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

Zhirui Dai; Tianxing Fan; Mani Amani; Jaemin Seo; Ki Myung Brian Lee; Hyondong Oh; Nikolay Atanasov

cs.ROMar 31, 2026

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

Authors: Zhirui Dai, Tianxing Fan, Mani Amani, Jaemin Seo, Ki Myung Brian Lee, Hyondong Oh, Nikolay Atanasov

Organizations: Department of Electrical and Computer Engineering, University of California San Diego, La Jolla, CA 92093, USA. · Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34051, Republic of Korea.

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.

Figures & tables

Panel of Figure 1 — Fig. 1 : Example of SDF estimation, visualized as a heatmap in (a), and mesh reconstruction in (b) of a large-scale environment, obtained using Kernel-SDF.

Panel of Figure 2 — Fig. 2 : (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.

Kernel	$k(r)$	$r(\hat{f})$
RBF	$\exp\left(-{r^{2}}/{2l^{2}}\right)$	$\sqrt{-2l^{2}\log\hat{f}}$
Matérn 3/2	$\left(1+{\sqrt{3}r}/{l}\right)\exp\left(-{\sqrt{3}r}/{l}\right)$	$-{l\log\hat{f}}/{\sqrt{3}}$

TABLE I : Kernel functions and their nonlinear transforms.

Metric	Method	room 0	room 1	room 2	office 0	office 1	office 2	office 3	office 4	C.L.	Cow	N.C.
F1 Score [ $<\delta$ ]% $\uparrow$	Ours	95.86	97.32	97.54	98.23	96.57	95.11	93.58	94.53	83.47	92.74	75.80
	FIESTA	76.71	76.49	78.15	77.64	82.43	75.45	79.73	78.19	19.37	18.03	54.73
	Voxblox	92.00	94.77	92.64	93.72	94.23	89.50	86.58	89.94	70.30	79.38	54.14
	iSDF	87.85	85.11	90.64	89.80	91.37	87.53	87.52	91.21	78.33	91.84	70.15
	VDB-GPDF	60.06	57.20	58.45	60.99	56.47	58.00	54.78	56.92	78.37	91.08	61.94
Recall [ $<\delta$ ]% $\uparrow$	Ours	92.94	95.51	95.50	97.53	95.54	92.10	90.15	90.06	97.42	98.98	75.10

TABLE II : Mesh reconstruction metrics on Replica, Cow&Lady (C.L. for the scene and Cow for the cow only), and Newer College (N.C.). When computing the ratio,

\delta=5

cm (20cm for Newer College). The best and second-best results are bold and underlined, respectively.

Panel of Figure 5 — Fig. 3 : Qualitative comparison of mesh reconstruction on Replica [ 32 ] (top) and Newer College [ 33 ] (bottom). Kernel-SDF achieves the best visual quality, accuracy, and completeness; e.g., the Replica bed-sheet textures are preserved where the baselines fail.

Panel of Figure 6 — Fig. 4 : Mesh reconstruction of the cow in Cow&Lady. Our method successfully recovers the detailed geometry (e.g., the horns, ears, and legs), which are missing in the baseline reconstructions.

Metric	Region	Method	room 0	room 1	room 2	office 0	office 1	office 2	office 3	office 4	C.L.	Cow	N.C.
SDF MAE $[\text{cm}]\downarrow$	All	Ours	1.691	1.548	1.638	1.576	1.562	1.712	1.595	1.699	4.628	2.367	22.989
		FIESTA	2.175	2.364	2.418	2.662	2.461	2.234	2.344	2.378	15.344	15.092	37.244
		Voxblox	3.128	2.539	2.733	3.017	2.725	3.470	3.740	3.077	6.297	4.000	32.030
		iSDF	3.745	4.230	4.259	3.890	4.226	5.094	4.245	4.147	7.327	4.258	64.304
		VDB-GPDF	2.925	2.900	2.778	2.902	2.968	2.861	3.176	2.929	4.683	3.056	21.783
	Near	Ours	1.741	1.706	1.777	1.665	1.624	1.755	1.697	1.809	3.572	2.442	8.813

TABLE III : SDF reconstruction metrics on Replica, Cow&Lady (C.L. for the scene and Cow for cow only), and Newer College (N.C.).

