Bayesian filtering for orientational distributions: A fourier approach

A Bayesian filter for rotation groups in 2D and 3D is derived. The prior, propagator, and measurement probability densities are all assumed to be bandlimited functions on SO(2) or SO(3), expressed as a Fourier series on these compact Lie groups. The posterior, which has a higher bandlimit, is computed and then low-pass filtered, resulting in a bandlimited approximation. The benefits and drawbacks of the Fourier approach presented here are examined in contrast to the Gaussian approach designed for small error covariances. While the Gaussian approach is much faster, it breaks down for large error covariances. The point where the Gaussian approach breaks down is analyzed with the Fourier method, indicating the range of error sizes where the switch to Fourier methods is required.