Neural Implicit Energy Optimization for LiDAR Mapping

Hrishikesh Viswanath; Md Ashiqur Rahman; Chi Lin; Damon Conover; Aniket Bera

cs.CVOct 20, 2026

Neural Implicit Energy Optimization for LiDAR Mapping

Authors: Hrishikesh Viswanath, Md Ashiqur Rahman, Chi Lin, Damon Conover, Aniket Bera

Organizations: Purdue University, USA · DEVCOM, Army Research Laboratory, USA

Abstract

Neural implicit representations provide memory-efficient signed distance field (SDF) parameterizations for large-scale LiDAR scenes, but the geometry recovered from sparse and often noisy inputs carries jitter and artifacts. Classical geometric optimization techniques resolve such artifacts with surface energy functionals, whose minimizers correspond to smooth surfaces. Motivated by this principle, we formulate neural LiDAR mapping as the optimization of energy functionals over the parameter space of neural implicit representations. Specifically, we regularize the signed distance field with an energy functional defined on the shape operator of its level set, which represents the curvature of the surface, thus encouraging smoothness along the surface. This adapts classical geometric priors to the scale of million-point LiDAR maps. Experiments on large-scale datasets show reconstructions with improved smoothness at preserved geometric fidelity, and no additional inference-time cost. The code is publicly available at https://github.com/HrishikeshVish/nieo

Figures & tables

Figure 2: Framework architecture: Our architecture consists of an implicit neural representation and a learnable octree, trained using the continuous smoothness energy as a geometric supervision signal.

Figure 3: Qualitative comparison across scenes. One column per scene ( Truck , Barn , Courthouse ): input cloud, Base , and ours. All three share a sampler, octree and decoder, so only the regularizer differs.

Figure 4: Neural implicit reconstruction via energy optimization. (Top) KITTI 00, bird’s-eye. (Bottom) Newer College, front view.

DATA	MODEL	ACC. [cm] $\downarrow$	COMP. [cm] $\downarrow$	C-L1 [m] $\downarrow$	ACC. RATIO [%] $\uparrow$	COMP. RATIO [%] $\uparrow$	F SCORE [%] $\uparrow$
Newer College	PUMA	7.70	15.37	0.107	89.77	80.31	84.77
	VDB FUSION	7.10	12.40	0.095	92.81	88.56	90.63
	NKSR	20.10	33.70	0.168	54.77	44.38	49.03
	PIN SLAM	11.04	9.30	0.137	62.92	96.67	76.22
	3QFP	6.72	10.50	0.097	93.63	93.35	93.48
	SHINE	6.70	10.00	0.084	93.80	93.60	93.70

Table 1: Quantitative results on NewerCollege , KITTI and MaiCity . Threshold

10

cm (MaiCity, KITTI) and

20

cm (NewerCollege). Bold is best.

Method	ACC. [cm] $\downarrow$	COMP. [cm] $\downarrow$	C-L1 [m] $\downarrow$	ACC. RATIO [%] $\uparrow$	F SCORE [%] $\uparrow$
Eikonal	6.70	10.00	0.084	93.80	93.70
Poisson	6.55	18.25	0.141	95.63	63.09
Energy Only	6.68	17.12	0.125	93.05	65.27
Eikonal + Dirichlet Energy	6.57	11.41	0.083	95.12	93.39
Eikonal + Surface Energy	5.68	11.18	0.082	97.08	94.41

Table 2: Ablation study evaluating the impact of different physics-informed formulations on the Newer College dataset.

