# Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces

arXiv: https://arxiv.org/abs/2010.11487 · HTML: https://mlsanity.com/paper/2010.11487/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2020.

## Built on by (most cited first, 7 of 7)

- [Uni-Fusion: Universal Continuous Mapping](https://mlsanity.com/paper/2303.12678/) (2023)
- [Distance and Collision Probability Estimation from Gaussian Surface Models](https://mlsanity.com/paper/2402.00186/) (2024)
- [Continuous-Time Gaussian Process Motion-Compensation for Event-vision Pattern Tracking with Distance Fields](https://mlsanity.com/paper/2303.02672/) (2023)
- [OREN: Octree Residual Network for Real-Time Euclidean Signed Distance Mapping](https://mlsanity.com/paper/2510.18999/) (2025)
- [From Distances to Trajectories: Real-Time Signed Distance Function Mapping and Distance-Accelerated Motion Planning for UAVs](https://mlsanity.com/paper/2607.19306/) (2026)
- [Informative Planning for Worst-Case Error Minimisation in Sparse Gaussian Process Regression](https://mlsanity.com/paper/2203.03828/) (2022)
- [Learning Scene-Level Signed Directional Distance Function with Ellipsoidal Priors and Neural Residuals](https://mlsanity.com/paper/2503.20066/) (2025)

## Abstract (arXiv)

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.