Batch Maximum Likelihood (ML) and Maximum A Posteriori (MAP) Estimation With Process Noise for Tracking Applications

Batch maximum likelihood (ML) and maximum a posteriori (MAP) estimation with process noise is now more than thirty-five years old, and its use in multiple target tracking has long been considered to be too computationally intensive for real-time applications. While this may still be true for general usage, it is ideally suited for special needs such as bias estimation, track initiation and spawning, long-term prediction of track states, and state estimation during periods of rapidly changing target dynamics. In this paper, we examine the batch estimator formulation for several cases: nonlinear and linear models, with and without a prior state estimate (MAP vs. ML), and with and without process noise. For the nonlinear case, we show that a single pass of an extended Kalman smoother-filter over the data corresponds to a Gauss-Newton step of the corresponding nonlinear least-squares problem. Even the iterated extended Kalman filter can be viewed within this framework. For the linear case, we develop a compact least squares solution that can incorporate process noise and the prior state when available. With these new views on the batch approach, one may reconsider its usage in tracking because it provides a robust framework for the solution of the aforementioned problems. Finally, we provide some examples comparing linear batch initiation with and without process noise to show the value of the new approach.