not multitarget trackers because they rest on the premise that targets are indistinguishable. Moreover, they require at least two levels of approximations: analytic approximations of the Bayes multitarget filter and particles approximation of the obtained recursion.
Inspired by [28, 4], this paper proposes a multitarget tracker for superpositional sensors which estimates target tracks and requires only one level of approximation. Our formulation is based on the same random finite set (RFS) framework that the superpositional CPHD filters [28, 4] were derived from. However, we used a special class of RFS models, called labelled RFS [31], which enables the estimation of target tracks as well as direct particle approximation of the (labeled) Bayes multitarget filter. To mitigate the depletion problem arising from sampling in high dimensional space we propose an efficient multitarget sampling strategy using the superpositional CPHD filter [4]. In particular, we will show how the recently introduced Labeled MultiBernoulli [32] and VoVo ^{1}^{1}1The VoVo density was originally called the Generalized Labeled MultiBernoulli density. However, for compactness we follow Mahler’s latest book [33] and call this the VoVo density [34] densities can be constructed from the superpositional approximate CPHD (SACPHD) filter. These densities are then used to design effective proposal distributions for the RFS multitarget particle filter. While both the CPHD and labeled RFS solutions require particle approximation, the latter has the advantage that it does not require particle clustering for the multitarget state estimation. The applicability of the proposed approach is verified through simulation analyses in a challenging closelyspaced multitarget scenario using radar power measurements with low signaltonoise ratio (SNR) [35].
The paper is organised as follows: in Section II we recall some definitions for Labeled RFSs and superpositional sensors. In Section III we discuss the multitarget particle filter, the labeled multitarget transition density, and the superpositional approximate CPHD. Two multitarget particle trackers using the Labeled MultiBernoulli (LMB) and the VoVo densities for the proposal distribution are presented in Section IV. Numerical results for a radar application are presented in Section V, while conclusions and future research directions are discussed in Section VI.
Ii Background
This section briefly presents background material on superpositional sensor model and the RFS framework which we adopt for the formulation of a multitarget tracking filter. Subsection IIA provides a summary of basic concepts in RFS. We present a concise description of the superpositional sensor model in subsection IIB and report a summary of key ideas on labled RFS needed for the derivation in subsection IIC.
Iia Multitarget Estimation
Suppose that at time , there are target states , each taking values in a state space . In the random finite set (RFS) framework, the multitarget state at time is represented by the finite set , and the multitarget state space is the space of all finite subsets of , denoted as . An RFS is simply a random variable that take values the space that does not inherit the usual Euclidean notion of integration and density. Mahler’s Finite Set Statistics (FISST) provides powerful yet practical mathematical tools for dealing with RFSs [36, 10] based on a notion of integration/density that is consistent with point process theory [37].
Similar to the standard state space model, the multitarget system model can be specified, for each time step , via the multitarget transition density and the multitarget likelihood function , using the FISST notion of integration/density. The multitarget posterior density (or simply multitarget posterior) contains all information about the multitarget states given the measurement history. The multitarget posterior recursion is direct generalisation of the standard posterior recursion [38], i.e.
(1)  
for , where , and is the measurement history with denoting the measurement vector at time . Target trajectories or tracks can be accommodated in the RFS formulation by incorporating a label in the target’s state vector [10, 31, 39]. The multitarget posterior (1) then contains all information on the random finite set of tracks, given the measurement history. In [39], the set of tracks are estimated by simulating from the multitarget posterior (1) using particle Markov Chain Monte Carlo (PMCMC) techniques [40].
Computing the multitarget posterior is prohibitively expensive for online applications. A more tractable alternative is the marginal at time known as the multitarget filtering density. For notational compactness we omit the dependence on the measurement history. Marginalizing the multitarget posterior recursion (1) yields the multitarget Bayes filter [36, 10],
(2)  
(3) 
where is the multitarget prediction density to time , and the integral is a set integral defined for any function by
(4) 
In [31, 34] an analytic solution to the multitarget Bayes filter (2), (3), known as the VoVo filter [33], was derived using labeled RFSs. Note that the majority of work in multitarget tracking is based on filtering, and often the term "multitarget posterior" is used in place of "multitarget filtering density".
IiB Superpositional Sensor
In a superpositional sensor model, the measurement is a nonlinear function of the sum of the contributions of individual targets and noise, i.e.
(5) 
where represents the contribution of the singletarget state to the sensor measurement ( is a nonlinear mapping in general), is the measurement noise, and is a nonlinear mapping. For example, a superpositional sensor model commonly used in radar is
(6) 
where is the pointspread function of target , is the (known) amplitude, is the phase noise, uniformly distributed on , and is circularly complex symmetric Gaussian noise. It is clear that this model takes on the form (5) by defining . In general, the multitarget likelihood function for the superpositional sensor model is the probability density of the measurement given the sum of the contributions of individual targets, i.e.
(7) 
The SACPHD filter filter presented in [4] is an approximation to the multitarget Bayes filter for a superpositional measurement model of the form
