Tim Zajic

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.

Bulk multitarget tracking using a first-order multitarget moment filter

In certain applications it is sometimes not necessary to detect and track individual targets with accuracy. Examples include applications with very large track densities, in which the overall distribution of forces is of greater interest than individual targets; or group target processing, in which detection and tracking of force-level objects (brigades, battalions, etc.) is of greater interest than detection and tracking of the individual targets which constitute them. The usual strategy is to attempt to detect and track individual targets first and then deduce group behavior from them. The approach described in this paper employs the opposite philosophy: it detects and tracks target groupings first and sorts out individual targets only as data quantity and quality permits. It is based on a multitarget statistical analog of the simplest approximate single-target filter: the constant-gain Kalman filter. Our approximate multitarget filter propagates a first-order statistical moment of the entire multitarget system. This moment, the probability hypothesis density (PHD), is the density function whose integral in any region of state space is the expected number of targets in the region. We describe the behavior of an implementation of the PHD filter in some simple bulk-tracking scenarios.

Joint tracking and identification with robustness against unmodeled targets

We report here on an application of a particle systems implementation of 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. The application we consider is the joint tracking and identification of multiple aircraft, with the observations consisting of noisy position measurements and high range resolution radar (HRRRR) signatures. We also take into consideration the presence of clutter and a probability of detection less than unity. Experimental results are presented.

Multitarget filtering using a multitarget first-order moment statistic

The theoretically optimal approach to multitarget detection, tracking, and identification is a suitable generalization of the recursive Bayes nonlinear filter. This approach will never be of practical interest without the development of drastic but principled approximation strategies. In single- target problems, the computationally fastest approximate filtering approach is the constant-gain Kalman filter. This filter propagates a first-order statistical moment of the single-target system (the posterior expectation) in the place of the posterior distribution. This paper describes an analogous strategy: propagation of a first-order statistical moment of the multitarget system. This moment, the probability hypothesis density (PHD), is the density function on single-target state space that is uniquely defined by the following property: its integral in any region of states space is the expected number of targets in that region. We describe recursive Bayes filter equations for the PHD that account for multiple sensors, missed detections and false alarms, and appearance and disappearance of targets.

Multitarget nonlinear filtering based on spectral compression and probability hypothesis density

The theoretically optimal approach to multitarget detection, tracking, and identification is a suitable generalization of the recursive Bayes nonlinear filter. However, this optimal filter is so computationally challenging that it must usually be approximated. We report on a novel approximation of a multi-target non-linear filtering based on the spectral compression (SPECC) non-linear filter implementation of Stein-Winter probability hypothesis densities (PHDs). In its current implementation, SPECC is a two-dimensional, four-state, FFT-based filter that is Bayes-Closed. It replaces a log-posterior or log-likelihood with an approximate log-posterior or log-likelihood, that is a truncation of a Fourier basis. This approximation is based on the minimization of the least-squares error of the log-densities. The ultimate operational utility of our approach depends on its computational efficiency and robustness when compared with similar approaches. Another novel aspect of the proposed algorithm is the propagation of a first-order statistical moment of the multitarget system. This moment, the probability hypothesis density (PHD) is a density function on single-target state space which is uniquely defined by the following property: its integral in any region of state space is the expected number of targets in that region. It is the expected value of the point process of the random track set (i.e., the density function whose integral in any region of state space is the actual number of targets in the region). The adequacy, and the accuracy of the algorithm when applied to simulated and real scenarios involving ground targets are demonstrated.

Scientific performance evaluation for distributed sensor management and adaptive data fusion

For the last two years at this conference, we have described the implementation of a unified, scientific approach to performance measurement for data fusion algorithms based on FINITE-SET STATISTICS (FISST). FISST makes it possible to directly extend Shannon-type information metrics to multisource, multitarget problems. In previous papers we described application of information Measures of Effectiveness (MoEs) to multisource-multitarget data fusion and to non-distributed sensor management. In this follow-on paper we show how to generalize this work to DISTRIBUTED sensor management and ADAPTIVE DATA FUSION.

User-defined information and scientific performance evaluation

For the past two years at this conference we have described results in the practical implementation of a unified, scientific approach to performance measurement for data fusion algorithms. Our approach is based on finite set statistics (FISST), a generalization of conventional statistics to multisource, multitarget problems. Finite-set statistics makes it possible to directly extend Shannon-type information metrics to multisource, multitarget problems in such a way that information can be defined and measured even though any given end-user may have conflicting or even subjective definitions of what information means. In this follow-on paper we describe the performance of FISST based metrics that take into account a users definition of information and develop a rigorous theory of partial information for multisource, multi-target problems.

Multisensor-multitarget sensor management with target preference

Multisensor-multitarget sensor management is viewed as a problem in nonlinear control theory. This paper applies newly developed theories for sensor management based on a Bayesian control-theoretic foundation. Finite-Set-Statistics (FISST) and the Bayes recursive filter for the entire multisensor-multitarget system are used with information-theoretic objective functions in the development of the sensor management algorithms. The theoretical analysis indicate that some of these objective functions lead to potentially tractable sensor management algorithms when used in conjunction with MHC (multi-hypothesis correlator)-like algorithms. We show examples of such algorithms, and present an evaluation of their performance against multisensor-multitarget scenarios. This sensor management formulation also allows for the incorporation of target preference, and experiments demonstrating the performance of sensor management with target preference will be presented.

Probabilistic track-to-track association

We consider the track-to-track association problem. This problem is often a key ingredient when seeking to integrate data from multiple sensors. We propose a probabilistic approach, inspired by the joint probabilistic data association, or JPDA, approach used in the data association problem. To solve the proposed model we adapt a recent deterministic polynomial-time approximation algorithm. We give consideration also to the setting in which one or more sensors may contain biases.

Confidence measure and performance evaluation for HRRR-based classifiers

The work presented here is a continuation of research first reported in Mahler, et. al. Our earlier efforts included integrating the Statistical Features algorithm with a Bayesian nonlinear filter, allowing simultaneous determination of target position, velocity, pose and type via maximum a posteriori estimation. We then considered three alternative classifiers: the first based on a principal component decomposition, the second on a linear discriminant approach, and the third on a wavelet representation. In addition, preliminary results were given with regards to assigning a measure of confidence to the output of the wavelet based classifier. In this paper we continue to address the problem of target classification based on high range resolution radar signatures. In particular, we examine the performance of a variant of the principal component based classifier as the number of principal components is varied. We have chosen to quantify the performance in terms of the Bhattacharyya distance. We also present further results regarding the assignment of confidence values to the output of the wavelet based classifier.