Hyperspherical Deterministic Sampling Based on Riemannian Geometry for Improved Nonlinear Bingham Filtering

Abstract

We present a novel geometry-driven scheme for generating equally weighted deterministic samples of Bingham distributions in arbitrary dimensions. Unlike existing approaches, our method provides flexibility in the sampling size with samples satisfying requirements of the unscented transform while approximating higher-order moments of the Bingham distribution. This is done by first using Dirac mixture approximation as a sampling scheme on the tangent plane at the mode with respect to the Bingham density via gnomonic projection. Subsequently, the tangent sigma points are retracted backwards to the hypersphere, after which an on-manifold moment correction is performed via Riemannian optimization. The proposed approach is further applied to quaternion Bingham filtering for recursive orientation estimations. Evaluation results show that the geometry-adaptive sampling scheme gives better tracking accuracy and robustness for nonlinear orientation estimations.