Issues in Target Tracking

In this lecture we discuss a number of concerns and items of interest that related to the tracking of targets in clutter. We begin with a discussion of performance evaluation methods. The familiar CRLB is shown to be adjusted in a straightforward way when deterministic trajectories are estimated in the presence of measurement-origin uncertainty (false alarms and missed detections); indeed, the same is true for nondeterministic ones, meaning those with process noise. We then mention the HYCA (hybrid conditional averaging) approach, which has considerable appeal in that it can predict the track-life as opposed to the accuracy of tracks that are kept. We then discuss track testing: the sequential probability ratio test (SPRT) for track acceptance, the Page test for track deletion, and here we most especially discuss variants (such as Shiryaev) for the case of fluctuating targets, as would be found in multi-static (fused) systems. Finally, we mention some management issues, specifically aspects of sampling time for multi-sensor systems, sensor placement with target tracking in mind, and the choice of waveform when the goal is tracking. I. PERFORMANCE EVALUATION A. The Static Case: The CRLB When Measurements are of Uncertain Origin 1) Introduction: In many estimation situations measurements are of uncertain origin. This is best exemplified by the targettracking situation in which at each scan a number mt of measurements are obtained, and it is not known which, if any, of these is target-originated. In several earlier papers the surprising observation was made that the Cramér-Rao lower bound (CRLB) for the estimation of a fixed parameter vector (e.g., initial position and velocity) that characterizes the target motion, for the special case multidimensional measurements in the presence of additive white Gaussian noise, is simply a multiple of that for the case with no uncertainty. That is, there is a scalar information-reduction factor. Details of this material are available in [24]. 2) The Multi-Parameter CRLB: In many estimation problems one is faced with the problem that one’s data is of uncertain origin. For example, in the target-tracking situation, the data set may consist of “hits” (threshold exceedances) indicative of a target’s presence in a particular location at a particular time; however, hits may be of spurious origin (that is, they are falsealarms), and indeed it may be that the true target is unrepresented in the data set at the time in question (a missed detection). At issue is how well one can estimate a fixed parameter vector that characterizes target motion given such uncertainty. The measure in which we are interested is, not unnaturally, the mean-squared error (MSE) of the estimate. There is a classical result for this, known as the Cramér-Rao lower bound (CRLB) (e.g. [20], [36]). Let us assume access to an observation Z which has probability density function (pdf) p(Z; x), meaning that the pdf depends on a parameter vector x which is to be estimated. Let us assume the existence of an unbiased estimator x̂(Z), meaning that we have E {x̂(Z)} = x (1) in which E denotes expectation parametrized by x. Then under fairly broad regularity conditions the CRLB has it that E { [x̂(Z)− x] [x̂(Z)− x] } ≥ J−1 (2) in which J ≡ E { [∇x log (p(Z; x))] [∇x log (p(Z; x))] } (3) is Fisher’s information matrix. Again under broad regularity conditions, if a maximum-likelihood estimator (MLE) for x exists, then it achieves the CRLB asymptotically. 3) Measurements of Uncertain Origin: The general multi-parameter CRLB is fairly standard; let us now turn to the case of particular interest. We define the aggregate observation Z = {Z(1),Z(2), . . . ,Z(T )} (4) in which the t observation is Z(t) = {zi(t)} i=1 (5) meaning, in the target-tracking situation, that there are mt individual observations which comprise it. Although mt is of course known to the estimator, to compute the MSE it is necessary to average over the possible values of mt, and hence we assume RTO-EN-SET-157(2011) 2 1 Issues in Target Tracking the existence of a probability mass function q(mt) controlling mt, and of a related measure (mt) denoting the probability that a particular measurement is target-originated, given that this true measurement is not missed. We further assume that