Method	Replica		Cow & Lady		Newer College
Method	FPT (s)	QT-1k (ms)	FPT (s)	QT-1k (ms)	FPT (s)	QT-1k (ms)
Kernel-SDF	0.19	2.80	0.11	4.43	0.17	3.04
VDB-GPDF	0.68	10.66	0.10	8.29	0.26	10.16
Voxblox	21.23	103.81	4.48	99.42	94.01	85.20
iSDF	2.49	0.29	0.84	0.27	1.38	0.28
FIESTA	0.05	0.05	0.02	0.03	0.28	0.04

TABLE IV : Timing metrics: average per-frame processing time (FPT) and 1k-point query time (QT-1k) on Replica room0 [ 32 ] , Cow&Lady [ 6 ] , and Newer College [ 33 ] . Best and second best are bold and underlined. Measured on an Intel i9-14900K and NVIDIA RTX 3090.

Panel of Figure 9 — Fig. 5 : Scalability of Kernel-SDF. (a) The serialized map size scales linearly with the environment volume across the eight Replica scenes ( $r=0.972$ ). (b)–(d) Per-component processing time over the full session (BHM front-end, SDF back-end, SDF prediction, and BHM prediction); all components stay stable with increasing map size and are independent of the global environment size.

Panel of Figure 10 — Fig. 6 : Kernel ablation on a 2D unit-circle dataset, empirically validating the softmin view of the log-GP back-end. UDF MAE vs. number of surface samples (left) $N$ and kernel scale $\lambda$ (equivalently softmin $\alpha$ ) (right). Except for the swept variable, defaults are $\lambda=100$ , $\alpha=100$ , and $N=100$ . The RBF kernel attains the lowest error and improves with larger scale and denser sampling.

Weighting	SDF MAE $[\text{cm}]\downarrow$	Grad. MAE $[\text{rad}]\downarrow$
Ours (w/ BHM var.)	1.691	0.1514
w/o BHM var. ( $\sigma^{2}{=}0$ )	1.731	0.1722
w/o BHM var. ( $\sigma^{2}{=}0.1$ )	1.734	0.1749

TABLE VI : Ablation on the BHM surface-uncertainty weighting (Replica room0, all query points). Ours uses per-sample, occupancy-derived variance

\sigma_{i}^{2}

; the alternatives replace it with a single constant variance, either

\sigma^{2}{=}0

(pure softmin) or

\sigma^{2}{=}0.1

Panel of Figure 12 — Fig. 7 : Consistency between SDF error and estimated SDF variance on a slice of Replica Office3 [ 32 ] : (a) predicted SDF, (b) absolute SDF error against ground truth, and (c) estimated SDF variance. Regions of higher error coincide with higher estimated variance, showing that Kernel-SDF effectively captures uncertainty.

Panel of Figure 13 — Fig. 9 : Effect of sensor noise on (left) mesh reconstruction and (right) SDF metrics on Replica room0 [ 32 ] . Kernel-SDF stays robust as noise increases, with only gradual degradation.

Panel of Figure 14 — Fig. 10 : Kernel-SDF enables safe bubble cover construction [ 21 ] and safe robot navigation. (a) A Clearpath Jackal robot with LiDAR sensor navigating in a lab environment. (b) Real-time Kernel-SDF estimate shown as a heatmap with safe bubble cover (union of orange circles) and planned trajectory (black line).

Panel of Figure 15 — Fig. E.1 : Noise-free ( $\sigma^{2}=0$ ) benchmark of the log-GP UDF on a 2D unit-circle. (a) Predicted UDF map at $\lambda=10$ ; the red curve is the unit circle. (b) Signed error (prediction minus ground truth): negative everywhere off the surface, a pure under-estimate. (c) Error vs. $\lambda$ : the bias stays negative and shrinks as $\lambda$ grows, matching the recovery term $-2l^{2}\log S$ .

