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.

翻译：随着三维高斯泼溅技术的日益普及及其应用范围从渲染扩展到三维重建，直接基于这种新型表示形式的几何处理应用需求也随之产生。虽然将高斯中心视为点云或进行网格化是一种可应用现有算法的选择，但这可能会忽略数据中存在的信息或带来不必要的计算开销。此外，高斯泼溅数据往往包含大量离群点，这些离群点虽不影响渲染质量，但在几何处理应用中必须正确处理以避免产生噪声结果。本研究提出一种基于马氏距离直接在高斯泼溅上计算拉普拉斯-贝尔特拉米算子的方法，该算子是几何处理中广泛使用的工具。尽管在概念上类似于点云拉普拉斯算子，但实验表明我们的方法对高斯泼溅中心编码的点云具有更高的计算精度，此外该算子还可用于优化过程中输出质量的评估。