HessianForge: Scalable LiDAR reconstruction with Physics-Informed Neural Representation and Smoothness Energy Constraints

TL;DR

This work addresses the challenge of producing smooth, watertight surface reconstructions from sparse LiDAR data. It introduces a physics-informed neural implicit representation that optimizes an $L_2$-Hessian energy (biharmonic PDE constraint) to enforce high-order surface smoothness, learned via a multi-scale octree-based input encoding and a set of per-level MLPs. A CUDA-accelerated test-time refinement via Laplacian-based least-squares optimization further improves mesh quality while preserving topology. Experiments on MaiCity, Newer College, and KITTI demonstrate improved accuracy and smoothness over state-of-the-art baselines, with careful analysis of Hessian energy overhead and ablations. The approach provides a scalable, geometry-aware solution for outdoor 3D mapping with practical implications for robust SLAM and large-scale environment modeling.

Abstract

Accurate and efficient 3D mapping of large-scale outdoor environments from LiDAR measurements is a fundamental challenge in robotics, particularly towards ensuring smooth and artifact-free surface reconstructions. Although the state-of-the-art methods focus on memory-efficient neural representations for high-fidelity surface generation, they often fail to produce artifact-free manifolds, with artifacts arising due to noisy and sparse inputs. To address this issue, we frame surface mapping as a physics-informed energy optimization problem, enforcing surface smoothness by optimizing an energy functional that penalizes sharp surface ridges. Specifically, we propose a deep learning based approach that learns the signed distance field (SDF) of the surface manifold from raw LiDAR point clouds using a physics-informed loss function that optimizes the $L_2$-Hessian energy of the surface. Our learning framework includes a hierarchical octree based input feature encoding and a multi-scale neural network to iteratively refine the signed distance field at different scales of resolution. Lastly, we introduce a test-time refinement strategy to correct topological inconsistencies and edge distortions that can arise in the generated mesh. We propose a \texttt{CUDA}-accelerated least-squares optimization that locally adjusts vertex positions to enforce feature-preserving smoothing. We evaluate our approach on large-scale outdoor datasets and demonstrate that our approach outperforms current state-of-the-art methods in terms of improved accuracy and smoothness. Our code is available at \href{https://github.com/HrishikeshVish/HessianForge/}{https://github.com/HrishikeshVish/HessianForge/}

Figures (8)

• Figure 1: We present a physics-informed neural implicit representation, optimized on $L_2$-Hessian Energy, for surface mapping from LiDAR point-clouds. The above figures represent a birds-eye view of KITTI sequence 00. Our approach achieves improved surface smoothness and accuracy compared to state-of-the art approaches (1.c.).
• Figure 2: Framework architecture: Split into learnable and non-learnable components, our architecture consists of a neural network architecture trained on $L_2$-Hessian loss and a CUDA-supported test-time refinement module, which includes least squares optimization of vertex Laplacians for fine-grained local smoothing.
• Figure 3: Smoothness: This figure contrasts the quality of reconstruction against baselines on Newer College. Our approach achieves higher precision, evidenced by smoother surface.
• Figure 4: Effect of Hessian loss and least squares optimization: The highlighted areas in (a) and (b) showcase improvements in smoothness due to Hessian energy minimization. (c) and (d) highlight per-vertex Laplacian values. Higher curvature is indicated by gray regions, which is reduced in (d).
• Figure 5: Computational overhead of autograd: The graphs quantify the computational overheads associated with autograd functionality. Combining autograd with FDM approximations reduces the computational overhead. These metrics were computed on RTX 3050 GPU.