# Unscented Kalman Filtering on Riemannian Manifolds

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## Abstract

In recent years there has been a growing interest in problems, where either the observed data or hidden state variables are confined to a known Riemannian manifold. In sequential data analysis this interest has also been growing, but rather crude algorithms have been applied: either Monte Carlo filters or brute-force discretisations. These approaches scale poorly and clearly show a missing gap: no generic analogues to Kalman filters are currently available in non-Euclidean domains. In this paper, we remedy this issue by first generalising the unscented transform and then the unscented Kalman filter to Riemannian manifolds. As the Kalman filter can be viewed as an optimisation algorithm akin to the Gauss-Newton method, our algorithm also provides a general-purpose optimisation framework on manifolds. We illustrate the suggested method on synthetic data to study robustness and convergence, on a region tracking problem using covariance features, an articulated tracking problem, a mean value optimisation and a pose optimisation problem.

## Keywords

Riemannian manifolds Unscented Kalman filter Filtering theory Optimisation on manifolds## Notes

### Acknowledgements

The authors would like to thank the anonymous reviewers for detailed comments, which substantially improved the quality of the manuscript. Furthermore, Søren Hauberg would like to thank the Villum Foundation for financial support.

## Supplementary material

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