  Most robot mapping techniques for lidar sensors tessellate the environment
into pixels or voxels and assume uniformity of the environment within them.
Although intuitive, this representation entails disadvantages: The resulting
grid maps exhibit aliasing effects and are not differentiable. In the present
paper, we address these drawbacks by introducing a novel mapping technique that
does neither rely on tessellation nor on the assumption of piecewise uniformity
of the space, without increasing memory requirements. Instead of representing
the map in the position domain, we store the map parameters in the discrete
frequency domain and leverage the continuous extension of the inverse discrete
cosine transform to convert them to a continuously differentiable scalar field
in the position domain, which we call DCT map. A DCT map assigns to each point
in space a lidar decay rate, which models the local permeability of the space
for laser rays. In this way, the map can describe objects of different laser
permeabilities, from completely opaque to completely transparent. DCT maps
represent lidar measurements significantly more accurate than grid maps,
Gaussian process occupancy maps, and Hilbert maps, all with the same memory
requirements, as demonstrated in our real-world experiments.
