The Jump Tracker: Nonlinear Bayesian Tracking with Adaptive Meshes and a Markov Jump Process Model

Many tracking problems lie outside of the traditional situation of linear (or linearizable) measurements and dynamics addressed by the Kalman filter and its variants. This is especially true in applications where target measurements are highly ambiguous and visibility is affected by unpredictable phenomena such as intermittent interference and low signal-to-noise ratios. Bayesian tracking provides the general solution to this more general class of problems. Conceptually, Bayesian tracking is straightforward: given the target measurements, apply Bayes´ rule to compute the probability density of the target location at any given time, all the while assuming a target motion model. Bayesian trackers are computationally expensive; there are two basic approaches to their implementation: sequential Monte Carlo methods such as particle filtering, and deterministic methods that compute the target density directly. This paper presents a new approach to the direct method that is theoretically and computationally novel in several ways. First, recent results from the theory of adaptive moving meshes are modified and applied, an approach that is distinctly different from previously published direct methods that use fixed meshes to solve the Fokker-Planck equation. Second, a straight-line motion model based on a Markov jump process for the velocity is assumed. Straight-line motion punctuated by jumps in target velocity may be a more suitable assumption for some target dynamics than the traditional random walk assumed in the Kalman filter and many Bayesian trackers. The resulting linear partial differential equation that describes the target position density is relatively easy to solve numerically, especially compared to the Fokker- Planck equation that results from the random walk motion assumption. The proposed Bayesian tracking algorithm is a promising alternative to competing methods. It is also shown that, like particle filters, the adaptive mesh approach necessarily suffers f- - rom the curse of dimensionality. Simulation results are shown using the example of bistatic radar.