2011 45th Annual Conference on Information Sciences and Systems, 2011
In Bayesian multi-target filtering knowledge of parameters such as clutter intensity and sensor f... more In Bayesian multi-target filtering knowledge of parameters such as clutter intensity and sensor field-of-view are of critical importance. Significant mismatches in clutter and sensor field of view model parameters results in biased estimates. In this paper we propose a multi-target filtering solution that can accommodate non-linear target model and unknown nonhomogeneous clutter intensity and sensor field-of-view. Our solution is based on the multi-target multi-Bernoulli filter that adaptively learns non-homogeneous clutter intensity and sensor field-of-view while filtering.
2006 9th International Conference on Information Fusion, 2006
Abstract The probability hypothesis density (PHD) recursion is a first moment approximation to th... more Abstract The probability hypothesis density (PHD) recursion is a first moment approximation to the multi-target Bayes filter which propagates the posterior intensity of the random finite set of targets in time. The cardinalized PHD (CPHD) recursion is a generalization of the PHD recursion, which jointly propagates the posterior intensity and the posterior cardinality distribution. Using the recently developed closed form solutions to both the PHD and CPHD recursions for linear Gaussian multi-target models, we present a comparative study of their ...
IEE Proceedings - Vision, Image, and Signal Processing, 2004
The problem of designing a transmit pulse such that after transmission over a dispersive channel ... more The problem of designing a transmit pulse such that after transmission over a dispersive channel the received pulse fits in a prescribed template can be formulated as a quadratic programming problem with affine semi-infinite constraints. In constrained optimisation, the optimal solution invariably lies on the boundary of the feasible set, and consequently perturbations on the optimal solution or the constraints means that the received pulses may no longer fit in the template. Perturbations to the optimal transmit pulse and the constraints arise, in practice, from errors in the implementation and uncertainty in the channel parameter, respectively. In the paper, a robust formulation is presented which ensures that the constraints are satisfied even in the presence of implementation errors and channel parameter uncertainty. The technique developed is applied to determine the optimal transmit pulse shape to be programmed on a commercial T1 (1.544 Mbit/s) line interface unit.
Let X and Y be two jointly distributed spatial Point Processes on X and Y respectively (both comp... more 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 probability hypothesis density (PHD) recursion propagates the posterior intensity of the ra... more — The probability hypothesis density (PHD) recursion propagates the posterior intensity of the random finite set of targets in time. The cardinalized PHD (CPHD) recursion is a generalization of the PHD recursion, which jointly propagates the posterior intensity and the posterior cardinality distribution. In general, the CPHD recursion is computationally intractable. This paper proposes a closed-form solution to the CPHD recursion under linear Gaussian assumptions on the target dynamics and birth process. Based on this solution, an effective multi-target tracking algorithm is developed. Extensions of the proposed closed form recursion to accommodate non-linear models are also given using linearization and unscented transform techniques. The proposed CPHD implementations not only sidestep the need to perform data association found in traditional methods, but also dramatically improve the accuracy of individual state estimates as well as the variance of the estimated number of targets when compared to the standard PHD filter. Our implementations only have a cubic complexity, but simulations suggest favourable performance compared to the standard JPDA filter which has a non-polynomial complexity.
— This paper presents a novel and mathematically rigorous Bayes recursion for tracking a target t... more — This paper presents a novel and mathematically rigorous Bayes recursion for tracking a target that generates multiple measurements with state dependent sensor field of view and clutter. Our Bayesian formulation is mathematically well-founded due to our use of a consistent likelihood function derived from random finite set theory. It is established that under certain assumptions, the proposed Bayes recursion reduces to the car-dinalized probability hypothesis density (CPHD) recursion for a single target. A particle implementation of the proposed recursion is given. Under linear Gaussian and constant sensor field of view assumptions, an exact closed form solution to the proposed recursion is derived, and efficient implementations are given. Extensions of the closed form recursion to accommodate mild non-linearities are also given using linearization and unscented transforms.
— The concept of a miss-distance, or error, between a reference quantity and its estimated/contro... more — The concept of a miss-distance, or error, between a reference quantity and its estimated/controlled value, plays a fundamental role in any filtering/control problem. Yet there is no satisfactory notion of a miss-distance in the well-established field of multi-object filtering. In this paper, we outline the inconsistencies of existing metrics in the context of multi-object miss-distances for performance evaluation. We then propose a new mathematically and intuitively consistent metric that addresses the drawbacks of current multi-object performance evaluation metrics.
