Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces

TL;DR

This paper introduces Log-Gaussian Process Implicit Surface (Log-GPIS), a probabilistic method for accurate Euclidean distance field and surface reconstruction that enables real-time applications without post-processing.

Contribution

The authors develop a novel approach that solves the Eikonal equation via a logarithmic transformation of GPIS, directly estimating distance fields and surface normals without sampling or post-processing.

Findings

Log-GPIS accurately recovers Euclidean distance fields.

It achieves comparable surface reconstruction results to state-of-the-art methods.

The method supports online computation suitable for real-time applications.

Abstract

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.