Random finite set solutions

Finite Set Statistics (FISST) is an approach which generalizes the single-object Bayes filter to multiple targets by using random finite sets (RFS) rather than ran-

2.5 Multiple object state estimation

dom vectors. It introduces concepts such as the set integral and probability gener- ating functionals which permit the direct manipulation of multi-object distributions without the need to use heuristics and data association [58]. The first practical filter produced with this framework is the Probability Hypothesis Density (PHD) Filter [56], which propagates the first order moment of the multi-target distribution to estimate both the number of objects of interest in the scene and their individual states. The resulting filter is not computationally burdensome and the resulting expressions for prediction and update are intuitive, which simplifies its implemen- tation and the interpretation of its results. A downfall of the PHD filter, however, is its strong assumption that the prior distribution is a Poisson multi-target pro- cess, which means that all objects are Independent Identically Distributed (IID) and that the cardinality distribution is Poisson. Poisson probability mass functions have means equal to their variances, which causes the estimate of the number of targets to be somewhat unstable when object appearance and disappearance are frequent. Recent developments have attempted to overcome this limitation by using more general distributions such as the negative binomial distribution [81].

The Probability Hypothesis Density (PHD) Filter [56] propagates the first mo- ment D(x) of the multi-target posterior, known as the PHD or intensity, which is a function defined on the single-target state space which indicates the expected num- ber of targets in any of its regions. If the multitarget probability distribution is denoted pk(X), then the PHD satisfies

Z

S

D(x) dx = Z

|X ∩ S|pk(X) δX = Nk(S), (2.35) where Nk(S) denotes the expected number of targets in set S and |S| is the cardi- nality of S. The integral of the form R . . . δX is a set integral [56]. The process and measurement models used by the PHD Filter are based on the following assumptions:

1. From time step k − 1 to time step k, each target xk−1 survives with proba- bility PS(xk−1) (the probability of survival ), evolving into xk ∼ fk(·|xk−1) or disappears with probability 1 − PS(xk−1).

2. New targets may appear at each time step according to an independent process.

3. Each target x produces a measurement z ∼ gk(·|x) with probability PD(x) (the probability of detection) or is not detected with probability 1 − PD(x).

4. False alarms are produced at each time step according to a certain clutter distribution.

2.5 Multiple object state estimation

These assumptions are synthesized in the following RFS process and measure- ment models: Xk = Γk∪ [ x∈Xk−1 Yk|k−1(x) (2.36) Zk = Kk∪ [ x∈Xk Θk(x) (2.37)

Where Γk and Kk are the birth and clutter random finite sets, respectively, and

Yk|k−1(x) =    {x0_{} , with probability P} S(x) and x0 ∼ fk(·|x) ∅, with probability 1 − PS(x) (2.38) Θk(x) =   

{z} , with probability PD(x) and z ∼ gk(·|x) ∅, with probability 1 − PD(x).

(2.39)

A derivation of the PHD filter starting from these assumptions can be found in [56]. The resulting prediction and update equations are the following:

Dk|k−1(x) = γ(x) + Z PS(x0)fk−1(x|x0)Dk−1(x0) dx0 (2.40) Dk(x) = (1 − PD(x))Dk|k−1(x) + X z∈Zk PD(x)gk(z|x)Dk|k−1(x) c(z) +R gk(z|x0)Dk|k−1(x0) dx0 , (2.41)

Here, γ(x) is the PHD of the birth process and c(z) is the PHD of the clutter process. In many applications, priors for newborn targets are not suitable to adequately describe appearing targets. In these cases, an alternative strategy can be used with measurement driven births [40]. In this approach, when no a priori information is available on where targets are likely to appear, measurements are used to determine likely positions new target appearances.

As in the case of the single target Bayes filter, an appropriate form for D(x) must be chosen in order to use equations (2.40) and (2.41) to implement a tractable filter. The most common approaches to do this are Gaussian mixture implementations [97] and SMC implementations [98], mirroring the single target tracking case.

In addition to estimating the state of populations of objects, the PHD filter has been extended to estimate more complex phenomena such as groups of objects with correlated motion (group targets) [90] and targets which can generate multi- ple measurements (extended targets) [89] by modeling them as independent spatial

2.5 Multiple object state estimation

cluster point processes. As a particular case of these filters, the single-cluster PHD filter was developed in order to estimate populations of objects conditioned on a single-object random variable, which is also unknown [91]. This lends itself well to problems where the population of objects is observed through a sensor which has unknown state, as it is necessary to estimate the state of the sensor in order to produce unbiased estimates of the observed population [73].

The single-cluster PHD filter has been used in many interesting applications. Ristic et al have used it in order to calibrate sensors using non-cooperative targets [73], while Lee et al. have applied to the robotics problem of simultaneous localiza- tion and mapping [52, 53, 54]. Schlangen et al have used it to solve the problem of estimating the position of intra-cellular proteins observed using fluorescence mi- croscopy, while simultaneously correcting for microscope drift [25, 82]. Hagen et al. applied it to image plane tracking of objects observed from a telescope, while simultaneously stabilizing it [33], and Houssineau et al. applied it to simultaneous 3D tracking and camera calibration from video data.

An important limitation of the PHD filter is that it only propagates the first moment of the multi-object posterior. Due to this, several attempts have been made to create more informative filters using the FISST framework by propagating more information. The Cardinalized Probability Hypothesis Density (CPHD) filter, for instance, propagates the cardinality distribution of the multi-object distribution alongside its spatial distribution, eliminating the need to assume a particular form for the cardinality distribution of the estimated densities. The result is that the estimated number of targets is more stable, but it adds to the computational burden of the method and the resulting expressions are more convoluted than the regular PHD filter [57].

More recent efforts have been oriented towards propagating the full multi-object posterior rather than its moments, under the rather general assumption of the pro- cess following a multi-Bernoulli distribution [58,100]. Indeed, multi-Bernoulli distri- butions do not assume that the multi-object population is IID, and their cardinality distribution is arbitrary.

An important limitation of approaches based on FISST is that as opposed to the classical framework, track identity is not directly propagated in the recursive estimation process due to sets being unordered. An interesting approach to over- coming this limitation was recently proposed alongside a study of conjugate priors for multi-object distributions, where filtering is done on labeled RFS [99]. This al- lows track identity to be preserved through time, but the resulting algorithm is computationally expensive.

2.5 Multiple object state estimation