Random Finite Set

—Bernoulli filters are a class of exact Bayesian filters for non-linear/non-Gaussian recursive estimation of dynamic systems, recently emerged from the random set theoretical framework. The common feature of Bernoulli filters is that they are designed for stochastic dynamic systems which randomly switch on and off. The applications are primarily in target tracking, where the switching process models target appearance or disappearance from the surveillance volume. The concept, however, is applicable to a range of dynamic phenomena, such as epidemics, pollution, social trends, etc. Bernoulli filters in general have no analytic solution and are implemented as particle filters or Gaussian sum filters. This tutorial paper reviews the theory of Bernoulli filters as well as their implementation for different measurement models. The theory is backed up by applications in sensor networks, bearings-only tracking, passive radar/sonar surveillance, visual tracking, monitoring/prediction of an epidemic and tracking using natural language statements. More advanced topics of smoothing, multi-target detection/tracking and sensor control are briefly reviewed with pointers for further reading.

— Random finite sets are natural representations of multi-target states and observations that allow multi-sensor multi-target filtering to fit in the unifying random set framework for Data Fusion. Although the foundation has been established in the form of Finite Set Statistics (FISST), its relationship to conventional probability is not clear. Furthermore, optimal Bayesian multi-target filtering is not yet practical due to the inherent computational hurdle. Even the Probability Hypothesis Density (PHD) filter, which propagates only the first moment (or PHD) instead of the full multi-target posterior, still involves multiple integrals with no closed forms in general. This article establishes the relationship between FISST and conventional probability that leads to the development of a sequential Monte Carlo (SMC) multi-target filter. In addition, a SMC implementation of the PHD filter is proposed and demonstrated on a number of simulated scenarios. Both of the proposed filters are suitable for problems involving non-linear non-Gaussian dynamics. Convergence results for these filters are also established.

In Bayesian multi-target filtering we have to contend with two notable sources of uncertainty, clutter and detection. Knowledge of parameters such as clutter rate and detection profile are of critical importance in multi-target filters such as the probability hypothesis density (PHD) and Cardinalized PHD (CPHD) filters. Significant mismatches in clutter and detection model parameters results in biased estimates. In practice these model parameters are often manually tuned or estimated off-line from training data. In this paper we propose PHD/CPHD filters that can accommodate model mismatch in clutter rate and detection profile. In particular we devise versions of the PHD/CPHD filters that can adaptively learn the clutter rate and detection profile while filtering. Moreover, closed form solutions to these filtering recursions are derived using Beta and Gaussian mixtures. Simulations are presented to verify the proposed solutions.

In autonomous applications, a vehicle requires reliable estimates of its location and information about the world around it. To capture prior knowledge of the uncertainties in a vehicle's motion response to input commands and sensor measurements, this fundamental task has been cast as probabilistic Simultaneous Localization and Map building (SLAM). SLAM has been investigated as a stochastic filtering problem in which sensor data is compressed into features, which are consequently stacked in a vector, referred to as the map. Inspired by developments in the tracking literature, recent research in SLAM has recast the map as a Random Finite Set (RFS) instead of a random vector, with huge mathematical consequences. With the application of recently formulated Finite Set Statistics (FISST), such a representation circumvents the need for fragile feature management and association routines, which are often the weakest component in vector based SLAM algorithms. This tutorial demonstrates that true sensing uncertainty lies not only in the spatial estimates of a feature, but also in its existence. This gives rise to sensor probabilities of detection and false alarm, as well as spatial uncertainty values. By re-addressing the fundamentals of SLAM under an RFS framework, it will be shown that it is possible to estimate the map in terms of true feature number, as well as location. The concepts are demonstrated with short range radar, which detects multiple features, but yields many false measurements. Comparison of vector, and RFS SLAM algorithms shows the superior robustness of RFS based SLAM to such realistic sensing defects.

