The Smart Sampling Kalman Filter with Symmetric Samples

Nonlinear Kalman Filters are powerful and widely-used
techniques when trying to estimate the hidden state of a stochastic
nonlinear dynamic system. In this paper, we extend the Smart Sampling
Kalman Filter (S2KF) with a new point symmetric Gaussian sampling
scheme. This not only improves the S2KF‘s estimation quality, but also
reduces the time needed to compute the required optimal Gaussian samples
drastically. Moreover, we improve the numerical stability of the sample
computation, which allows us to accurately approximate a
thousand-dimensional Gaussian distribution using tens of thousands of
optimally placed samples. We evaluate the new symmetric S2KF by
computing higher-order moments of standard normal distributions and
investigate the estimation quality of the S2KF when dealing with
symmetric measurement equations. Finally, extended object tracking based
on many measurements per time step is considered. This high-dimensional
estimation problem shows the advantage of the S2KF being able to use an
arbitrary number of samples independent of the state dimension, in
contrast to other state-of-the-art sample-based Kalman Filters.