Scene	Method	F Score $\uparrow$	CD- $L_{1}$ (mm) $\downarrow$	F Score n $\uparrow$	$\Delta E$	Vert. (M)
Truck	Base	52.4	8.64	45.8	—	6.42
	RegSDF [ 66 ]	51.6	8.95	49.4	2.48 †	3.40 †
	NeuralSSD [ 63 ]	51.7	8.79	46.3	2.88 †	2.00 †
	GradHess [ 58 ]	52.0	8.71	46.0	1.07	5.21
	Sing. Hessian [ 60 ]	52.5	8.65	45.9	0.56	5.63
	DiGS [ 8 ]	52.3	8.65	45.2	0.81	6.04

Table 3: Reconstruction metrics and field characteristics. Quantitative comparisons against neural reconstruction baselines. † denotes settings with lower vertex count and lower energies, but higher Chamfer distance and lower F-scores than baseline, indicating oversmoothing.

Panel of Figure 5 — Figure 5: Cost and performance of the energy. (Left) Wall-clock for 40K steps against differentiation order. (Middle) Accuracy against step on Truck . (Right) Peak GPU memory against Hessian point count.

Method	01	02	quad e	math e	ug e	cloister	stairs	Avg.
F-LOAM [ 57 ]	6.74	-	0.40	0.26	0.09	7.69	-	3.04
KISS-ICP [ 56 ]	0.62	1.88	0.10	0.07	0.33	0.30	-	0.55
SuMa [ 7 ]	2.03	3.65	0.28	0.16	0.09	0.20	1.85	1.18
MULLS [ 37 ]	2.51	8.39	0.12	0.35	0.86	0.41	-	2.11
MD-SLAM [ 15 ]	-	1.74	0.25	-	-	0.36	0.34	-
SC-LeGO-LOAM [ 27 ]	-	1.30	0.09	-	-	0.20	3.20	-

Table 4: SLAM accuracy (ATE RMSE, m) against LiDAR odometry (above) and LiDAR SLAM (below) on Newer College. Ours is an energy-optimized PIN-SLAM variant.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Energy optimization in SDF fields. A noisy field with artifacts (left) becomes smooth on minimizing the energy functional (right).

Scene	Finest $\ell$ (m)	$C$ (M)	Feat. mem (MiB)	$\alpha$
Ignatius	0.005	7.41	226	$-2.40$
Truck	0.0075	16.88	515	$-2.28$
Caterpillar	0.010	16.99	518	$-2.37$
Barn	0.020	19.71	602	$-2.24$
Courthouse	0.050	11.72	358	$-2.09$

Appendix

Table 5: Cost follows octree occupancy, not scene size. Feature memory is

C\times d\times 4

bytes at

d{=}8

Scene	Method	F-Score $\uparrow$	CD- $L_{1}$ $\downarrow$	F-Score n $\uparrow$	Energy Reduction	Verts (M)
Ignatius	Base	67.7	3.09	59.7	—	2.03
	RegSDF [ 66 ]	59.0	3.55	49.1	3.30	1.49
	NeuralSSD [ 63 ]	61.6	3.45	51.4	3.56	1.29
	GradHess [ 58 ]	69.1	3.05	59.7	0.97	1.78
	Singular Hessian [ 60 ]	68.9	3.05	61.6	0.91	1.80
	DiGS [ 8 ]	69.6	3.05	61.7	0.72	1.96

Appendix

Table 6: Reconstruction quality and field regularity, remaining scenes. Columns and protocol are identical to Tab. 3 ;

WEIGHT SCALE	$e^{-5}$	$e^{-7}$	$e^{-9}$	$e^{-11}$
Avg. Scaled Energy Magnitude	2.86	7.8 $e^{-1}$	9.7e -2	$1.5e^{-3}$
Reconstruction F Score [%]	5.44	64.73	93.95	91.44

Appendix

Table 7: Ablation of the energy weight on Newer College . Scales are set proportional to the nominal Eikonal loss magnitude.

Figure 7: Energy-optimized neural points. Smoothness energy embedded in PIN-SLAM on Newer College ; estimated trajectory in blue.

Figure 8: Baseline weight profiles. F-score against the scale applied to each regularizer’s own declared weight, octree and decoder. A scale of

1

is the weight published by the corresponding work; the declared weights differ by orders of magnitude between methods, and stoch_precond_sigma_m is a perturbation radius in metres rather than a loss weight, so the shared axis is relative.