—Quantitative analysis of the dynamics of tiny cellular and sub-cellular structures, known as particles, in time-lapse cell microscopy sequences requires the development of a reliable multi-target tracking method capable of tracking numerous similar targets in the presence of high levels of noise, high target density, complex motion patterns and intricate interactions. In this paper, we propose a framework for tracking these structures based on the random finite set Bayesian filtering framework. We focus on challenging biological applications where image characteristics such as noise and background intensity change during the acquisition process. Under these conditions, detection methods usually fail to detect all particles and are often followed by missed detections and many spurious measurements with unknown and time-varying rates. To deal with this, we propose a bootstrap filter composed of an estimator and a tracker. The estimator adaptively estimates the required meta parameters for the tracker such as clutter rate and the detection probability of the targets, while the tracker estimates the state of the targets. Our results show that the proposed approach can outperform state-of-the-art particle trackers on both synthetic and real data in this regime.

—In multi-object inference, the multi-object probability density captures the uncertainty in the number and the states of the objects as well as the statistical dependence between the objects. Exact computation of the multi-object density is generally intractable and tractable implementations usually require statistical independence assumptions between objects. In this paper we propose a tractable multi-object density approximation that can capture statistical dependence between objects. In particular, we derive a tractable Generalized Labeled Multi-Bernoulli (GLMB) density that matches the cardinality distribution and the first moment of the labeled multi-object distribution of interest. It is also shown that the proposed approximation minimizes the Kullback-Leibler divergence over a special tractable class of GLMB densities. Based on the proposed GLMB approximation we further demonstrate a tractable multi-object tracking algorithm for generic measurement models. Simulation results for a multi-object Track-Before-Detect example using radar measurements in low signal-to-noise ratio (SNR) scenarios verify the applicability of the proposed approach.

—Most tracking algorithms in the literature assume that the targets always generate measurements independently of each other, i.e. the sensor is assumed to have infinite resolution. Such algorithms have been dominant because addressing the presence of merged measurements increases the computational complexity of the tracking problem, and limitations on computing resources often make this infeasible. When merging occurs, these algorithms suffer degraded performance, often due to tracks being terminated too early. In this paper, we use the theory of random finite sets (RFS) to develop a principled Bayesian solution to tracking an unknown and variable number of targets in the presence of merged measurements. We propose two tractable implementations of the resulting filter, with differing computational requirements. The performance of these algorithms is demonstrated by Monte Carlo simulations of a multi-target bearings-only scenario, where measurements become merged due to the effect of finite sensor resolution.

A forward-backward Probability Hypothesis Density (PHD) smoother involving forward filtering followed by backward smoothing is proposed. The forward filtering is performed by Mahler's PHD recursion. The PHD backward smoothing recursion is derived using Finite Set Statistics (FISST) and standard point process theory. Unlike the forward PHD recursion, the proposed backward PHD recursion is exact and does not require the previous iterate to be Poisson. In addition, assuming the previous iterate is Poisson, the cardinality distribution and all moments of the backward-smoothed multi-target density are derived. It is also shown that PHD smoothing alone does not necessarily improve cardinality estimation. Using an appropriate particle implementation we present a number of experiments to investigate the ability of the proposed multi-target smoother to correct state as well as cardinality errors.

—An analytic solution to the multi-target Bayes re-cursion known as the δ-Generalized Labeled Multi-Bernoulli (δ-GLMB) filter has been recently proposed in [1]. As a sequel to [1], this paper details efficient implementations of the δ-GLMB multi-target tracking filter. Each iteration of this filter involves an update operation and a prediction operation, both of which result in weighted sums of multi-target exponentials with intractably large number of terms. To truncate these sums, the ranked assignment and K-th shortest path algorithms are used in the update and prediction, respectively, to determine the most significant terms without exhaustively computing all of the terms. In addition, using tools derived from the same framework, such as probability hypothesis density filtering, we present inexpensive (relative to the δ-GLMB filter) look-ahead strategies to reduce the number of computations. Characterization of the L1-error in the multi-target density arising from the truncation is presented.

