Ronald P. S. Mahler

PHD filters of higher order in target number

The multitarget recursive Bayes nonlinear filter is the theoretically optimal approach to multisensor-multitarget detection, tracking, and identification. For applications in which this filter is appropriate, it is likely to be tractable for only a small number of targets. In earlier papers we derived closed-form equations for an approximation of this filter based on propagation of a first-order multitarget moment called the probability hypothesis density (PHD). In a recent paper, Erdinc, Willett, and Bar-Shalom argued for the need for a PHD-type filter which remains first-order in the states of individual targets, but which is higher-order in target number. In this paper we show that this is indeed possible. We derive a closed-form cardinalized PHD (CPHD) filter, which propagates not only the PHD but also the entire probability distribution on target number.

Multitarget miss distance via optimal assignment

The concept of miss distance-Euclidean, Mahalanobis, etc.-is a fundamental, far-reaching, and taken-for-granted element of the engineering theory and practice of single-target systems. In this paper we introduce a comprehensive L/sub p/-type theory of distance metrics for multitarget (and, more generally, multiobject) systems. We show that this theory extends, and provides a rigorous theoretical basis for, an intuitively appealing optimal-assignment approach proposed by Drummond for evaluating the performance of multitarget tracking algorithms. We describe tractable computational approaches for computing such metrics based on standard optimal assignment or convex optimization techniques. We describe the potentially far-reaching implications of these metrics for applications such as performance evaluation and sensor management. In the former case, we demonstrate the application of multitarget miss-distance metrics as measures of effectiveness (MoEs) for multitarget tracking algorithms.

Particle-systems implementation of the PHD multitarget-tracking filter

We report here on the implementation of a particle systems approximation to the probability hypothesis density (PHD). The PHD of the multitarget posterior density has the property that, given any volume of state space, the integral of the PHD over that volume yields the expected number of targets present in the volume. As in the single target setting, upon receipt of an observation, the particle weights are updated, taking into account the sensor likelihood function, and then propagated forward in time by sampling from a Markov transition density. We also incorporate resampling and regularization into our implementation, introducing the new concept of cluster resampling.

CPHD Filtering With Unknown Clutter Rate and Detection Profile

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 result in biased estimates. In practice, these model parameters are often manually tuned or estimated offline 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.

Robust Multi-Bernoulli Filtering

In Bayesian multi-target filtering knowledge of parameters such as clutter intensity and detection probability profile are of critical importance. Significant mismatches in clutter and detection model parameters results in biased estimates. In this paper we propose a multi-target filtering solution that can accommodate non-linear target models and an unknown non-homogeneous clutter and detection profile. Our solution is based on the multi-target multi-Bernoulli filter that adaptively learns non-homogeneous clutter intensity and detection probability while filtering.

“Statistics 102” for Multisource-Multitarget Detection and Tracking

This tutorial paper summarizes the motivations, concepts and techniques of finite-set statistics (FISST), a system-level, “top-down,” direct generalization of ordinary single-sensor, single-target engineering statistics to the realm of multisensor, multitarget detection and tracking. Finite-set statistics provides powerful new conceptual and computational methods for dealing with multisensor-multitarget detection and tracking problems. The paper describes how “multitarget integro-differential calculus” is used to extend conventional single-sensor, single-target formal Bayesian motion and measurement modeling to general tracking problems. Given such models, the paper describes the Bayes-optimal approach to multisensor-multitarget detection and tracking: the multisensor-multitarget recursive Bayes filter. Finally, it describes how multitarget calculus is used to derive principled statistical approximations of this optimal filter, such as PHD filters, CPHD filters, and multi-Bernoulli filters.

Closed-Form Solutions to Forward–Backward Smoothing

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.

Approximate multisensor CPHD and PHD filters

The probability hypothesis density (PHD) filter and cardinalized probability hypothesis density (CPHD) filter are principled approximations of the general multitarget Bayes recursive filter. Both filters are single-sensor filters. Since their multisensor generalizations are computationally intractable, a further approximation - iterating their corrector equations, once for each sensor - has been used instead. This approach is theoretically unpleasing because it is not invariant under reordering of the sensors, and because it is implicitly based on strong simplifying assumptions. The purpose of this paper is to derive multisensor PHD and CPHD filters that (1) are invariant under sensor reordering, (2) require much weaker simplifying assumptions, and (3) are potentially computationally tractable (at least in the case of the multisensor CPHD filter).

SLAM Gets a PHD: New Concepts in Map Estimation

Having been referred to as the Holy Grail of autonomous robotics research, simultaneous localization and mapping (SLAM) lies at the core of most the autonomous robotic applications. This article explains the recent advances in the representations of robotic sensor measurements and the map itself as well as their consequences on the robustness of SLAM. Fundamentally, the concept of a set-based measurement and map state representation allows all of the measurement information, spatial and detection, to be incorporated into joint Bayesian SLAM frameworks. Modeling measurements and the map state as random finite sets (RFSs) rather than the traditionally adopted random vectors is not merely a triviality of notation. It will be demonstrated that a set-based framework circumvents the necessity for any fragile data association and map management heuristics, which are necessary in vector-based solutions.

Forward-Backward Probability Hypothesis Density Smoothing

A forward-backward probability hypothesis density (PHD) smoother involving forward filtering followed by backward smoothing is proposed. The forward filtering is performed by Mahlers 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.