G-EDF-Loc | Continuous Gaussian Distance Field for Robust 6DoF Localization

Snail Dataset — Urban detail view 1

Abstract

G-EDF-Loc is a robust 6-DoF localization framework built on direct, CPU-based scan-to-map registration. It leverages G-EDF, a continuous, memory-efficient 3D distance field that models the Euclidean Distance Field (EDF) with a Block-Sparse Gaussian Mixture Model, guaranteeing global C¹ continuity across block transitions. Its analytical, Eikonal-consistent gradients enable high-fidelity reconstruction and real-time localization. On large-scale datasets, G-EDF-Loc performs competitively against state-of-the-art methods and remains exceptionally robust even under severe odometry degradation or without IMU priors.

Method

Distance Field

Gaussian Mixture Approximation

The distance field is represented as a weighted sum of K anisotropic Gaussian kernels. Each kernel has:

w_k — amplitude weight (can be negative)
μ_k — center position in 3D
Σ_k — diagonal covariance matrix

Negative weights carve sharp valleys near surfaces, enabling precise zero-crossings.

Block-Sparse Architecture

Space is partitioned into 1m³ cubes. Each trains an independent GMM, enabling parallel processing at scale.

Adaptive Complexity

Gaussian count scales with geometric complexity. Simple regions use fewer kernels; complex areas get more.

C¹ Blending

Adjacent blocks share overlap margins blended via Smoothstep: α(t) = 3t² − 2t³, ensuring gradient continuity.

Eikonal Gradient

Closed-form ∇d̂(x) satisfies ‖∇d̂‖ ≈ 1 throughout the volume for stable optimization.

Results

< 3cm Mean Absolute Error

≈ 1.0 Gradient Magnitude

EDF Gradient

Comparison on New College dataset (z = 3.0m) with 1.0m³ blocks and δ = 0.25m overlap.

Gradient without blending — Without Blending

Quantitative results on reconstruction fidelity. Distance errors are in meters (↓ lower is better). Gradient magnitude (‖∇d̂‖) indicates Eikonal consistency (≈ 1.0 is ideal).
Dataset	Distance Error (m)			Gradient ‖∇d̂‖
Dataset	MAE ↓	Med. ↓	Std ↓	Mean	Std ↓
New College	0.033	0.018	0.044	0.984	0.089
Snail	0.035	0.018	0.049	0.979	0.108

Localization

G-EDF-Loc is a direct, CPU-based 6-DoF scan-to-map registration pipeline built on top of the G-EDF representation. An Error-State Kalman Filter (ESKF) predicts the pose, deskews incoming LiDAR scans, and provides the initial guess for a non-linear optimization that aligns the scan directly against the continuous distance field, with no feature extraction or point-to-point matching required.

ESKF Prior & Deskewing

A 15-DoF Error-State Kalman Filter fuses IMU data to predict pose and deskew LiDAR sweeps, though the system remains fully operational without any IMU input.

Direct Gradient-Based Alignment

Evaluates the continuous distance field d̂(x) and its analytical gradient directly at each scan point, avoiding nearest-neighbor search and feature matching entirely.

Coarse-to-Fine Optimization

A wide-to-narrow Cauchy robust loss schedule expands the basin of attraction for large initial errors, then sharpens to reject outliers for high-precision convergence.

Dynamic Outlier Masking

Points outside the mapped volume are softly ignored (zero gradient) rather than discarded, re-entering the optimization once back in range.

Robustness Study

G-EDF-Loc is benchmarked against Fast-GICP and NDT (via the hdl_localization package) on the Newer College and Snail datasets, using an identical tracking pipeline for all methods. Four conditions are evaluated per trajectory: nominal IMU-aided tracking (Inertial), complete absence of IMU (No IMU), and two levels of injected odometry noise (Low / High).

bc (Snail) park (Newer College)

Estimated trajectories (red) against ground truth (black), Fast-GICP (blue), and NDT (green) across the four evaluated conditions.

park trajectory - High Noise — High Noise

Select a trajectory and a condition to compare Position (m) and Rotation (°) RMSE and average per-scan processing time (ms) across methods. "–" indicates divergence.

bc sl st ss quad-medium quad-hard park

Inertial No IMU Low Noise High Noise

bc — Inertial. Best results in **bold**, second-best underlined.
Method	Pos ↓ (m)	Rot ↓ (°)	Time ↓ (ms)

Under nominal conditions, all three methods achieve State-of-the-Art accuracy at the centimeter level. The gap widens dramatically under degraded odometry: Fast-GICP diverges on several trajectories without IMU (e.g. park, sl, st, ss) and exhibits extreme processing-time spikes (up to 1.37 s/scan on bc – No IMU), while G-EDF-Loc maintains bounded, real-time performance and stable convergence across every tested condition, including the complete absence of IMU priors.

Citation

If you find our work useful in your research, please consider citing:

BibTeX

@inproceedings{maese2026gedfloc,
    author={Maese, José E. and Coto-Elena, Lucía and Merino, Luis and Caballero, Fernando},
    title={{G-EDF-Loc: 3D Continuous Gaussian Distance Field for Robust Gradient-Based 6DoF Localization}},
    booktitle={2026 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)},
    year={2026}
}