Gaussian Mixture-Based Point Mass Filtering With Applications to Terrain-Relative Navigation

The accuracy of the point mass filter (PMF) relies on the precise placement of grid points. Since the approximated probability distributions are evaluated only at these points, suboptimal grid placement can result in an inaccurate representation of the posterior distribution. This work addresses this issue by introducing a variant of the PMF that represents the propagated grid points as a Gaussian mixture, enabling a Gaussian sum filter (GSF) update before grid construction. The GSF update improves the accuracy of the posterior mean and covariance estimates, leading to better grid placement. In addition, an extension is presented, using kernel density estimation techniques to improve filter performance in low process noise scenarios. A comparative analysis is conducted between the proposed approach, the standard PMF, and other PMF variants. Using a bivariate example, the proposed method shows a better approximation of the posterior distribution compared to the other filters. Furthermore, two sequential filtering problems are used to analyze the performance of the filter, the first involving the Ikeda map and the second focusing on terrain-relative navigation. The results show that the proposed method provides more accurate and consistent filtering compared to the other PMF variants considered.