(8) 
where is distributed according to , a zero mean Gaussian with covariance . Hence, the likelihood function for the superpositional measurement is
(9) 
Similar to the CPHD filter (for the standard sensor model) [41], the SACPHD filter filter [4] is an analytic approximation of the Bayes multitarget filter (2), (3) based on independently and identically distributed (iid) cluster RFS. A brief review of the SACPHD filter is given in subsection IIIC. Both filters recursively propagate the cardinality distribution and the PHD of the posterior multitarget RFS. The CPHD filter can be implemented with Gaussian mixtures or particles [42], while only the particle implementation is available for the SACPHD filter [4]. Particle implementations of PHD/CPHD filter in general require clustering to extract multitarget estimates, which can introduce additional errors under challenging scenarios.
IiC Labeled RFS
To perform tracking in the RFS framework we use the labeled RFS model which incorporates a unique label in the target’s state vector to identify its trajectory [10]. In this model, the singletarget state space is a Cartesian product , where is the feature/kinematic space and is the (discrete) label space. A finite subset set of has distinct labels if and only if and its labels have the same cardinality. An RFS on with distinct labels is called a labeled RFS [31, 34].
For the rest of the paper, we use the standard inner product notation . We denote a generalization of the Kroneker delta and the inclusion function that take arbitrary arguments such as sets, vectors, by
We also write in place of when = . Singletarget states are represented by lowercase letters, e.g. , while multitarget states are represented by uppercase letters, e.g. , , symbols for labeled states and their distributions are bolded to distinguish them from unlabeled ones, e.g. , , , etc, spaces are represented by blackboard bold e.g. , , , etc.
An important class of labeled RFS distribution is the generalized labeled multiBernoulli distribution [31], known as the VoVo distribution [33], which is the basis of an analytic solution to the Bayes multitarget filter [34]. Under the standard multitarget measurement model, the VoVo distribution is a conjugate prior that is also closed under the ChapmanKolmogorov equation. If we start with a VoVo initial prior, then the multitarget posterior at any time is a also a VoVo distribution. Let be the projection , let denote the distinct label indicator, and , denote the multiobject exponential, where is a realvalued function, with by convention. A VoVo density is a labeled RFS density on
(10) 
where is a discrete index set, and satisfy:
(11)  
(12) 
The VoVo density (10) can be interpreted as a mixture of multiobject exponentials. Each term in (10) consists of a weight that depends only on the labels of , and a multiobject exponential that depends on the entire . The Labeled MultiBernoulli (LMB) family is a special case of the VoVo density with one term of the form:
(13)  
(14)  
(15) 
where , , is a given set of parameters with representing the existence probability of track , and the probability density of the kinematic state of track given its existence [31]. Note that for an LMB the index space has only one element, in which case the superscript is not needed. The LMB family is the basis of the LMB filter, a principled and efficient approximation of the Bayes multitarget tracking filter, which is highly parallelizable and capable of tracking large number of targets [43].
Iii Bayesian multitarget tracking for superpositional sensor
In this section we describe the classical particle Bayes multitarget filter [37], which has very high computational complexity in general. Fortunately, using labeled targets greatly simplifies the multitarget transition density and drastically reduces the computational complexity. Subsection IIIA presents a summary of the classical multitarget particle filter and Subsection IIIB details the labeled multitarget transition density that reduces the computational complexity. Subsection IIIC reviews the equations of the superpositional CPHD filter that is used to following section to construct LMB/VoVo efficient proposal distribution for the multitarget particle filter.
Following [31, 34], to ensure distinct labels we assign each target an ordered pair of integers , where is the time of birth and is a unique index to distinguish targets born at the same time. The label space for targets born at time is denoted as , and a target born at time , has state . The label space for targets at time (including those born prior to ), denoted as , is constructed recursively by (note that and are disjoint). A multitarget state at time , is a finite subset of . For completeness, the Bayes multitarget tracking filter, i.e. the multitarget Bayes recursion (2), (3) for labeled RFS, is provided below
(16)  
(17) 
Iiia Particle Bayes multitarget filter
The propagation of the multitarget posterior involves the evaluation of multiple set integrals and hence the computational requirement is much more intensive than singletarget filtering. Particle filtering techniques permit recursive propagation of the set of weighted particles that approximate the posterior. Central in Monte Carlo methods is the notion of approximating the integrals of interest using random samples. While the FISST density is not a density (in the RadonNikodym context), it can be converted into a probability density (with respect to a particular dominating measure) by cancelling out the unit of measurement [37]. Monte Carlo approximations of the integrals of interest can then be constructed using random samples. The singletarget particle filter can thus be directly generalised to the multitarget case. In the multitarget context however, each particle is a finite set and the particles themselves can thus be of varying dimensions. Following [37], suppose that at time , a set of weighted particles representing the multitarget posterior is available, i.e.
(18) 
Note that is the Diracdelta concentrated at (different from the Kroneckerdelta that takes values of either 1 or 0). The particle filter proceeds to approximate the multitarget posterior at time by a new set of weighted particles as follows
Multitarget Particle Filter
For time