— It is shown analytically that the Multi-Target Multi-Bernoulli (MeMBer) recursion, proposed by ... more — It is shown analytically that the Multi-Target Multi-Bernoulli (MeMBer) recursion, proposed by Mahler, has a significant bias in the number of targets. To reduce the cardinality bias, a novel multi-Bernoulli approximation to the multi-target Bayes recursion is derived. Under the same assumptions as the MeMBer recursion, the proposed recursion is unbiased. In addition, a Sequential Monte Carlo (SMC) implementation (for generic models) and a Gaussian mixture (GM) implementation (for linear Gaussian models) are proposed. The latter is also extended to accommodate mildly non-linear models by linearization and the unscented transform.
— A new recursive algorithm is proposed for jointly estimating the time-varying number of targets... more — 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.
— Random finite sets are natural representations of multi-target states and observations that all... more — 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.
—The problem of jointly detecting multiple objects and estimating their states from image observa... more —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 context is sensor control for multi-object Bayes filtering in the framework of partially obse... more 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 Bayesia... more 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 k... more 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 probability hypothesis density (PHD) filter is an attractive approach to tracking an unknown ... more 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.
Performance evaluation of multi-target tracking algorithms is of great practical importance in th... more Performance evaluation of multi-target tracking algorithms is of great practical importance in the design, parameter optimisation and comparison of tracking systems. The goal of performance evaluation is to measure the distance between two sets of tracks: the ground truth tracks and the set of estimated tracks. This paper proposes a mathematically rigorous metric for this purpose. The basis of the proposed distance measure is the recently formulated consistent metric for performance evaluation of multi-target filters, referred to as the OSPA metric. Multi-target filters sequentially estimate the number of targets and their position in the state-space. The OSPA metric is therefore defined on the space of finite sets of vectors. The distinction between filtering and tracking is that tracking algorithms output tracks, and a track represents a labeled temporal sequence of state estimates, associated with the same target. The metric proposed in this paper is therefore defined on the space of finite sets of tracks. Numerical examples demonstrate that the proposed metric behaves in a manner consistent with our expectations.
—This paper proposes an integrated Bayesian framework for feature-based simultaneous localisation... more —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.
In this paper we derive a forward-backward smoother for joint target detection and estimation and... more 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 forward-backward Probability Hypothesis Density (PHD) smoother involving forward filtering foll... more 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.
In Bayesian multi-target filtering we have to contend with two notable sources of uncertainty, cl... more 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.
2011 45th Annual Conference on Information Sciences and Systems, 2011
In Bayesian multi-target filtering knowledge of parameters such as clutter intensity and sensor f... more In Bayesian multi-target filtering knowledge of parameters such as clutter intensity and sensor field-of-view are of critical importance. Significant mismatches in clutter and sensor field of view model parameters results in biased estimates. In this paper we propose a multi-target filtering solution that can accommodate non-linear target model and unknown nonhomogeneous clutter intensity and sensor field-of-view. Our solution is based on the multi-target multi-Bernoulli filter that adaptively learns non-homogeneous clutter intensity and sensor field-of-view while filtering.
2006 9th International Conference on Information Fusion, 2006
Abstract The probability hypothesis density (PHD) recursion is a first moment approximation to th... more Abstract The probability hypothesis density (PHD) recursion is a first moment approximation to the multi-target Bayes filter which propagates the posterior intensity of the random finite set of targets in time. The cardinalized PHD (CPHD) recursion is a generalization of the PHD recursion, which jointly propagates the posterior intensity and the posterior cardinality distribution. Using the recently developed closed form solutions to both the PHD and CPHD recursions for linear Gaussian multi-target models, we present a comparative study of their ...
IEE Proceedings - Vision, Image, and Signal Processing, 2004
The problem of designing a transmit pulse such that after transmission over a dispersive channel ... more The problem of designing a transmit pulse such that after transmission over a dispersive channel the received pulse fits in a prescribed template can be formulated as a quadratic programming problem with affine semi-infinite constraints. In constrained optimisation, the optimal solution invariably lies on the boundary of the feasible set, and consequently perturbations on the optimal solution or the constraints means that the received pulses may no longer fit in the template. Perturbations to the optimal transmit pulse and the constraints arise, in practice, from errors in the implementation and uncertainty in the channel parameter, respectively. In the paper, a robust formulation is presented which ensures that the constraints are satisfied even in the presence of implementation errors and channel parameter uncertainty. The technique developed is applied to determine the optimal transmit pulse shape to be programmed on a commercial T1 (1.544 Mbit/s) line interface unit.
