Multimodal Circular Filtering Using Fourier Series

Recursive filtering with multimodal likelihoods and transition densities on periodic manifolds is, despite the compact domain, still an open problem. We propose a novel filter for the circular case that performs well compared to other state-of-the-art filters adopted from linear domains. The filter uses a limited number of Fourier coefficients of the square root of the density. This representation is preserved throughout filter and prediction steps and allows obtaining a valid density at any point in time. Additionally, analytic formulae for calculating Fourier coefficients of the square root of some common circular densities are provided. In our evaluation, we show that this new filter performs well in both unimodal and multimodal scenarios while requiring only a reasonable number of coefficients.