For sample and set
(19) 
Normalize the weights:
Resampling Step

Resample to get
The importance sampling density is a multitarget density and is a sample from an RFS. It is implicit in the above algorithm description that
(20) 
so that the weights are welldefined. Convergence results for the multitarget particle filter are given in [37].
Notice that the entire posterior can be computed by modifying the pseudocode of the multitarget particle filter so that is used in place of and is used in place of . This would in principle solve the so called mixed labelling problem [44]. However, this is computationally demanding because it requires recomputing the whole history of each multitarget particle [39]. Alternatively, forwardbackward smoothing can be used to approximate the entire posterior [10]. In this paper we focus on designing efficient proposal distributions for the multitarget particle filter approximating the filtering recursion. In future work we will consider the application of the proposed approach to the problem of estimating the full posterior.
The main practical problem with the multitarget particle filter is the need to perform importance sampling in very high dimensional spaces if many targets are present. In [45, 46, 32], the transition density is used as the proposal, i.e. . While this avoids the evaluation of the transition density, it suffers from particle depletion even for a small number of targets. This problem is compounded with superpositional sensor due to less informative measurements arising from low SNR. A naive choice of importance density such as the transition density will typically lead to an algorithm whose efficiency decreases exponentially with the number of targets for a fixed number of particles [37]. The problem with using a proposal other than the transition density, is that the weights are difficult to evaluate due to the combinatorial nature of the transition density for unlabled RFS. Fortunately, for labeled RFS the transition density simplifies to a form that is inexpensive to evaluate.
IiiB Labeled multitarget transition density
The multitarget transition model for labeled RFS is summarised as follows. Given a multitarget state at time , each state either continues to exist at the next time step with probability and evolves to a new state with probability density , or dies with probability . In addition, the set of new targets born at time is distributed according to the LMB distribution
(21) 
where and are given parameters of the multitarget birth density , defined on . Note that if contains any element with . The birth model (21) covers both labeled Poisson and labeled multiBernoulli [31]. The multitarget state , at time , is the superposition of surviving targets and new born targets. The model uses the standard assumption that targets evolve independently of each other and that births are independent of surviving targets.
It was shown in [31] that the multitarget transition density is given by
(22) 
where
(23)  
(24) 
Unlike the general multitarget transition density (see [36, 10]), the special case for labeled RFS (22) does not contain any combinatorial sums. It is simply a product of terms corresponding to the surviving targets and new targets. Consequently, numerical complexity of the weight update in the multitarget particle filter drastically reduces.
IiiC Superpositional Approximate CPHD filter
In this section we recall the approximate CPHD for superpositional measurements of the following form:
(25) 
where is the multitarget state at time , is zeromean white Gaussian noise, and is a possibly nonlinear function of the single state vector . Notice that the model in eq. (25) can be used to approximate the radar power measurement eq. (61) assuming a Gaussian noise in power. Obviously the model in eq. (25) is a strong approximation of eq. (61). However, it allows using the update step of the SACPHD filter to evaluate measurement updated intensity function and cardinality distribution for the target set. In turn, the information in the updated and , along with the targets labels from the previous step and birth process, can be used to construct an approximate posterior density using the VoVo and/or LMB distributions in eq. (10) and (13). Finally, the obtained approximate posterior is used as a proposal distribution for the multiobject particle filter.
The vector measurement in eq. (25) usually represents an array of sensors for SACPHD filter, e.g. acoustic amplitude sensors, radiofrequency tomography, etc. For application of the SACPHD filter to tracking using radar power returns, the vector measurement contains the radar power returns from the set of cells being interrogated by the radar at time . Hence, where is the number of cells being interrogated by the radar. Following [4], standard CPHD formulas are used for the predicted cardinality distribution and PHD, while the update step of the SACPHD filter is given by:
(26)  
(27) 
