PCRLB for tracking in cluttered environments: measurement sequence conditioning approach

We consider the problem of calculating the posterior Cramer-Rao lower bound (PCRLB) for tracking in cluttered domains in which there can be both missed detections and false alarms. We present a novel formulation of the PCRLB in which we initially determine a bound conditional on the sequence of measurements available. We then create an unconditional bound as a weighted average of these conditional PCRLBs. This new bound is proven to be less optimistic than the standard formulation of the PCRLB for cluttered environments recently developed (Zhang and Willett, 2001 and Hernandez et al., 2002) and will therefore better predict optimal estimator performance. At each stage, the conditional PCRLB must evaluate the effect of the uncertain measurements, and we extend previous work (Hernandez et al., 2002) to show that the measurement origin uncertainty manifests itself as a single information reduction factor (IRF) that is dependent on the number of measurements available. We also present some useful approximations when the false alarm rate is low. Simulations then consider the problems of 1) determining the CRLB for the point of impact of a ballistic missile, and 2) determining the PCRLB for tracking a nearly constant-velocity (NCV) target in a high clutter environment. In each case, we compare the new bound with the standard approach, and as expected the new CRLB/PCRLB can be seen to be less optimistic. Moreover, in case 1) we compare the new CRLB with a heuristic bound specially constructed for this problem, and a maximum likelihood estimator (MLC). The new bound both compares favorably with the heuristic bound, and shows close agreement with the performance of the MLE. The new bound is therefore an accurate predictor of filter performance in this case. In example 2) we demonstrate some interesting features of the new theory. Of particular interest we determine both precisely when the new bound will be significantly greater than the standard bound and when the two bounds will be virtually identical. This is useful in determining when the new approach, with its greater computational burden, should be preferred to the established approach. We conclude that the novel PCRLB formulation introduced herein represents an exciting development in the determination of RMSE performance bounds in cluttered environments.