Figure 9: Qualitative comparison, remaining scenes. One column per scene ( Ignatius , Caterpillar , Redwood ): input point cloud, Base , and ours. Redwood is indoor RGB-D, and its world frame is the first camera pose, so it is re-oriented to a level floor before framing.

Figure 10: Qualitative comparison on the LiDAR benchmarks. Rows: Newer College , MaiCity . Columns: input cloud (coloured by height), SHINE [ 67 ] , 3QFP [ 50 ] , ours. SHINE is retrained here from the authors’ released configuration, so all three share the input, poses and a $10$ cm marching-cubes grid.

Figure 11: Level-set curvature $\|S\|_{F}$ on Truck . Range $24$ – $476$ m -1 .

Figure 12: Level-set curvature $\|S\|_{F}$ on Barn . Range $7.5$ – $150$ m -1 .

Figure 13: Level-set curvature $\|S\|_{F}$ on Courthouse .

Figure 14: Level-set curvature $\|S\|_{F}$ on Caterpillar .

Figure 15: Level-set curvature $\|S\|_{F}$ on Redwood . Absolute values are not comparable with the Tanks-and-Temples scenes: this is an indoor RGB-D reconstruction at a much coarser threshold.

Figure 16: Qualitative effect of each energy term. A close-up of the hood panel of Truck under four objectives, all sharing a sampler, octree and decoder. The unregularized surface is pitted; directly regularizing via Hessian energy exhibits oversmoothing, resulting in a hole; the curvature energy removes it while the surface stays closed.

Parameter	KITTI	MaiCity	Newer College
Octree Leaf Voxel Size	0.3 m	0.2 m	0.2 m
Voxel Downsampling	0.05 m	0.05 m	0.05 m
Batch Size	4096	4096	4096
Iterations	40,000	20,000	20,000
Learning Rate	0.05	0.01	0.01
SDF Loss Type	BCE	BCE	BCE

Appendix

Table 8: Hyperparameter configurations across evaluation datasets.

Parameter	Truck	Ignatius	Caterpillar	Barn	Courthouse	Redwood
Input
Frame stride ( every_frame )	5	2	6	3	3	16
Crop radius $r_{\mathrm{pc}}$ (m)	8	8	8	15	40	8
Min. range (m)	0.1	0.1	0.1	0.1	0.1	0.2
Voxel downsample (m)	0.008	0.005	0.01	0.01	0.025	0.01
Global $z$ shift (m)	7.8	7.3	-157.65	21.3	17.79	–

Appendix

Table 9: Per-scene training and evaluation parameters. Read from the configuration each run wrote into its own output directory. Shared by all ten methods on that scene; parameters that differ between methods are in Tab. 10 . Conventions: Sec. E.3 .

Method	Quantity	Truck	Ignatius	Caterpillar	Barn	Courthouse	Redwood
Base	Learning rate	0.01	0.01	0.01	0.01	0.01	0.01
RegSDF	Learning rate	0.01	0.01	0.01	0.01	0.01	0.01
RegSDF	Weight	0.003	0.003	0.003	0.003	0.003	0.003
NeuralSSD	Learning rate	0.01	0.01	0.01	0.01	0.01	0.01
NeuralSSD	Weight	$3\mathrm{e}{-5}$	$3\mathrm{e}{-5}$	$3\mathrm{e}{-5}$	$3\mathrm{e}{-5}$	$3\mathrm{e}{-5}$	$3\mathrm{e}{-5}$
GradHess	Learning rate	0.01	0.01	0.01	0.01	0.01	0.01

Appendix

Table 10: Per-method parameters. The only quantities that differ between methods on a scene; the rest are in Tab. 9 . Each method enables one regularizer on the common backbone. Each weight is the setting that maximized F-score in the sweep of Fig. 8 , applied unchanged to every scene.

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