Smoothing Algorithms for the Probability Hypothesis Density Filter

The probability hypothesis density (PHD) filter is a recursive algorithm for solving multi-target tracking problems. The method consists of replacing the expectation of a random variable used in the traditional Bayesian filtering equations, by taking generalized expectations using the random sets formalism. In this approach, a set of observations is used to estimate the state of multiple and unknown number of targets. However, since the observations does not contains all information required to infer the multi-target state, the PHD filter is constructed upon several approximations that diminishes the underlying combinatorial problem. More specifically, the PHD filter is a first-order moment approximation to a full posterior density, so there is an inherent loss of information. In dynamic estimation problems, smoothing is used gather more information about the system state in order to improve the error of the filtering estimates. In other words, instead of using all previous and the most recent observations, we can also use future information in order to estimate the current multi-target state. In this report, we derive two smoothing strategies for the PHD filter and provide their sequential Monte Carlo implementations. Real world tracking applications like urban surveillance, radar and sonar usually involve finding more than one moving object in the scene. In these cases, it remains a great challenge for the standard single-target Bayesian filter to effectively track multiple targets. The dynamic system in consideration must take into account targets appearing and disappearing possibly at random times, while also coping with the possibility of clutter and nondetected targets. Moreover, additional uncertainty in the dynamic and the sensor model can make the single target likelihood function a poor characterization of the real system, so the performance of multiple coupled filters can be far from optimal [41]. 1 A Bayesian method for dealing with multiple and unknown numbers of targets was developed by Ronald P. Mahler [23] and the resulting algorithm was called the probability hypothesis density (PHD) filter. The method uses a first-order moment approximation for the posterior distribution of a random set, or equivalently a multi-dimensional random point process. A previous formulation of the multi-target problem using the point process formalism was proposed in [48], where a time-varying finite set of observations is represented as a random counting measure being defined as a point process. Instead of considering the multiple target observation model as a list of measurements, the approach proposed by Washburn used a representation in terms of random scattered points. The method was theoretically developed assuming a known number of targets, and had a naturally appealing interpretation in terms of radar or sonar returns being monitored in a screen. Further explorations of the point processes approach considered an unknown and time-varying number of targets that has to be estimated alongside the individual target states. In this context, the approach taken by Miller, Sristava and Grenader considered a continuous-time model for the estimation and recognition of multiple targets [27]. The approach taken by Mahler was theoretically developed using a re-formulation of point process theory named finite-set statistics (FISST) [22] and significantly differs from the previous works, since it considered random finite sets (RFS) evolving in discrete time. In FISST, the multi-target and multi-sensor data fusion problem is treated as an estimation problem of a single “global target” observed by a single “global sensor”, both being represented by random sets. Given that the expected value of a RFS cannot be mathematically expressed using the standard Bayesian filtering equations, filtering and update equations for the PHD filter were developed using set derivatives and set integrals of belief mass functions [15]. The expectation of a random set requires set integration, which can be calculated by considering all singletons (sum of indicator functions) that almost surely belongs to the random set [2]. In FISST, the expectation of a random set is calculated (in a similar way as generating functions of random variables) by taking functional derivatives of a sum of bounded functions that characterize the distribution of a point process [25]. This derivation leads to analytical forms for the Bayesian prediction and update equations of the recursive multi-target densities. However, an alternative derivation can be also achieved if the state-space is discretized into a finite number of disjoint bins containing a single target. The multi-target density is then recovered by taking the sum of the probabilities of all bins and the volume of each bin taking infinitesimally small values. This technique was called the physicalspace approach and was explored in [11] and [16]. More recent formulations suggest that filtering and update equations can be also obtained by means of transformations of a point processes (see references [37] and [39],[38]). The posterior multi-target density is not (in general) a Poisson process, but it 2 can be approximated to be Poisson distributed by taking the product of all normalized and unit-less marginal single-target densities.