  Bayesian estimation is a vital tool in robotics as it allows systems to
update the belief of the robot state using incomplete information from noisy
sensors. To render the state estimation problem tractable, many systems assume
that the motion and measurement noise, as well as the state distribution, are
all unimodal and Gaussian. However, there are numerous scenarios and systems
that do not comply with these assumptions. Existing non-parametric filters that
are used to model multimodal distributions have drawbacks that limit their
ability to represent a diverse set of distributions. In this paper, we
introduce a novel approach to nonparametric Bayesian filtering to cope with
multimodal distributions using harmonic exponential distributions. This
approach leverages two key insights of harmonic exponential distributions: a)
the product of two distributions can be expressed as the element-wise addition
of their log-likelihood Fourier coefficients, and b) the convolution of two
distributions can be efficiently computed as the tensor product of their
Fourier coefficients. These observations enable the development of an efficient
and exact solution to the Bayes filter up to the band limit of a Fourier
transform. We demonstrate our filter's superior performance compared with
established nonparametric filtering methods across a range of simulated and
real-world localization tasks.