Let X and Y be two jointly distributed spatial Point Processes on X and Y respectively (both comp... more 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 probability hypothesis density (PHD) recursion propagates the posterior intensity of the ra... more — The probability hypothesis density (PHD) recursion propagates the posterior intensity of the random finite set of targets in time. The cardinalized PHD (CPHD) recursion is a generalization of the PHD recursion, which jointly propagates the posterior intensity and the posterior cardinality distribution. In general, the CPHD recursion is computationally intractable. This paper proposes a closed-form solution to the CPHD recursion under linear Gaussian assumptions on the target dynamics and birth process. Based on this solution, an effective multi-target tracking algorithm is developed. Extensions of the proposed closed form recursion to accommodate non-linear models are also given using linearization and unscented transform techniques. The proposed CPHD implementations not only sidestep the need to perform data association found in traditional methods, but also dramatically improve the accuracy of individual state estimates as well as the variance of the estimated number of targets when compared to the standard PHD filter. Our implementations only have a cubic complexity, but simulations suggest favourable performance compared to the standard JPDA filter which has a non-polynomial complexity.
— This paper presents a novel and mathematically rigorous Bayes recursion for tracking a target t... more — This paper presents a novel and mathematically rigorous Bayes recursion for tracking a target that generates multiple measurements with state dependent sensor field of view and clutter. Our Bayesian formulation is mathematically well-founded due to our use of a consistent likelihood function derived from random finite set theory. It is established that under certain assumptions, the proposed Bayes recursion reduces to the car-dinalized probability hypothesis density (CPHD) recursion for a single target. A particle implementation of the proposed recursion is given. Under linear Gaussian and constant sensor field of view assumptions, an exact closed form solution to the proposed recursion is derived, and efficient implementations are given. Extensions of the closed form recursion to accommodate mild non-linearities are also given using linearization and unscented transforms.
— The concept of a miss-distance, or error, between a reference quantity and its estimated/contro... more — The concept of a miss-distance, or error, between a reference quantity and its estimated/controlled value, plays a fundamental role in any filtering/control problem. Yet there is no satisfactory notion of a miss-distance in the well-established field of multi-object filtering. In this paper, we outline the inconsistencies of existing metrics in the context of multi-object miss-distances for performance evaluation. We then propose a new mathematically and intuitively consistent metric that addresses the drawbacks of current multi-object performance evaluation metrics.
— It is shown analytically that the Multi-Target Multi-Bernoulli (MeMBer) recursion, proposed by ... more — It is shown analytically that the Multi-Target Multi-Bernoulli (MeMBer) recursion, proposed by Mahler, has a significant bias in the number of targets. To reduce the cardinality bias, a novel multi-Bernoulli approximation to the multi-target Bayes recursion is derived. Under the same assumptions as the MeMBer recursion, the proposed recursion is unbiased. In addition, a Sequential Monte Carlo (SMC) implementation (for generic models) and a Gaussian mixture (GM) implementation (for linear Gaussian models) are proposed. The latter is also extended to accommodate mildly non-linear models by linearization and the unscented transform.
— A new recursive algorithm is proposed for jointly estimating the time-varying number of targets... more — 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.
— Random finite sets are natural representations of multi-target states and observations that all... more — 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.
—The problem of jointly detecting multiple objects and estimating their states from image observa... more —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 context is sensor control for multi-object Bayes filtering in the framework of partially obse... more 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 Bayesia... more 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 k... more 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 probability hypothesis density (PHD) filter is an attractive approach to tracking an unknown ... more 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.
Performance evaluation of multi-target tracking algorithms is of great practical importance in th... more Performance evaluation of multi-target tracking algorithms is of great practical importance in the design, parameter optimisation and comparison of tracking systems. The goal of performance evaluation is to measure the distance between two sets of tracks: the ground truth tracks and the set of estimated tracks. This paper proposes a mathematically rigorous metric for this purpose. The basis of the proposed distance measure is the recently formulated consistent metric for performance evaluation of multi-target filters, referred to as the OSPA metric. Multi-target filters sequentially estimate the number of targets and their position in the state-space. The OSPA metric is therefore defined on the space of finite sets of vectors. The distinction between filtering and tracking is that tracking algorithms output tracks, and a track represents a labeled temporal sequence of state estimates, associated with the same target. The metric proposed in this paper is therefore defined on the space of finite sets of tracks. Numerical examples demonstrate that the proposed metric behaves in a manner consistent with our expectations.
—This paper proposes an integrated Bayesian framework for feature-based simultaneous localisation... more —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.
In this paper we derive a forward-backward smoother for joint target detection and estimation and... more 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 forward-backward Probability Hypothesis Density (PHD) smoother involving forward filtering foll... more 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.
In Bayesian multi-target filtering we have to contend with two notable sources of uncertainty, cl... more 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.
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