Geometry-driven Deterministic Sampling for Nonlinear Bingham Filtering

2020 54th Annual Conference on Information Sciences and Systems (CISS)

2020

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We want to approximate general multivariate probability density functions by deterministic sample sets. For optimal sampling, the closeness to the given continuous density has to be assessed. This is a difficult challenge in multivariate settings. Simple solutions are restricted to the one-dimensional case. In this paper, we propose to employ one-dimensional density projections. These are the Radon transforms of the densities. For every projection, we compute their cumulative distribution function. These Projected Cumulative Distributions (PCDs) are compared for all possible projections (or a discrete set thereof). This leads to a tractable distance measure in multivariate space. The proposed approximation method is efficient as calculating the distance measure mainly entails sorting in one dimension. It is also surprisingly simple to implement.

Section: Reduction To Univariate Casementioning

confidence: 99%

Deterministic Sampling of Multivariate Densities based on Projected Cumulative Distributions

2020 54th Annual Conference on Information Sciences and Systems (CISS)

2020

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“…In [ 23 ], deterministic samples were drawn from typical circular distributions via optimal quadratic quantification based on the Voronoi cells. For unit hyperspheres, major efforts have been dedicated to the Bingham distribution, where the basic UT-based sampling scheme in [ 24 ] (

samples as for

) was extended for arbitrary sample sizes, first in the principal directions [ 25 ] and then for the entire hyperspherical domain [ 26 ]. The sampling paradigm was DMA-based, with an on-manifold optimizer minimizing the statistical divergence of the samples to the underlying distribution under the moment constraints up to the second order.…”

Section: Introductionmentioning

confidence: 99%

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

Pfaff

2021

Sensors

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In this work, we present a novel scheme for nonlinear hyperspherical estimation using the von Mises–Fisher distribution. Deterministic sample sets with an isotropic layout are exploited for the efficient and informative representation of the underlying distribution in a geometrically adaptive manner. The proposed deterministic sampling approach allows manually configurable sample sizes, considerably enhancing the filtering performance under strong nonlinearity. Furthermore, the progressive paradigm is applied to the fusing of measurements of non-identity models in conjunction with the isotropic sample sets. We evaluate the proposed filtering scheme in a nonlinear spherical tracking scenario based on simulations. Numerical results show the evidently superior performance of the proposed scheme over state-of-the-art von Mises–Fisher filters and the particle filter.

“…These methods are all based on moment matching for fitting prior and posterior densities to given propagated or reweighted samples during the prediction and update steps, respectively. Samples are drawn from corresponding continuous densities that preserve moments up to a certain order [13], [14].…”

Section: Introductionmentioning

confidence: 99%

“…Within the scope of deterministic sampling for hyperspherical continuous distributions, approaches generating configurable numbers of samples were only proposed for the Bingham distribution. In [13], [14], spherical geometry was exploited to establish a tangent space around the mode. There, samples are drawn via a sampling scheme originally proposed for multivariate Gaussian distributions in [12].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear von Mises–Fisher Filtering Based on Isotropic Deterministic Sampling

Pfaff

2020 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI)

2020

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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.