Gaussian Linear Functional Manifold Method for Massive Point Cloud Data

In this study, we develop a Structured framework for Gaussian Splatting (3DGS) with LiDAR integration (Structured-Li-GS). It is a lightweight Gaussian Splatting pipeline that leverages LiDAR-inertial-visual SLAM. Structured-Li-GS achieves high-quality 3D reconstructions with fewer Gaussians by training on accurate, dense, colorized point clouds. Gaussian primitives are anchored using sub-sampled point clouds, and their ellipsoidal parameters are initialized from local surface geometry. Our training strategy integrates a comprehensive set of loss terms, including photometric, flattening, offset, depth, and normal losses, guided by the dense point cloud, enabling accurate reconstruction without Gaussian densification. This approach produces up-to-scale, high-fidelity results with a moderate model size. For experimental validation, we develop a custom hardware-synchronized LiDAR-camera handheld scanner. Experiments on both benchmark datasets and our real-world in-house dataset demonstrate that Structured-Li-GS surpasses state-of-the-art methods while using fewer Gaussians.

High-quality digital terrain models derived from airborne laser scanning (ALS) data are essential for a wide range of geospatial analyses, and their generation typically relies on robust ground filtering (GF) to separate point clouds across diverse landscapes into ground and non-ground parts. Although current deep-learning-based GF methods have demonstrated impressive performance, especially in specific challenging terrains, their cross-scene generalization remains limited by two persistent issues: the context-detail dilemma in large-scale processing due to limited computational resources, and the random misclassification of tall objects arising from classification-only optimization. To overcome these limitations, we propose SparseGF, a height-aware sparse segmentation framework enhanced with context compression. It is built upon three key innovations: (1) a convex-mirror-inspired context compression module that condenses expansive contexts into compact representations while preserving central details; (2) a hybrid sparse voxel-point network architecture that effectively interprets compressed representations while mitigating compression-induced geometric distortion; and (3) a height-aware loss function that explicitly enforces topographic elevation priors during training to suppress random misclassification of tall objects. Extensive evaluations on two large-scale ALS benchmark datasets demonstrate that SparseGF delivers robust GF across urban to natural terrains, achieving leading performance in complex urban scenes, competitive results on mixed terrains, and moderate yet non-catastrophic accuracy in densely forested steep areas. This work offers new insights into deep-learning-based GF research and encourages further exploration toward truly cross-scene generalization for large-scale environmental monitoring.

The eigendecomposition of the Laplace--Beltrami Operator (LBO) is fundamental to geometric analysis, yet computing its low-frequency eigenmodes remains a significant bottleneck due to the high cost of iterative solvers on large-scale data. To amortize this cost, we introduce the Neural Eigenspace Operator (NEO), a feed-forward framework designed to predict the spectrum directly from point clouds. Crucially, NEO circumvents the ill-posed nature of standard eigenvector regression, which suffers from intrinsic sign flips and rotation ambiguities, by learning the stable, invariant low-frequency subspace instead. Specifically, the network predicts a redundant set of basis functions whose span robustly covers the target eigenspace, allowing for the recovery of accurate eigenpairs via a lightweight Rayleigh--Ritz refinement. To handle irregular sampling, we propose a mass-aware neural operator that incorporates per-point area weights into attention-based aggregation, improving robustness to non-uniform densities and enabling zero-shot generalization across resolutions. Our approach achieves near-linear runtime scaling and substantial wall-clock speedups over iterative solvers at comparable accuracy, and exhibits strong zero-shot transfer to high-resolution point clouds. The resulting eigenpairs support standard spectral geometry tasks, while the raw basis functions provide effective point-wise features for downstream learning. Code: https://github.com/Adversarr/NEO.