Multivariate Angular Filtering Using Fourier Series
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Filtering for multiple, possibly dependent angular variates on higher-dimensional manifolds such as the hypertorus is challenging as solutions from the circular case cannot easily be extended. In this paper, we present an approach to recursive multivariate angular estimation based on Fourier series. Since only truncated Fourier series can be used in practice, implications of the approximation errors need to be addressed. While approximating the density directly can lead to negative function values in the approximation, this problem can be solved by approximating the square root of the density. As this comes at the cost of additional complexity in the algorithm, we present both a filter based on approximating the density and a filter based on approximating its square root and closely regard the trade-offs. While the computational effort required for the filters grows exponentially with increasing number of dimensions, our approach is more accurate than a sampling importance resampling particle filter when comparing configurations of equal run time.