The δ-generalized labeled multi-Bernoulli (δ-GLMB) tracker is the first multiple hypothesis tracking (MHT)-like tracker that is provably Bayes-optimal. However, in its basic form, the δ-GLMB provides no mechanism for adaptively initializing targets at their first appearance from unlabeled measurements. By introducing a new multitarget likelihood function that accounts for new target appearance, a data-driven δ-GLMB tracker is derived that automatically initializes new targets in the tracker measurement update. Monte Carlo results of simulated multitarget tracking problems demonstrate improved multitarget tracking accuracy over comparable adaptive birth methods.

—This paper proposes a generalization of the multi-Bernoulli filter called the labeled multi-Bernoulli filter that outputs target tracks. Moreover, the labeled multi-Bernoulli filter does not exhibit a cardinality bias due to a more accurate update approximation compared to the multi-Bernoulli filter by exploiting the conjugate prior form for labeled Random Finite Sets. The proposed filter can be interpreted as an efficient approximation of the δ-Generalized Labeled Multi-Bernoulli filter. It inherits the advantages of the multi-Bernoulli filter in respect of particle implementation and state estimation. It also inherits advantages of the δ-Generalized Labeled Multi-Bernoulli filter in that it outputs (labeled) target tracks and achieves better performance.

This paper presents a novel Bayesian method to track multiple targets in an image sequence without explicit detection. Our method is formulated based on finite set representation of the multi-target state and the recently developed multi-Bernoulli filter. Experimental results on sport player and cell tracking studies show that our method can automatically track numerous targets, and it outperforms the state-of-the-art in terms of false positive (false alarm) and false negative (missing) rates as detection error measures, and in terms of label switching rate and lost tracks ratio as tracking error measures.

The conventional GMPHD/CPHD filters require the PHD for target births to be a Gaussian mixture, which is potentially inefficient because careful selection of the mixture parameters may be required to ensure good performance. Here we present approximations which allow part of the birth PHD to be uniformly distributed, obviating the need to use large Gaussian mixtures to model target births. The benefits of this approach are demonstrated by simulations on a bearings-only filtering scenario.

—In this paper, we extend the notion of Cauchy-Schwarz divergence to point processes and establish that the Cauchy-Schwarz divergence between the probability densities of two Poisson point processes is half the squared L 2-distance between their intensity functions. Extension of this result to mixtures of Poisson point processes and, in the case where the intensity functions are Gaussian mixtures, closed form expressions for the Cauchy-Schwarz divergence are presented. Our result also implies that the Bhattacharyya distance between the probability distributions of two Poisson point processes is equal to the square of the Hellinger distance between their intensity measures. We illustrate the result via a sensor management application where the system states are modeled as point processes.

The probability hypothesis density (PHD) filter is an attractive approach to tracking an unknown and time-varying number of targets in the presence of data association uncertainty, clutter, noise, and detection uncertainty. The PHD filter admits a closed form solution for a linear Gaussian multi-target model. However, this model is not general enough to accommodate maneuvering targets that switch between several models. In this paper, we generalize the notion of linear jump Markov systems to the multiple target case to accommodate births, deaths and switching dynamics. We then derive a closed form solution to the PHD recursion for the proposed linear Gaussian jump Markov multi-target model. Based on this an efficient method for tracking multiple maneuvering targets that switch between a set of linear Gaussian models is developed. An analytic implementation of the PHD filter using statistical linear regression technique is also proposed for targets that switch between a set of nonlinear models. We demonstrate through simulations that the proposed PHD filters are effective in tracking multiple maneuvering targets.

—This paper proposes an integrated Bayesian framework for feature-based simultaneous localisation and map building (SLAM) in the general case of uncertain feature number and data association. By modeling the measurements and feature map as random finite sets (RFSs), a formulation of the feature-based SLAM problem is presented that jointly estimates the number and location of the features, as well as the vehicle tra-jectory. More concisely, the joint posterior distribution of the set-valued map and vehicle trajectory is propagated forward in time as measurements arrive, incorporating both data association and feature management into a single recursion. Furthermore, the Bayes optimality of the proposed approach is established. A first order solution, coined the probability hypothesis density (PHD) SLAM filter, is derived, which jointly propagates the posterior PHD of the map and the posterior distribution of the vehicle trajectory. A Rao-Blackwellised implementation of the PHD-SLAM filter is proposed based on the Gaussian mixture PHD filter (for the map) and a particle filter (for the vehicle trajectory). Simulated and experimental results demonstrate the merits of the proposed approach, particularly in situations of high clutter and data association ambiguity.

