Filtering from observations on Stiefel manifolds

• We propose a general algorithm to estimate angular velocity for incomplete datasets.
• A particle filter implementation is given with validation on real data.
• The proposed approach is based on a global linearization of a manifold-valued signal named the anti-development signal.
• A interpolation scheme (theorem) is provided for data sampled on the rotation group.

This paper addresses the problem of filtering processes on the rotation group SO(n)SO(n) when partial observations, i.e. processes on the Stiefel manifold, are only available. In particular, we consider the problem of estimating the angular velocity of an object when only partial observation of its orientation is available. Interpolation schemes are proposed to overcome the discrete nature of observations and a practical numerical solution (particle filtering) is developed for filtering on the Stiefel manifold. The proposed solution to the filtering problem is based on the anti-development signal concept, which allows to overcome the non-additivity nature of the noise present in the observed signal. The solutions proposed are general and can be transposed to many manifolds. However, the presented examples and simulations consider the case of partially observed signals taking values on the special orthogonal group when one or more components are missing. In this particular case, the observed process takes its values in the Stiefel manifold. A comparison of the proposed particle filter with the extended Kalman filter is presented. Results on synthetic and real (sport science) data show the behavior of the proposed algorithm.