Proc. SPIE. 6699, Signal and Data Processing of Small Targets 2007
KEYWORDS: Weapons, Statistical analysis, Detection and tracking algorithms, Matrices, Error analysis, Computer simulations, Personal digital assistants, Monte Carlo methods, Data processing, Expectation maximization algorithms
The Probabilistic Multi-Hypothesis Tracker (PMHT) has been demonstrated to be an effective multi-target
tracker while retaining linear computational complexity in the number of measurements and targets. However
PMHT only provides a point estimate for target tracks. The "covariance" returned by the PMHT is a byproduct
of applying the Expectation-Maximization algorithm to maximize the PMHT likelihood function and
is not intended to be the track estimate covariance. In this paper we derive a consistent covariance estimator
for PMHT. By re-introducing the constraint that the sum of the PMHT weights (posterior probabilities that a
measurement is target-originated) across measurements sum to unity, a covariance based on Probabilistic Data
Association (PDA) principles is derived. We show through simulations that the resulting covariance provides a
consistent covariance for the PMHT track estimates.
There has been some work both in the statistics and engineering literature that gives the posterior covariance
for ML Gaussian-mixture estimation, and the PMHT can be viewed as a tracker whose genesis is of
MAP Gaussian-mixture estimation with a Gaussian prior. The expressions and calculations are, unfortunately,
complicated. Consequently we also report on a novel and intuitive way to derive these via calculus.
In many active sonar tracking applications, targets frequently undergo fading detection performance in which the target's detection probability can shift suddenly from a high to a low value. This characteristic is a function of the undersea environment. Using a multistatic active sonar problem, we examine the performance of track management (initiation and termination) routines where the target detection probability is based on an underlying Hidden Markov Model (HMM) with high and low detection states. Using a likelihood ratio test, we develop the optimum track initiation performance as measured by a System Operating Characteristic, similar to a Receiver Operating Characteristic, which plots the probability of initiating a true track versus the probability of initiating a false track. We show that near-optimal performance can be attained using track initiation logic that differentiates the measurements as to receiver source in an "M detections of N scans from C sensors" type of rule. Performance can further be improved by using a composite track initiation test that combines two or three such rules in a logical OR operation. We next show that the use of a Shiryaev-Roberts test for track termination yields the quickest detection of false tracks for a given duration of true target tracks when compared to a Page test and rule-based tests of the form "M or fewer detections from K scans".
Advances in characterizing the angle measurement covariance for phased array monopulse radar systems that use adaptive beamforming to null out a jammer source allow for the use of improved sensor models in tracking algorithms. Using a detection probability likelihood function consisting of a Gaussian sum that incorporates negative contact measurement information, three tracking systems are compared when used to track a maneuvering target passing into and through standoff jammer interference. Each tracker differs in how closely it replicates sensor performance in terms of accuracy of measurement covariance and the use of negative information. Only the tracker that uses both the negative contact information and corrected angle measurement covariance is able to consistently reacquire the target when it exits the jammer interference.
In underwater tracking, such as with an unmanned undersea vehicle (UUV) or torpedo it is advantageous to track a target covertly. The Maximum Likelihood-Probabilistic Data Association (ML-PDA) tracking algorithm, which has been demonstrated to establish and maintain track in low SNR/high clutter environments, is used to develop track information in a covert tracking application. By combining intermittent sensor data (active sonar) from the UUV with that of the launch platform (passive sonar) in the ML-PDA track algorithm, fewer active transmissions are required to establish and maintain a given track accuracy thereby reducing the chance of target alertment. We show that this is a viable operating model and demonstrate how sensor placement affects track accuracy, including determination of best sensor placement and requirements on active transmissions to maintain minimum tracking accuracy.