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NIEO

Neural Implicit Energy Optimization for LiDAR Mapping
WACV 2027
Hrishikesh Viswanath1 Md Ashiqur Rahman1 Chi Lin1 Damon Conover2 Aniket Bera1
1Department of Computer Science, Purdue University 2DEVCOM Army Research Laboratory
Paper (coming soon) Code BibTeX
Newer College (quad). Left: SHINE-Mapping. Right: the same octree and decoder trained with our curvature energy.

TL;DR

We cast LiDAR mapping as energy optimization: the reconstruction is formulated as a low-energy configuration of a neural signed distance field, one that minimizes the curvature energy of its level set while simultaneously satisfying the Eikonal constraint. This suppresses geometric jitter and high-frequency artifacts, giving smooth, geometrically consistent reconstructions of million-point LiDAR sweeps over open surfaces spanning hundreds of metres, with no extra cost at inference.

Surface Curvature of the Reconstruction

Ignatius coloured by level-set curvature for each method
Each mesh is coloured by \(\|S\|_F\), the curvature of its level set, with black bands marking iso-contours. The colour scale is shared across panels and anchored on the unregularized Base, so a lighter panel is smoother than Base and a darker one is rougher. Ignatius.

Method

NIEO pipeline
A sparse multi-resolution feature octree and a small decoder predict the signed distance; the curvature energy is a geometric supervision signal on that field.

Energy over the curvature of the level set

A LiDAR sweep observes surfaces, so we smooth the surface rather than the ambient field. For a neural SDF \(u\) with unit normal \(n = \nabla u / \|\nabla u\|\) and tangent projector \(P = I - nn^\top\), the shape operator and its energy are

$$S = \frac{P\,\mathcal{H}(u)\,P}{\|\nabla u\|}, \qquad E_S(u) = \int_\Sigma \|S\|_F \, dA,$$

where the eigenvalues of \(S\) are the principal curvatures, so \(\|S\|_F^2 = \kappa_1^2 + \kappa_2^2\). Unlike a raw Hessian penalty, \(E_S\) charges only for genuine bending: it ignores normal-direction curvature and changes in \(\|\nabla u\|\). A gradient–Hessian alignment term covers exactly that complementary block, so the two never double-count, and a minimal-surface term \(E_M\) penalizes spurious area:

$$\mathcal{L} = \lambda_1 \mathcal{L}_{\text{bce}} + \lambda_2 \mathcal{L}_{\text{Eikonal}} + \lambda_S E_S + \lambda_A E_A + \lambda_M E_M .$$

The energy changes only the training objective. Architecture, sampling and inference are those of SHINE-Mapping, so there is no inference-time cost.

Results

Against prior regularizers

Every method shares one sampler, octree and decoder, so only the regularizer differs. Circles mark the same regions in every panel. Ignatius.

Compared regularizers in the paper and code: RegSDF, NeuralSSD, GradHess, Neural-Singular-Hessian, DiGS, SIREN, DiffCD and Stochastic Preconditioning.

Removing high-frequency artifacts

Base (left) vs. ours (right).
Base morphing onto ours. Each Base vertex moves to its closest point on our surface; red marks the largest displacement.

Across scales

Qualitative comparison on Truck, Barn and Courthouse
One column per scene (Truck, Barn, Courthouse): input cloud, unregularized Base, and ours.
Qualitative comparison on Newer College and MaiCity
LiDAR benchmarks. Rows: Newer College, MaiCity. Columns: input cloud (coloured by height), SHINE, 3QFP, ours.

Citation

@inproceedings{viswanath2027nieo,
  title     = {Neural Implicit Energy Optimization for LiDAR Mapping},
  author    = {Viswanath, Hrishikesh and Rahman, Md Ashiqur and Lin, Chi and Conover, Damon and Bera, Aniket},
  booktitle = {Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV)},
  year      = {2027}
}