—We propose a Bayesian multi-target batch processing algorithm capable of tracking an unknown number of targets that move close and/or cross each other in a dense clutter environment. The optimal Bayes multi-target tracking problem is formulated in the random finite set framework and a Particle Marginal Metropolis-Hastings (PMMH) technique which is a combination of Metropolis-Hastings (MH) algorithm and sequential Monte Carlo methods is applied to compute the multi-target posterior distribution. The PMMH technique is used to design a high dimensional proposal distributions for the MH algorithm and allows the proposed batch process multi-target tracker to handle a large number of tracks in a computationally feasible manner. Our simulations show that the proposed tracker reliably estimates the number of tracks and their trajectories in scenarios with a large number of closely spaced tracks in a dense clutter environment albeit, more expensive than on-line methods.

— We propose a closed form Gaussian sum smoother and, more importantly, closed form smoothing solutions for increasingly complex problems arising from practice, including tracking in clutter, joint detection and tracking (in clutter), and multiple target tracking (in clutter) via the Probability Hypothesis Density. The solutions are based on the corresponding forward-backward smoothing recursions that involve forward propagation of the filtering densities, followed by backward propagation of the smoothed densities. The key to the exact solutions is the use of alternative forms of the backward propagations, together with standard Gaussian identities. Simulations are also presented to verify the proposed solutions.

—It was recently demonstrated that the Gaussian Mixture Cardinalised Probability Hypothesis Density (GMCPHD) filter can be used when the clutter density is unknown. Here we examine the performance of this filter, and as one would expect, it does not do as well as the conventional GMCPHD with matched clutter density. To improve the performance, we propose a bootstrap filtering scheme, and demonstrate by simulations on a bearings-only multi-target filtering scenario, that it is capable of performing almost as well as the matched GMCPHD filter.

—The objective of multi-object estimation is to simultaneously estimate the number of objects and their states from a set of observations in the presence of data association uncertainty, detection uncertainty, false observations and noise. This estimation problem can be formulated in a Bayesian framework by modeling the (hidden) set of states and set of observations as random finite sets (RFSs) that covers thinning, Markov shifts and superposition. A prior for the hidden RFS together with the likelihood of the realization of the observed RFS gives the posterior distribution via the application of Bayes rule. We propose a new class of RFS distributions that is conjugate with respect to the multi-object observation likelihood and closed under the Chapman-Kolmogorov equation. This result is tested via a Bayesian multi-target tracking example.

In this paper we derive a forward-backward smoother for joint target detection and estimation and propose a sequential Monte Carlo implementation. We model the target by a Bernoulli random finite set since the target can be in one of two 'present' or 'absent' modes. Finite Set Statistics is used to derive the smoothing recursion. Our results indicate that smoothing has two distinct advantages over just using filtering: Firstly, we are able to more accurately identify the appearance and disappearance of a target in the scene and secondly, we can provide improved state estimates when the target exists.

— A new recursive algorithm is proposed for jointly estimating the time-varying number of targets and their states from a sequence of observation sets in the presence of data association uncertainty, detection uncertainty, noise and false alarms. The approach involves modelling the respective collections of targets and measurements as random finite sets and applying the probability hypothesis density (PHD) recursion to propagate the posterior intensity, which is a first order statistic of the random finite set of targets, in time. At present, there is no closed form solution to the PHD recursion. This work shows that under linear, Gaussian assumptions on the target dynamics and birth process, the posterior intensity at any time step is a Gaussian mixture. More importantly, closed form recursions for propagating the means, covariances and weights of the constituent Gaussian components of the posterior intensity are derived. The proposed algorithm combines these recursions with a strategy for managing the number of Gaussian components to increase efficiency. This algorithm is extended to accommodate mildly nonlinear target dynamics using approximation strategies from the extended and unscented Kalman filters.

