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Progressive von Mises–Fisher Filtering Using Isotropic Sample Sets for Nonlinear Hyperspherical Estimation^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. General Conventions of Notations

#### 2.2. The von Mises–Fisher Distribution

#### 2.3. Geometric Structure of Hyperspherical Manifolds

## 3. Isotropic Deterministic Sampling

Algorithm 1: Isotropic Deterministic Sampling |

#### 3.1. Numerical Solution for Equation (5)

`solve`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:

#### 3.2. Example

## 4. Progressive Unscented von Mises–Fisher Filtering

#### 4.1. Task Formulation

#### 4.2. Prediction Step for Nonlinear von Mises–Fisher Filtering

#### 4.3. Deterministic Progressive Update Using Isotropic Sample Sets

Algorithm 2: Isotropic Progressive Update |

## 5. Evaluation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Illustration of isotropic deterministic sampling with $(\lambda ,\tau )=(3,10)$ for a von Mises–Fisher distribution ($\kappa =4$) on ${\mathbb{S}}^{2}$. (

**A**) Equal partitioning in ${\mathbb{T}}_{\underline{\nu}}{\mathbb{S}}^{2}$ with regard to its local basis. (

**B**) Scaling with the UT-preserving interval in ${\mathbb{T}}_{\underline{\nu}}{\mathbb{S}}^{2}$. (

**C**) Exponential map from ${\mathbb{T}}_{\underline{\nu}}{\mathbb{S}}^{2}$ to ${\mathbb{S}}^{2}$ for placing planet samples on hyperspherical orbits.

**Figure 2.**Illustration of the proposed isotropic deterministic sampling schemes with von Mises–Fisher distributions on ${\mathbb{S}}^{2}$ of different parameterizations in Section 3.2. Samples (red dots) are uniformly weighted and dotted with sizes proportional to weights.

**Figure 3.**Illustration of the deterministic progressive update using isotropic sample sets. Sizes of red dots are proportional to their weights. The same isotropic sampling configuration, $(\lambda ,\tau )=(2,10)$, is deployed for both the single-step and the progressive updates.

**Figure 4.**Error over sample numbers (log scale) for the evaluated filters. The configurations with five samples for UvMFF and Prog-UvMFF are based on the original UT-based sampling method in [15].

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**MDPI and ACS Style**

Li, K.; Pfaff, F.; Hanebeck, U.D.
Progressive von Mises–Fisher Filtering Using Isotropic Sample Sets for Nonlinear Hyperspherical Estimation. *Sensors* **2021**, *21*, 2991.
https://doi.org/10.3390/s21092991

**AMA Style**

Li K, Pfaff F, Hanebeck UD.
Progressive von Mises–Fisher Filtering Using Isotropic Sample Sets for Nonlinear Hyperspherical Estimation. *Sensors*. 2021; 21(9):2991.
https://doi.org/10.3390/s21092991

**Chicago/Turabian Style**

Li, Kailai, Florian Pfaff, and Uwe D. Hanebeck.
2021. "Progressive von Mises–Fisher Filtering Using Isotropic Sample Sets for Nonlinear Hyperspherical Estimation" *Sensors* 21, no. 9: 2991.
https://doi.org/10.3390/s21092991