Recovering Sub-Voxel Geometry: Learned Implicit Quadrics for Large-Scale 3D Scenes

Large-scale 3D scene representations based on discrete voxelization inherently discard sub-voxel surface structure, limiting geometric fidelity and the accuracy of downstream physics-aware sensor simulation. We introduce QuadricUNet, a neural architecture that represents scene surfaces using compact, continuous second-order polynomial surfaces (quadrics) instead of discrete voxels. Given only a sparse semantic occupancy grid, QuadricUNet predicts the 10 coefficients of a quadric surface for each occupied voxel, without requiring dense point clouds, surface normals, or explicit meshes as input. Because each surface is a low-degree implicit polynomial, geometric operations act directly on the predicted coefficients. We develop a vectorized Newton-Raphson projection for dense surface sampling that is more than 11× faster than Marching Cubes sampling the same field, and an analytical ray-quadric intersection for LiDAR simulation that eliminates the need for mesh extraction. On large-scale outdoor scenes, QuadricUNet achieves a Chamfer distance of 4.91 cm and 90.25% F1@10 cm on SemanticKITTI, and 4.93 cm and 93.37% F1@10 cm on the Waymo Open Dataset, improving on Marching Cubes and NKSR when these baselines operate on the same voxel-only input. For LiDAR simulation, the analytical representation achieves a 0.0814 m depth MAE, and an SPVCNN segmentation model trained on its simulated sweeps reaches 0.48 macro-mIoU on real sweeps, compared with 0.37 and 0.33 for sweeps simulated from Marching Cubes and NKSR meshes (0.64 when trained on real sweeps). These results show that continuous quadric-based geometry supports both higher-fidelity reconstruction and sensor simulation from sparse scene representations.