Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces

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. To derive the proposed representation, Varadhan's formula is exploited to approximate the non-linear Eikonal partial differential equation (PDE) of the Euclidean distance field~(EDF) by the logarithm of the screen Poisson equation that is linear. We show that members of the Matern covariance family directly satisfy this linear PDE. Thus, our key contribution is the realisation that the regularised Eikonal equation can be simply solved by applying the logarithmic function to a GPIS formulation to recover the accurate EDF and, at the same time, the implicit surface. 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.