Kernel-SDF: An Open-Source Library for Real-Time Signed Distance Function Estimation using Kernel Regression

Zhirui Dai^1,*, Tianxing Fan^1,*, Mani Amani¹, Jaemin Seo²,

Ki Myung Brian Lee¹, Hyondong Oh², Nikolay Atanasov¹

¹Existential Robotics Lab, University of California San Diego

² Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST)

^*Equal contribution

IEEE Robotics and Automation Letters (RA-L), 2026.

Paper Appendix arXiv BibTeX Code

An example of 3D SDF mapping using the proposed Kernel-SDF method. Left: Visualization of the learned SDF as a heatmap, where warmer colors indicate higher SDF values (free space) and cooler colors indicate lower SDF values (occupied space), as well as the voxels (colored by height) that indicate the locations where local BHMs and GPs are instantiated. Right: The online mesh reconstructed in real-time from the surface estimation front-end of Kernel-SDF.

Abstract: Accurate and efficient scene representation is crucial for robotic tasks such as motion planning, manipulation, and navigation. Signed distance functions (SDFs) have emerged as a powerful representation for encoding distance to obstacle boundaries, enabling efficient collision-checking and trajectory optimization. However, existing methods are limited for large-scale uncertainty-aware SDF estimation from streaming sensor data: voxel-based approaches have fixed resolution and lack uncertainty quantification, neural network methods require significant training time, and Gaussian process (GP) methods struggle with scalability, sign estimation, and uncertainty calibration. In this letter, we develop an open-source library, Kernel-SDF, using kernel regression to learn SDF with calibrated uncertainty in real-time. It combines a front-end learning a continuous occupancy field via kernel regression with a back-end that estimates accurate SDF via GP regression using samples from the front-end surface boundaries. Kernel-SDF provides accurate SDF, gradient, uncertainty, and mesh construction in real-time. Evaluations show it achieves superior accuracy over existing methods while maintaining real-time performance, making it suitable for robotics tasks requiring reliable uncertainty-aware geometry.

Methodology

Overview

(a) Overview of Kernel-SDF: the front-end estimates surface points and normals from point clouds via BHM and marching cubes; these train the back-end, which learns SDF and its gradient using multiple GPs in an octree. (b) For each ray hitting or crossing the sampling region, we generate occupied and free samples to update the BHM. (c) Weight sync between neighboring BHMs: managed weights are synced to unmanaged weights. (d) Surface points colored by BHM log-odds; the ground has lower log-odds than the walls.

SDF Prediction Back-End

The SDF prediction back-end uses multiple GPs to learn the SDF from the surface points estimated by the front-end. Each GP is associated with a voxel in a hierarchical tree that partitions the space.

Given a set of surface points $\mathcal{D}_\text{surf}=\{ \mathbf{x}_i, \sigma^2_i \}_{i=1}^{N}$, where $\sigma^2_i$ is the uncertainty of the surface point $\mathbf{x}_i$ estimated by the front-end, we can train the corresponding GP for predicting the shortest distance $|d(\mathbf{x})|$ by regressing a surrogate function $f$, which should be monotonically decreasing with respect to the distance $|d(\mathbf{x})|$ to the nearest surface point, such that $f\rightarrow 0$ as $|d(\mathbf{x})| \rightarrow \infty$ and $f \rightarrow 1$ as $|d(\mathbf{x})|\rightarrow 0$, which is consistent with zero-mean prior GP regression that as the query point goes far away from the training set, the posterior mean goes to zero. With the trained GP, we can predict the unsigned distance $|d(\mathbf{x}_\ast)|=r(\hat{f})$, where $r(\cdot)$ is the inverse function of the surrogate function $f(\cdot)$. The sign of the SDF can be determined by querying the occupancy probability from the front-end BHM at the query point $\mathbf{x}_\ast$.

Surface Estimation Front-End

The role of the surface estimation front-end is to provide reliable estimates of on-surface points, their uncertainty, and optionally the corresponding normals. We use the Bayesian Hilbert map (BHM) for this purpose, although other front-ends may also be used.

Given a dataset $\mathcal{D}_\text{BHM}=\{ \mathbf{x}_k, y_k \}_{k=1}^{K}$ is generated from the sensor measurement $\mathcal{S}_t=(\mathbf{R}_t, \mathbf{o}_t, \mathcal{P}_t)$ by sampling occupied samples as $(\mathbf{R}_t\mathbf{p}_i+\mathbf{o}_t, 1)$ and free-space samples as $(\lambda \mathbf{R}_t \mathbf{p}_i + \mathbf{o}_t, 0)$ for $\lambda \sim \mathcal{U}(0, 1)$ and $\forall \mathbf{p}_i \in \mathcal{P}_t$. We can update the BHM with an Expectation-Maximization algorithm, which enables incremental updates of the BHM. Hence, the BHMs can filter out sensor noise by combining multiple noisy measurements over time and adapt to dynamic changes in the environment.

High-quality Mesh Reconstruction

Replica Dataset with Simulated Depth Sensor Noise

Ground Truth

Ours

FIESTA

Voxblox

iSDF

VDB-GPDF

The Cow in Cow and Lady Dataset

Ground Truth

Ours

FIESTA

Voxblox

iSDF

VDB-GPDF

Newer College Dataset

Ground Truth

Ours

FIESTA

Voxblox

iSDF

VDB-GPDF

Robust to Sensor Noise

Calibrated Uncertainty Quantification

SDF Error vs. Estimated Variance

(a) Predicted SDF

(b) SDF Error

A key advantage of Kernel-SDF is its calibrated SDF uncertainty. Qualitatively, on a slice of Replica Office3, regions of higher SDF error coincide with regions of higher estimated variance: (a) predicted SDF, (b) absolute SDF error against ground truth, and (c) estimated SDF variance. This shows that Kernel-SDF effectively captures uncertainty.

Uncertainty Calibration

Replica room0

Cow & Lady

Newer College

Reliability curves of the predicted SDF uncertainty: for each target confidence level (x-axis), the fraction of query points whose error falls within it (y-axis). Empirical coverage closely tracks the ideal diagonal (ECE $0.052$, $0.031$, $0.011$), confirming calibration. Across Replica room0, Cow & Lady, and Newer College, the normalized error $z = (u - \hat{u}) / \hat{\sigma}$ satisfies $\mathbb{E}[z^2] = 1.05, 1.03, 1.04$ (within $5.5\%$ of $1$), showing the predicted variance closely matches the true error variance.

BibTeX

@article{dai2026kernelsdf,
    title={Kernel-SDF: An Open-Source Library for Real-Time Signed Distance Function Estimation using Kernel Regression},
    author={Dai, Zhirui and Fan, Tianxing and Amani, Mani and Seo, Jaemin and Lee, Ki Myung Brian and Oh, Hyondong and Atanasov, Nikolay},
    journal={IEEE Robotics and Automation Letters},
    year={2026},
    publisher={IEEE},
}

Acknowledgement

This work was supported by ARL DCIST CRA W911NF-17-2-0181, NSF FRR CAREER 2045945, the Ministry of Trade, Industry, and Energy (MOTIE), Korea, under the Strategic Technology Development Program supervised by the Korea Institute for Advancement of Technology (KIAT) [Grant No. P0026052], and the Korea Institute of Planning and Evaluation for Technology in Food, Agriculture, Forestry (IPET) through the Smart Farm Innovation Technology Development Program, funded by the Ministry of Agriculture, Food and Rural Affairs (MAFRA) (RS-2025-02219411).