The context is sensor control for multi-object Bayes filtering in the framework of partially observed Markov decision processes. The current information state is represented by the multi-object probability density function (PDF), while the reward function, associated with each sensor control (action), is the information gain measured by the alpha or Rényi divergence. Assuming that both the predicted and updated state can be represented by independent identically distributed (IID) cluster random finite sets (RFS's) or, as a special case, the Poisson RFS's, the paper derives the analytic expressions of the corresponding Rényi divergence based information gains. The implementation of Rényi divergence via the sequential Monte Carlo method is presented. The performance of the proposed reward function is demonstrated by a numerical example, where a moving range-only sensor is controlled to estimate the number and the states of several moving objects using the PHD filter.

The problem addressed in this paper is information theoretic sensor control for recursive Bayesian multi-object state-space estimation using random finite sets. The proposed algorithm is formulated in the framework of partially observed Markov decision processes where the reward function associated with different sensor actions is computed via the Rényi or alpha divergence between the multi-object prior and the multi-object posterior densities. The proposed algorithm in implemented via the sequential Monte Carlo method. The paper then presents a case study where the problem is to localise an unknown number of sources using a controllable moving sensor which provides range-only detections. Four sensor control reward functions are compared in the study and the proposed scheme is found to perform the best.

The standard formulation of the PHD and CPHD filters assumes that the target birth intensity is known a priori. In situations where the targets can appear anywhere in the surveillance volume this is clearly inefficient, since the target birth intensity needs to cover the entire state space. This paper presents a new extension of the PHD and CPHD filters, which distinguishes between the persistent and the newborn targets. This extension enables us to adaptively design the target birth intensity at each scan using the received measurements. Sequential Monte-Carlo implementations of the resulting PHD and CPHD filters are presented and their performance studied numerically. The proposed measurement driven birth intensity improves the estimation accuracy of both the number of targets and their spatial distribution. Index Terms Bayesian multi-object filtering, random finite set, probability hypothesis density (PHD), particle filter

The problem of jointly detecting multiple objects and estimating their states from image observations is formulated in a Bayesian framework by modeling the collection of states as a random finite set. Analytic characterizations of the posterior distribution of this random finite set are derived for various prior distributions under the assumption that the regions of the observation influenced by individual objects do not overlap. These results provide tractable means to jointly estimate the number of states and their values from image observations. As an application, we develop a multi-object filter suitable for image observations with low signal to noise ratio. A particle implementation of the multi-object filter is proposed and demonstrated via simulations.

The problem of jointly detecting multiple objects and estimating their states from image observations is formulated in a Bayesian framework by modeling the collection of states as a random finite set. Analytic characterizations of the posterior distribution of this random finite set are derived for various prior distributions under the assumption that the regions of the observation influenced by individual objects do not overlap. These results provide tractable means to jointly estimate the number of states and their values from image observations. As an application, we develop a multi-object filter suitable for image observations with low signal to noise ratio. A particle implementation of the multi-object filter is proposed and demonstrated via simulations.

Let X and Y be two jointly distributed spatial Point Processes on X and Y respectively (both complete separable metric spaces). We address the problem of estimating X, which is the hidden Point Process (PP), given the realisation y of the observed PP Y. We characterise the posterior distribution of X when it is marginally distributed according to a Poisson and Gauss-Poisson prior and when the transformation from X to Y includes thinning, displacement and augmentation with extra points. These results are then applied in a filtering context when the hidden process evolves in discrete time in a Markovian fashion. The dynamics of X considered are general enough for many target tracking applications.

The problem of jointly detecting multiple objects and estimating their states from image observations is formulated in a Bayesian framework by modeling the collection of states as a random finite set. Analytic characterizations of the posterior distribution of this random finite set are derived for various prior distributions under the assumption that the regions of the observation influenced by individual objects do not overlap. These results provide tractable means to jointly estimate the number of states and their values from image observations. As an application, we develop a multi-object filter suitable for image observations with low signal to noise ratio. A particle implementation of the multi-object filter is proposed and demonstrated via simulations.