Second-order multi-object filtering with target interaction using determinantal point processes

The probability hypothesis density (PHD) filter is a practical alternative to the optimal Bayesian multi-target filter based on random finite sets. It propagates the PHD function, the first order moment of the posterior multi-target density, from which the number of targets as well as their individual states can be extracted. Furthermore, the sequential Monte Carlo (SMC) approximation of the PHD filter (also known as particle-PHD filter) is available in the literature in order to overcome its intractability. However, the PHD filter keeps no track of the target identities and hence cannot produce track-valued estimates of individual targets. This work consider the use of an improved implementation, of the particle-PHD filter that gives the track-valued estimates of individual targets and propose a novel way for doing so. The improved PHD filter combines the particles approximation of the posterior PHD function and the peak extraction from the posterior PHD particles to create the target identities of the individual estimates. The improved PHD filter does not affect the convergence results of the particle-PHD filter

Multiple target tracking requires data association that operates in conjunction with filtering. When multiple targets are closely spaced, the conventional approaches (as, e.g., MHT/assignment) may not give satisfactory results. This is mainly because of the difficulty in deciding what the number of targets is. Recently, the probability hypothesis density (PHD) filter has been proposed and particle filtering techniques have been developed to implement the PHD filter. In the particle PHD filter, the track labeling problem is not considered, i.e., the PHD is obtained only for a frame at a time, and it is very difficult to perform the multipeak extraction, particularly in high clutter environments. A track labeling method combined with the PHD approach, as well as considering the finite resolution, is proposed here for multitarget tracking, i.e., we keep a separate tracker for each target, use the PHD in the resolution cell to get the estimated number and locations of the targets at each time step, and then perform the track labeling ("peak-to-track" association), whose results can provide information for PHD peak extraction at the next time step. Besides, by keeping a separate tracker for each target, our approach provides more information than the standard particle PHD filter. For example, in group target tracking, if we are interested in the motion of a specific target, we can track this target, which is not possible for the standard particle PHD filter, since the standard particle PHD filter does not keep track labels. Using our approach, multitarget tracking can be performed with automatic track initiation, maintenance, spawning, merging, and termination

Multi-target tracking is a common problem with many applications. In most of these the expected number of targets is not known a priori, so that it has to be estimated from the measured data. Poisson point processes (PPPs) are a very useful theoretical model for diverse applications. One of those is multi-target tracking of an unknown number of targets, leading to the intensity filter (iFilter) and the probability hypothesis density (PHD) filter. This chapter presents a sequential Monte Carlo (SMC) implementation of the iFilter. In theory it was shown that the iFilter can estimate a clutter model from the measurements and thus does not need it as a priori knowledge, like the PHD filter does. Our studies show that this property holds not only in simulations but also in real world applications. In addition it can be shown that the performance of the PHD filter decreases substantially if the a priori knowledge of the clutter intensity is chosen incorrectly.

This paper explores application of the Probability Hypothesis Density (PHD) filter to the estimation of a threat level pertaining to an object population. Specifically, it develops explicit and compact expression for computation of its variance, a second-order statistical moment that quantifies the dispersion of the threat level around its mean value. The behaviour of the statistic is demonstrated through simulation examples.

We evaluate the use of a probability hypothesis density (PHD) filter in a bearings-only tracking application. The main feature of a PHD filter is that it propagates the first-order statistical moment of a multisource posterior distribution. Multisource estimation using a PHD filter has been shown to reliably track multiple simulated targets in the bearings-only case. In this paper we evaluate the utility of the sequential Monte-Carlo PHD filter for tracking surface ships using bearings-only data acquired from a Bluefin-21 unmanned underwater vehicle in Boston Harbor. The unmanned underwater vehicle was equipped with a rigidly mounted planar hydrophone array that measures the bearing angle to sources of acoustic noise, of which shipping traffic is the dominant source. We further evaluate several target maneuvering models, including clockwise and counter-clockwise coordinated turns. The combination of the coordinated turn models with a constant velocity model is used in a multiple model PHD filter. The results of the multiple model PHD filter are compared to the results of a PHD filter using only a constant velocity model.

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.

The probability hypothesis density (PHD) filter is an effective means to track multiple targets in that it avoids explicit data associations between the measurements and targets. However, the target birth intensity as a prior is assumed to be known before tracking in a traditional target-tracking algorithm; otherwise, the performance of a conventional PHD filter will decline sharply. Aiming at this problem, a novel target birth intensity scheme and an improved measurement-driven scheme are incorporated into the PHD filter. The target birth intensity estimation scheme, composed of both PHD pre-filter technology and a target velocity extent method, is introduced to recursively estimate the target birth intensity by using the latest measurements at each time step. Second, based on the improved measurement-driven scheme, the measurement set at each time step is divided into the survival target measurement set, birth target measurement set, and clutter set, and meanwhile, the survival and birth target measurement sets are used to update the survival and birth targets, respectively. Lastly, a Gaussian mixture implementation of the PHD filter is presented under a linear Gaussian model assumption. The results of numerical experiments demonstrate that the proposed approach can achieve a better performance in tracking systems with an unknown newborn target intensity.

We propose a filtering framework for multitarget tracking that is based on the probability hypothesis density (PHD) filter and data association using graph matching. This framework can be combined with any object detectors that generate positional and dimensional information of objects of interest. The PHD filter compensates for missing detections and removes noise and clutter. Moreover, this filter reduces the growth in complexity with the number of targets from exponential to linear by propagating the first-order moment of the multitarget posterior, instead of the full posterior. In order to account for the nature of the PHD propagation, we propose a novel particle resampling strategy and we adapt dynamic and observation models to cope with varying object scales. The proposed resampling strategy allows us to use the PHD filter when a priori knowledge of the scene is not available. Moreover, the dynamic and observation models are not limited to the PHD filter and can be applied to any Bayesian tracker that can handle state-dependent variances. Extensive experimental results on a large video surveillance dataset using a standard evaluation protocol show that the proposed filtering framework improves the accuracy of the tracker, especially in cluttered scenes.

