Progressive Bayesian Filtering with Coupled Gaussian and Dirac Mixtures

Nonlinear filtering is the most important aspect in state estimation with real-world systems. While the Kalman filter provides a simple though optimal estimate for linear systems, feasible filters for general systems are still subject of intensive research. The previously proposed Progressive Gaussian Filter PGF42 marked a new milestone, as it was able to efficiently compute an optimal Gaussian approximation of the posterior density in nonlinear systems [1]. However, for highly nonlinear systems where true posteriors are “banana-shaped” (e.g., cubic sensor problem) or multimodal (e.g., extended object tracking), even an optimal Gaussian approximation is an inadequate representation. Therefore, we generalize the established framework around the PGF42 from Gaussian to Gaussian mixture densities that are better able to approximate arbitrary density functions. Our filter simultaneously holds approximate Gaussian mixture and Dirac mixture representations of the same density, what we call coupled discrete and continuous densities (CoDiCo). For conversion between discrete and continuous representation, we employ deterministic sampling and the expectation-maximization (EM) algorithm, which we extend to deal with weighted particles.