Laplace-Beltrami Operator for Gaussian Splatting

TL;DR

This work develops a Laplace-Beltrami operator directly on 3D Gaussian splatting by leveraging both centers and covariance through a Mahalanobis-based neighborhood. The authors define a Gaussian LBO that incorporates variance information, relate its spectrum to the stable geometry of the representation, and demonstrate improved performance on heat diffusion, geodesic distance, curvature, and shape matching compared to baseline point-cloud and mesh-based approaches. They also introduce Gaussian filtering and adaptive training guided by the LBO spectrum to reduce outliers and improve geometry during optimization, and provide a MB08-based dataset variant to facilitate further geometry processing in 3D Gaussian splatting. The method enables geometry processing tasks directly on 3DGS without mesh extraction, offering robust, topology-resilient tools for smoothing, matching, and convergence monitoring in 3D scene representations.

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.

Figures (11)

• Figure 1: We introduce a novel method to accurately compute the Laplace-Beltrami operator directly on 3D Gaussian splats, leveraging both the center and the covariance information. The operator can be directly applied on 3D Gaussian splats, without the need for mesh extraction, for multiple applications such as surface smoothing, as shown above. We can transform the original scene (left) by computing the Laplace-Beltrami eigenfunctions (middle, both original and smoothed), leading to a smoothed scene of the 3D Gaussians splats (right), which presents transformation on the original rendering (left) and the computed Laplace-Beltrami eigenfunction (middle).
• Figure 2: Discretization of the Laplace Beltrami operator on a mesh (a) and a point cloud (b). In the mesh the connectivity is directly given and can be used to compute properties like angles. The point cloud Laplacian relies on a approximation of tangent plane at a point on which the neighborhood is projected (blue).
• Figure 3: Difference between the (a) Euclidean distance and (b) Mahalanobis distance. In (a) and (b) the marked points both have the same distance to the center, however, in the Euclidean and Mahalanobis distances, respectively. The Mahalanobis distance allows weighting directions differently, which we do in our methods using the variance of the Gaussians.
• Figure 4: Example of the inside of the point cloud before(left), after (middle) filtering using the training from GOF yu2024gaussian, and using adaptive training (right).
• Figure 5: Statistical analysis over all the objects on the loss of eigenvalues. The index represents the order of the eigenvalue by magnitude. The line represent the average of the loss and one std is used for the confidence interval.