We propose an approach for tracking a varying number of simultaneously active acoustic wideband signal sources in an acoustic enclosure. Relying on the assumption of W-disjoint orthogonality, the method uses narrowband position estimates of the sources for the targets. The instantaneous position estimates form clusters rather than single points, as would be required for a conventional Probability Hypothesis Density (PHD) filter. Therefore, we model the position estimates as extended targets and use a special form of the PHD filter, the extended target Gaussian mixture PHD filter, for tracking the targets. This allows to model target birth and death, which correspond to speech onset and end of utterance, respectively. With this model and by using the well-developed theory of Finite Set Statistics (FISST)-based multi-target tracking, we provide a comprehensive, strictly Bayesian treatment of the problem of tracking wideband acoustic sources using narrowband position estimates. We validate the results by tracking a varying number of targets in an enclosure simulated with the image-source method.

The probability hypothesis density (PHD) filter is a first moment approximation to the evolution of a dynamic point process which can be used to approximate the optimal filtering equations of the multiple-object tracking problem. We show that, under reasonable assumptions, a sequential Monte Carlo (SMC) approximation of the PHD filter converges in mean of order \(p \geq 1\), and hence almost surely, to the true PHD filter. We also present a central limit theorem for the SMC approximation, show that the variance is finite under similar assumptions and establish a recursion for the asymptotic variance. This provides a theoretical justification for this implementation of a tractable multiple-object filtering methodology and generalises some results from sequential Monte Carlo theory.

In this paper we propose a Track Before Detect (TBD) filter for Direction Of Arrival (DOA) tracking of multiple targets from phased-array observations. The phased-array model poses a new problem since each target emits a signal, called source signal. Existing methods consider the source signal as part of the system state. This is inefficient, especially for particle approximations of posteriors, where samples are drawn from the higher-dimensional posterior of the extended state. To address this problem, we propose a novel Marked Poisson Point Process (MPPP) model and derive the Probability Hypothesis Density (PHD) filter that adaptively estimates target DOAs. The PPP models variations of both the number and the location of points representing targets. The mark of a point represents the source signal, without the need of an extended state. Recursive formulas for the MPPP PHD filter are derived with simulations showcasing improved performance over state-of-the art methods.

The work on 3D human pose estimation has been through a significant amount of progress in recent years, particularly due to the widespread availability of commodity depth sensors. However, most pose estimation methods follow a tracking-as-detection approach which does not explicitly handle occlusions, thus introducing outliers and identity association issues when multiple targets are involved. To address these issues, we propose a new method based on Probability Hypothesis Density (PHD) filter. In this method, the PHD filter with a novel clutter intensity model is used to remove outliers in the 3D head detection results, followed by an identity association scheme with occlusion detection for the targets. Experimental results show that our proposed method greatly mitigates the outliers, and correctly associates identities to individual detections with low computational cost.

This paper is focus on multiple target tracking with arbitrary clutter process by using the probability hypothesis density (PHD) filter. Based on the Random Finite Set (RFS) framework, the clutter process in the classical PHD filter is modeled as Poisson RFS, which is only reasonable for some scenarios in reality. The PHD filter suitable for arbitrary clutter process has not yet been implemented. In this paper, an arbitrary clutter PHD filter, as well as its particle filter implementation, are derived by using the probability generating functional (PGFL) based on the RFS framework. Simulation results show that the proposed arbitrary clutter PHD filter can be used for multiple target tracking with arbitrary clutter process.

Determinantal point processes (DPPs) have recently proved to be a useful class of models in several areas of statistics, including spatial statistics, statistical learning and telecommunications networks. They are models for repulsive (or regular, or inhibitive) point processes, in the sense that nearby points of the process tend to repel each other. We consider two ways to quantify the repulsiveness of a point process, both based on its second-order properties, and we address the question of how repulsive a stationary DPP can be. We determine the most repulsive stationary DPP, when the intensity is fixed, and for a given $R>0$ we investigate repulsiveness in the subclass of $R$-dependent stationary DPPs, that is, stationary DPPs with $R$-compactly supported kernels. Finally, in both the general case and the $R$-dependent case, we present some new parametric families of stationary DPPs that can cover a large range of DPPs, from the stationary Poisson process (the case of no interaction) to the most repulsive DPP.

Randomized Numerical Linear Algebra (RandNLA) uses randomness to develop improved algorithms for matrix problems that arise in scientific computing, data science, machine learning, etc. Determinantal Point Processes (DPPs), a seemingly unrelated topic in pure and applied mathematics, is a class of stochastic point processes with probability distribution characterized by sub-determinants of a kernel matrix. Recent work has uncovered deep and fruitful connections between DPPs and RandNLA which lead to new guarantees and improved algorithms that are of interest to both areas. We provide an overview of this exciting new line of research, including brief introductions to RandNLA and DPPs, as well as applications of DPPs to classical linear algebra tasks such as least squares regression, low-rank approximation and the Nystrom method. For example, random sampling with a DPP leads to new kinds of unbiased estimators for least squares, enabling more refined statistical and inferential understanding of these algorithms; a DPP is, in some sense, an optimal randomized algorithm for the Nystrom method; and a RandNLA technique called leverage score sampling can be derived as the marginal distribution of a DPP. We also discuss recent algorithmic developments, illustrating that, while not quite as efficient as standard RandNLA techniques, DPP-based algorithms are only moderately more expensive.