# Laplace-Beltrami Operator for Gaussian Splatting

arXiv: https://arxiv.org/abs/2502.17531 · HTML: https://mlsanity.com/paper/2502.17531/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Builds on (most cited first, 16 of 16)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields](https://mlsanity.com/paper/2111.12077/) (2021)
- [2D Gaussian Splatting for Geometrically Accurate Radiance Fields](https://mlsanity.com/paper/2403.17888/) (2024)
- [SuGaR: Surface-Aligned Gaussian Splatting for Efficient 3D Mesh Reconstruction and High-Quality Mesh Rendering](https://mlsanity.com/paper/2311.12775/) (2023)
- [Ref-NeRF: Structured View-Dependent Appearance for Neural Radiance Fields](https://mlsanity.com/paper/2112.03907/) (2021)
- [SC-GS: Sparse-Controlled Gaussian Splatting for Editable Dynamic Scenes](https://mlsanity.com/paper/2312.14937/) (2023)
- [PGSR: Planar-based Gaussian Splatting for Efficient and High-Fidelity Surface Reconstruction](https://mlsanity.com/paper/2406.06521/) (2024)
- [A Survey on 3D Gaussian Splatting](https://mlsanity.com/paper/2401.03890/) (2024)
- [SplattingAvatar: Realistic Real-Time Human Avatars with Mesh-Embedded Gaussian Splatting](https://mlsanity.com/paper/2403.05087/) (2024)
- [Segment Any 3D Gaussians](https://mlsanity.com/paper/2312.00860/) (2023)
- [NeuSG: Neural Implicit Surface Reconstruction with 3D Gaussian Splatting Guidance](https://mlsanity.com/paper/2312.00846/) (2023)
- [GSDF: 3DGS Meets SDF for Improved Rendering and Reconstruction](https://mlsanity.com/paper/2403.16964/) (2024)
- [3DGSR: Implicit Surface Reconstruction with 3D Gaussian Splatting](https://mlsanity.com/paper/2404.00409/) (2024)
- [Isospectralization, or how to hear shape, style, and correspondence](https://mlsanity.com/paper/1811.11465/) (2018)
- [Finsler-Laplace-Beltrami Operators with Application to Shape Analysis](https://mlsanity.com/paper/2404.03999/) (2024)
- [Neural Laplacian Operator for 3D Point Clouds](https://mlsanity.com/paper/2409.06506/) (2024)

## Built on by (most cited first, 5 of 5)

- [Sinkhorn Normalization of Diffusion Kernels](https://mlsanity.com/paper/2507.06161/) (2025)
- [Proxy-Free Gaussian Splats Deformation with Splat-Based Surface Estimation](https://mlsanity.com/paper/2511.19542/) (2025)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)
- [Confidence matters: Leveraging Multi-view Geometric Priors for GS-based Reconstruction](https://mlsanity.com/paper/2608.06117/) (2026)
- [GaussPDE: Graph-Based Partial Differential Equation-Driven Rendering for 3D Gaussian Splatting](https://mlsanity.com/paper/2609.27264/) (2026)

## Abstract (arXiv)

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.