The Harmonic Exponential Filter for Nonparametric Estimation on Motion Groups

TL;DR

The paper introduces the Harmonic Exponential Filter (HEF), a nonparametric Bayesian filtering framework for motion groups (notably SE(2)) that leverages harmonic exponential distributions and Fourier analysis to handle multimodal beliefs. By representing densities in log-density spectral space, HEF turns products into additive operations and convolutions into tensor products in the Fourier domain, yielding an asymptotically exact filter up to the Fourier band limit. The method offers significant efficiency advantages over histogram-based approaches and outperforms classic filters (EKF, histogram, PF) in multimodal localization tasks across simulated and real-world datasets. The work demonstrates strong expressivity for complex distributions and discusses future directions toward differentiable, end-to-end trainable filtering and SLAM integration, with practical implications for robust robotics localization and state estimation.

Abstract

Bayesian estimation is a vital tool in robotics as it allows systems to update the robot state belief 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 unimodal and Gaussian. However, there are numerous scenarios and systems that do not comply with these assumptions. Existing nonparametric filters that are used to model multimodal distributions have drawbacks that limit their ability to represent a diverse set of distributions. This paper introduces a novel approach to nonparametric Bayesian filtering on motion groups, designed to handle 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 asymptotically exact solution to the Bayes filter up to the band limit of a Fourier transform. We demonstrate our filter's performance compared with established nonparametric filtering methods across simulated and real-world localization tasks.

Figures (7)

• Figure 1: The product (a) and convolution (b) operations between two distributions play an important role in recursive Bayesian filtering. The examples here are shown with harmonic exponential distributions on the circle $S^1$.
• Figure 2: The relationship between probability density $p(g)$, log-density $\text{ln}(p(g))$, and their spectral coefficients $\eta$ and $M$. We model the log-density $\text{ln}(p(g))$ using $\eta$. To recover the $p(g)$, we compute the inverse FFT of $\eta$ and exponentiate the result. The spectral coefficients $M$ are obtained by taking the FFT of $p(g)$ and are used to compute convolutions.
• Figure 3: Analysis of the harmonic exponential distribution and histogram distributions in modeling a mixture of two von Mises distributions on $S^1$. (a) depicts how each method reproduces the underlying mixture. (b) shows how $D_{KL}$ evolves for each method as the number of parameters used increases, and shows the effect of the concentration parameter $\kappa$ on the fidelity of each method, as measured by $D_{KL}$.
• Figure 4: Ability of the HEF and other filtering approaches to represent the banana distribution. (a) shows the prior and motion model, which are modeled with a rectangular and Gaussian distribution, respectively. Both EKF and IEKF use a Gaussian prior. (b) presents the posterior distribution of the four approaches after five prediction steps. The HEF, PF, and IEKF are able to capture the banana distribution.
• Figure 5: Sample results from the simulated environment. (a) shows the posterior belief for the baselines and the HEF. (b) depicts the steps of the HEF: the predicted belief (top) is $\overline{bel}_t$ from \ref{['eq:predict']}, the range-only measurement likelihood (middle) is $p_{z_t \vert x_t}$, and the posterior belief (bottom) is $bel_t$ from \ref{['eq:udpate']}.