Nonlinear von Mises–Fisher Filtering Based on Isotropic Deterministic Sampling

Abstract

We present a novel deterministic sampling approach for von Mises-Fisher distributions of arbitrary dimensions. Following the idea of the unscented transform, samples of configurable size are drawn isotropically on the hypersphere while preserving the mean resultant vector of the underlying distribution. Based on these samples, a von Mises-Fisher filter is proposed for nonlinear estimation of hyperspherical states. Compared with existing von Mises-Fisher-based filtering schemes, the proposed filter exhibits superior hyperspherical tracking performance.

Progressive von Mises–Fisher Filtering Using Isotropic Sample Sets for Nonlinear Hyperspherical Estimation

“…In practice, the Newton’s method specified above with the proposed initialization results in convergence below the error threshold within five steps, which is faster than our implementation in [ 37 ] by two orders of magnitude, thereby guaranteeing efficient sampling performance for online estimation. We now consider the following example to illustrate the efficacy of the proposed isotropic sampling scheme on von Mises–Fisher distributions of various configurations.…”

“…The proposed sampling method yields isotropic deterministic sample sets of arbitrary sizes that represent the underlying uncertainty more comprehensively for the unscented transform. As shown in our preceding work [ 37 ], the current unscented von Mises–Fisher filtering scheme is thus considerably enhanced for nonlinear estimation. However, for nonlinear and non-identity measurement models, the current paradigm simply reweights prior samples based on the likelihoods for the moment-matching of the posterior estimates.…”

“…Instead of deploying a universal numerical solver (e.g., the function in Matlab) to solve Equation ( 5 ) as in our preceding work [ 37 ], we now provide a tailored Newton’s method with iterative steps of a closed form. For that, the first derivative of the Dirichlet kernel is provided as follows: …”

“…However, for nonlinear and non-identity measurement models, the current paradigm simply reweights prior samples based on the likelihoods for the moment-matching of the posterior estimates. Although superior efficiency was shown over the approach using random samples, large sizes of deterministic samples are still desirable under strong nonlinearity [ 37 ] or with peaky likelihoods to avoid degeneration. To alleviate this issue, we propose the progressive unscented von Mises–Fisher filter using isotropic sample sets.…”

Cooperative Parameter Tracking on the Unit Sphere Using Distributed Adapt-Then-Combine Particle Filters and Parallel Transport

“…[16] and [17]. Furthermore, to account for the fact that, unlike in [9,[12][13][14][15], the state vector in [6] (as in [18]) was constrained to S L−1 , we used a Von Mises-Fisher (vMF) distribution [19] to model the dynamic evolution of the state on the manifold.…”