Further improvement of the adaptive level of detail transform: Splitting in direction of the nonlinearity

In earlier work, we have presented a novel approach to nonlinear, non-Gaussian tracking problems. The approach was based on keeping a bank of unscented Kalman filters, which were split and merged in order to adapt the level of detail of the filtering density according to the nonlinearity of the tracking problem. More recently, that approach has been refined and generalized to a general method for nonlinear transformations of Gaussian mixture random variables. Here, we further extend it by the following aspects: we consider splitting a Gaussian distribution into three components rather than two; and we show how splitting can be performed in direction of the nonlinearity, which in simulations gave a 25% reduction of the mean squared error, compared to the previous implementation of the split and merge unscented Gaussian mixture filter. In addition to that, we show how splitting can be implemented efficiently, through Cholesky downdates.