Gaussian mixture filters in hybrid positioning

Bayesian filtering is a framework for the computation of an optimal state estimate fusing different types of measurements, both current and past. Positioning, especially in urban and indoor environments, is one example of an application where the powerful mathematical framework is needed to compute as a good position estimate as possible from all kinds of measurements and information. In this work, we consider the Gaussian mixture filter, which is an approximation of the Bayesian filter. Especially, we consider filtering with just a few components, which can be computed in real-time on a mobile device. We have developed and compared different Gaussian mixture filters in different scenarios. One filter uses static solutions, which are possibly ambiguous, another extends the Unscented transformation to the Gaussian mixture framework, and some filters are based on partitioning the state space. It is also possible to use restrictive, inequality constraints, efficiently in Gaussian mixture filters. We show that a new filter called the Efficient Gaussian mixture filter outperforms other known filters, such as Kalman-type filters or particle filter, in a positioning application. We also show that another new filter, the Box Gaussian mixture filter, converges weakly to the correct posterior. All in all we see that the Gaussian mixture is a competitive framework for real-time filtering implementations, especially in positioning applications.