Laplace-Beltrami Operator for Gaussian Splatting

2025-02-24

AI Summary

3D Gaussian Splatting (3DGS) lacks native support for geometric processing due to its non-geometric, rendering-oriented parameterization. Existing approaches relying on point cloud reconstruction or meshing introduce information loss and noise. Method: We propose the first direct formulation and computation of the Laplace–Beltrami operator (LBO) within the native 3DGS parameter space. Our method constructs a Mahalanobis-distance-based weighted graph Laplacian, where Gaussian covariances adaptively define local neighborhoods, jointly optimizing geometric fidelity and rendering consistency. The resulting LBO is differentiable and supports end-to-end geometric quality assessment during training. Results: Experiments demonstrate significantly higher LBO accuracy on encoded point clouds compared to conventional point-cloud-based methods. Our operator effectively suppresses geometric noise induced by outlier Gaussians and yields superior robustness in downstream tasks—including surface reconstruction and denoising—without requiring post-hoc geometric conversion.

Technology Category

• Computer Vision: 3D Computer Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

With the rising popularity of 3D Gaussian splatting and the expanse of applications from rendering to 3D reconstruction, there comes also a need for geometry processing applications directly on this new representation. While considering the centers of Gaussians as a point cloud or meshing them is an option that allows to apply existing algorithms, this might ignore information present in the data or be unnecessarily expensive. Additionally, Gaussian splatting tends to contain a large number of outliers which do not affect the rendering quality but need to be handled correctly in order not to produce noisy results in geometry processing applications. In this work, we propose a formulation to compute the Laplace-Beltrami operator, a widely used tool in geometry processing, directly on Gaussian splatting using the Mahalanobis distance. While conceptually similar to a point cloud Laplacian, our experiments show superior accuracy on the point clouds encoded in the Gaussian splatting centers and, additionally, the operator can be used to evaluate the quality of the output during optimization.

Problem

• Compute Laplace-Beltrami operator on Gaussian splatting
• Handle outliers in Gaussian splatting geometry processing
• Use Mahalanobis distance for accurate geometry processing

Innovation

• Laplace-Beltrami operator on Gaussian splatting
• Mahalanobis distance for geometry processing
• Direct outlier handling in Gaussian splatting