  Robots reason about the environment through dedicated representations.
Popular choices for dense representations exploit Truncated Signed Distance
Functions (TSDF) and Octree data structures. However, TSDF is a projective
signed distance obtained directly from depth measurements that overestimates
the Euclidean distance. Octrees, despite being memory efficient, require tree
traversal and can lead to increased runtime in large scenarios. Other
representations based on Gaussian Process (GP) distance fields are appealing
due to their probabilistic and continuous nature, but the computational
complexity is a concern. In this paper, we present an online efficient mapping
framework that seamlessly couples GP distance fields and the fast-access VDB
data structure. This framework incrementally builds the Euclidean distance
field and fuses other surface properties, like intensity or colour, into a
global scene representation that can cater for large-scale scenarios. The key
aspect is a latent Local GP Signed Distance Field (L-GPDF) contained in a local
VDB structure that allows fast queries of the Euclidean distance, surface
properties and their uncertainties for arbitrary points in the field of view.
Probabilistic fusion is then performed by merging the inferred values of these
points into a global VDB structure that is efficiently maintained over time.
After fusion, the surface mesh is recovered, and a global GP Signed Distance
Field (G-GPDF) is generated and made available for downstream applications to
query accurate distance and gradients. A comparison with the state-of-the-art
frameworks shows superior efficiency and accuracy of the inferred distance
field and comparable reconstruction performance. The accompanying code will be
publicly available. https://github.com/UTS-RI/VDB_GPDF
