Continuous Occupancy Map Fusion with Fast Bayesian Hilbert Maps

Mapping the occupancy of an environment is central for robot autonomy. Traditional occupancy grid maps discretise the environment into independent cells, neglecting important spatial correlations, and are unable to capture the continuous nature of the real world. With these drawbacks of grid maps in mind, Hilbert Maps (HM) and more recently Bayesian Hilbert Maps (BHMs), were introduced as a continuous representation of the environment. In this paper we propose a method to merge Bayesian Hilbert Maps built by a team of robots in a decentralised manner. The training of BHMs requires the inversion of a large covariance matrix, incurring cubic complexity. We introduce an approximation, Fast Bayesian Hilbert Maps (Fast-BHM), which reduces the time complexity to below quadratic. Such speed-ups allow the building and merging of Bayesian Hilbert Map models to be practical, opening the door for multi-robot Hilbert Map systems which can be much faster and more robust than an individual robot. By merging several individual Fast-BHMs in a decentralised manner we obtain a unified model of the environment which is itself a Fast-BHM. We conduct experiments to show that global Fast-BHM models do not deteriorate after repeated merging and training. We then empirically demonstrate, due to its the compact representation, fused Fast-BHMs outperform fusion methods involving discretising continuous representations, when the amount of information communicated is limited.