Panel of Figure 16 — Fig. F.1 : 2D consistency demo of minimum-fusion local GPs versus a single global GP. Two overlapping local GPs (a), (b) are minimum-fused (c) and compared against a global GP trained on all samples (d). The fusion is almost identical to the global GP: the absolute UDF difference (e) is at the $10^{-8}$ level and the gradient angular error (f) is below $10^{-6}\,$ rad across the whole domain, including the GP-switching boundary. The black curve marks the zero-level set (the unit circle).

Panel of Figure 17 — Fig. H.1 : To construct the tree efficiently, we implement two key optimizations: (a) voxels are updated in ray-sorted order to maximize the CPU cache hit rate; and (b) this ordering is generated via a stride-2 parallel reduction. In this scheme, threads first process ray batches in parallel, then odd-indexed threads ( $2i-1$ ) merge results from their even-indexed counterparts ( $2i$ ). Finally, these partially merged results are aggregated sequentially into thread 0.

Panel of Figure 18 — Fig. J.1 : Sharp transition of the BHM occupancy field at the ground-truth surface vertices of the Replica room0 scene. (a) Occupancy log-odds gradient norm $\|\nabla\log\frac{q}{1-q}\|$ , with the equivalent transition-band width (top axis, median $\approx 3.6\,$ mm). (b) Occupancy probability $q$ at the surface: $97\%$ of points have $q<0.1$ or $q>0.9$ (shaded), so the sign is effectively deterministic.

Metric	Region	Method	room 0	room 1	room 2	office 0	office 1	office 2	office 3	office 4
F1 Score $[\%]\uparrow$	All	Ours	98.25	97.69	97.66	97.00	96.52	97.92	97.79	97.91
		FIESTA	97.74	97.69	97.66	96.74	96.53	98.21	97.97	97.83
		Voxblox	96.77	97.91	96.82	97.24	97.35	95.33	94.97	95.46
		iSDF	97.93	98.43	98.26	98.60	98.09	98.34	97.63	98.14
		VDB-GPDF	86.85	86.07	91.54	85.26	86.34	87.90	84.04	87.94
	Near	Ours	94.51	93.43	93.26	91.71	91.33	93.83	93.49	93.26

TABLE J.1 : Sign metrics. Results are reported as the F1 score, precision, recall and accuracy for all points (All), points near the surface (Near) and points far from the surface (Far) on the Replica dataset. The best, second best and third best results are bold, underlined and curly-underlined, respectively.

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3D Reconstruction

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Key part of robotics, augmented reality, and digital inspection is dense 3D reconstruction from depth observations. Traditional volumetric fusion techniques, including truncated signed distance functions (TSDF), enable efficient and deterministic geometry reconstruction; however, they depend on heuristic weighting and fail to transparently convey uncertainty in a systematic way. Recent neural implicit methods, on the other hand, get very high fidelity but usually need a lot of GPU power for optimization and aren't very easy to understand for making decisions later on. This work presents BayesFusion-SDF, a CPU-centric probabilistic signed distance fusion framework that conceptualizes geometry as a sparse Gaussian random field with a defined posterior distribution over voxel distances. First, a rough TSDF reconstruction is used to create an adaptive narrow-band domain. Then, depth observations are combined using a heteroscedastic Bayesian formulation that is solved using sparse linear algebra and preconditioned conjugate gradients. Randomized diagonal estimators are a quick way to get an idea of posterior uncertainty. This makes it possible to extract surfaces and plan the next best view while taking into account uncertainty. Tests on a controlled ablation scene and a CO3D object sequence show that the new method is more accurate geometrically than TSDF baselines and gives useful estimates of uncertainty for active sensing. The proposed formulation provides a clear and easy-to-use alternative to GPU-heavy neural reconstruction methods while still being able to be understood in a probabilistic way and acting in a predictable way. GitHub: https://mazumdarsoumya.github.io/BayesFusionSDF

Abstract

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