  In this letter, we introduce the Log-Gaussian Process Implicit Surface
(Log-GPIS), a novel continuous and probabilistic mapping representation
suitable for surface reconstruction and local navigation. Our key contribution
is the realisation that the regularised Eikonal equation can be simply solved
by applying the logarithmic transformation to a GPIS formulation to recover the
accurate Euclidean distance field (EDF) and, at the same time, the implicit
surface. To derive the proposed representation, Varadhan's formula is exploited
to approximate the non-linear Eikonal partial differential equation (PDE) of
the EDF by the logarithm of a linear PDE. We show that members of the Matern
covariance family directly satisfy this linear PDE. The proposed approach does
not require post-processing steps to recover the EDF. Moreover, unlike
sampling-based methods, Log-GPIS does not use sample points inside and outside
the surface as the derivative of the covariance allow direct estimation of the
surface normals and distance gradients. We benchmarked the proposed method on
simulated and real data against state-of-the-art mapping frameworks that also
aim at recovering both the surface and a distance field. Our experiments show
that Log-GPIS produces the most accurate results for the EDF and comparable
results for surface reconstruction and its computation time still allows online
operations.