where is the noise covariance, is the predicted average number of targets and:
(28)  
(29)  
(30)  
(31)  
(32)  
(33) 
where is the normalized predicted intensity, and , and are the variance, second factorial moment and third factorial moment of the predicted cardinality distribution . The equations of the superpositional approiximate CPHD filter can be implemented efficiently using SMC methods. In the following section we describe how the updated PHD and cardinality distribution from the SACPHD filter can be used to design efficient proposal distributions for multitarget tracking.
Iv Efficient Proposal Distributions based on Superpositional Approximate CPHD filter
In this section we detail the CPHDbased proposal distribution and the multitarget particle filter equations. In superpositional multitarget filtering, the multitarget posterior generally cannot be written as a product of independent densities because the target states are statistically dependent through the measurement update. This means that an effective particle approximation of the posterior distribution is of great interest. Unfortunately, designing an effective multiobject proposal distribution is not a simple task when using superpositional sensors. In this section we exploit the SACPHD filter to construct a relatively inexpensive LMB based proposal as well as more accurate VoVo based proposal. The basic idea is to obtain the updated PHD and cardinality distribution at time from the SACPHD filter and construct a proposal distribution that exploits the approximate posterior information contained in both the cardinality distribution and the state samples from .
Assume a particle representation of the posterior distribution is available at time . Then, the cardinality distribution and the PHD of the unlabeled multitarget state at time are given by [37]:
(34)  
(35) 
where denotes the kinematic part of each , and is the Diracdelta concentrated at . The superpositional CPHD is then used to obtain the update cardinality distribution and PHD using the measurement collected at time . Notice that differently from standard unlabeled CPHD filtering, there is a natural labeling/clustering of particles due to the existing labels at time and the chosen cluster process with implicit cluster labels for the birth model. In fact, let be the updated PHD at time
(36) 
Then we can rewrite the (unlabeled) PHD as a sum over all labels of labeled PHD terms , i.e.
(37) 
where
(38) 
is the contribution to the PHD of track . Note that the above is not the PHD of a labeled RFS but the PHD mass from a specific label representing a survival or birth target. This means that at time we can extract clusters of particles from the posterior PHD. Furthermore, a continuous approximation to each cluster can be obtained by evaluating sample mean and covariance for a Gaussian approximation to . Alternatively, it is possible to use kernel density estimation (KDE), however this will not be considered in this paper. For , let and denote the sample mean and covariance corresponding to the PHD cluster ). Hence, we approximate the PHD clusters as follows
(39) 
where is the PHD mass of the cluster. For the sake of explicitness, in our exposition we denote the PHD mass of survival targets as and the PHD mass of newly born targets as ,
(40)  
(41) 
In practice we constrain the survival and birth probabilities and . The constraint is imposed to avoid the complete loss of a track due to errors in the CPHD update while the constraint is required since the PHD in each track cluster can exceed .
The obtained posterior cardinality and posterior target clusters can be used to construct a proposal distribution . In the following subsections we detail two strategies for constructing the proposal as an LMB density of the form (13) and as a VoVo density of the form (10).
Iva LMB Proposal Distributions
In this subsection we describe how to construct a multitarget proposal distribution for the multitarget particle tracker by using an LMB density, i.e.
(42) 
where and are the LMB proposals for survival targets and birth targets, respectively. Specifically, the survival and birth proposals are constructed using the CPHD updated birth and survival probabilities and the Gaussian clusters . For the survival proposal we have,
(43)  
(44) 
while for the birth proposal we have,
(45)  
(46) 
In summary, the multitarget proposal distribution in eq. (19) is constructed using two LMB densities for the existing and newly appeared targets, respectively. A pseudocode of the multitarget particle filter using the LMB proposal for sampling is given below. Notice that we used the following definitions for grouping of labels in each particle
(47)  
(48)  
(49)  
(50) 
where for each particle , is the set of survived labels, is the set of death labels, is the set of labels for newly born targets, and is the set of labels that did not generate a new targets.
MultiTarget Particle Filter
with LMB Proposal Distribution
Initialize particles
For

