  This paper introduces a novel method to estimate distance fields from noisy
point clouds using Gaussian Process (GP) regression. Distance fields, or
distance functions, gained popularity for applications like point cloud
registration, odometry, SLAM, path planning, shape reconstruction, etc. A
distance field provides a continuous representation of the scene defined as the
shortest distance from any query point and the closest surface. The key concept
of the proposed method is the transformation of a GP-inferred latent scalar
field into an accurate distance field by using a reverting function related to
the kernel inverse. The latent field can be interpreted as a smooth occupancy
map. This paper provides the theoretical derivation of the proposed method as
well as a novel uncertainty proxy for the distance estimates. The improved
performance compared with existing distance fields is demonstrated with
simulated experiments. The level of accuracy of the proposed approach enables
novel applications that rely on precise distance estimation: this work presents
echolocation and mapping frameworks for ultrasonic-guided wave sensing in
metallic structures. These methods leverage the proposed distance field with a
physics-based measurement model accounting for the propagation of the
ultrasonic waves in the material. Real-world experiments are conducted to
demonstrate the soundness of these frameworks.
