Continuous-Discrete Filtering and Smoothing on Submanifolds of Euclidean Space

TL;DR

This paper addresses filtering and smoothing for state variables evolving on submanifolds of Euclidean space, introducing formal expressions and projection methods, including von Mises-Fisher based filters and smoothers.

Contribution

It develops a theoretical framework for filtering and smoothing on submanifolds, extending classical results and proposing projection-based algorithms for practical implementation.

Findings

Derived formal prediction and smoothing equations for submanifold states.

Showed that projection filters on submanifolds can match Euclidean case equations.

Implemented von Mises-Fisher based filters and smoothers for directional data.

Abstract

In this paper the issue of filtering and smoothing in continuous discrete time is studied when the state variable evolves in some submanifold of Euclidean space, which may not have the usual Lebesgue measure. Formal expressions for prediction and smoothing problems are derived, which agree with the classical results except that the formal adjoint of the generator is different in general. For approximate filtering and smoothing the projection approach is taken, where it turns out that the prediction and smoothing equations are the same as in the case when the state variable evolves in Euclidean space. The approach is used to develop projection filters and smoothers based on the von Mises-Fisher distribution.