For

For each

Generate

If generate


For each

Generate

If generate


Evaluate the transition kernel
(51) 
Evaluate the proposal distribution

Evaluate the multiobject likelihood

Update the particle weight using eq. (19)


Normalize the weights and resample as usual
IvB VoVo Proposal Distributions
The LMB proposal distribution leads to an efficient sampling strategy for the multitarget particle filter. However, the LMB proposal does not exploit all the information from the SACPHD filter since the cardinality distribution of the LMB proposal does not match the cardinality distribution from the CPHD prediction/update. Matching of the cardinality distribution is important if we are interested in designing a proposal distribution that is efficient in low SNR scenarios. Generally, for low SNR the MultiBernoulli cardinality distribution is not sufficiently informative, so that being able to estimate a more general cardinality distribution becomes fundamental. This reasoning is true also in classical multitarget tracking with detection measurements, e.g. the CPHD filter outperforms the MultiBernoulli filter in low SNR [10]. Hence, we seek a proposal distribution that matches the CPHD cardinality exactly while exploiting the weights of individual labeled target clusters as computed from the approximate posterior PHD. A single component VoVo density can be used for this purpose,
(52) 
We now specify the component weight and the multiobject exponential , needed to match the CPHD updated cardinality distribution and to account for the weights of individual target clusters. Clearly, the singletarget densities are obtained straightforwardly from Gaussian clusters, i.e.
(53) 
The weight is then chosen to preserve the CPHD cardinality distribution, and for a given cardinality, to sample labels proportionally to the product of the posterior PHD masses of any possible label combinations. Specifically, from the posterior PHD mass of each cluster we construct approximate “existence” probabilities as
(54) 
The cardinality of the set of labels , including birth and survival labels, grows exponentially in time. Moreover, in any practical implementation the use of a finite sample approximation coupled with resampling strategies typically leads to a much smaller unique labels set at each time . Thus, eq. (54) is implemented by considering only labels from resampled particles at time ,
(55)  
(56) 
The weight is then defined as
(57) 
where denotes the set of “existence” probabilities for all current tracks and is the elementary symmetric function of order . The construction of the proposal in (52) leads a simple and efficient strategy for sampling. Specifically, to sample from (52) we,

sample the cardinality of the newly proposed particle according to the distribution ,

sample labels from using the distribution defined by ,

for each we sample the kinematic part from .
A detailed pseudocode for implementation is reported below.
MultiTarget Particle Filter
with VoVo Proposal Distribution
Initialize particles
For

For

Sample the cardinality for the new particle

Sample the set of labels uniformly from

For each generate

For evaluate the transition kernel

Evaluate the proposal distribution

Evaluate the multiobject likelihood

Update the particle weight using
