Laplace-Beltrami Operator for Gaussian Splatting

With the rising popularity of 3D Gaussian splatting and the expanse of applications from rendering to 3D reconstruction, the need for geometry processing methods tailored directly to this representation becomes increasingly apparent. While existing approaches convert the centers of Gaussians to a point cloud or mesh to use them in existing algorithms, this conversion might discard valuable information present in the Gaussian parameters or introduce unnecessary computational overhead. Additionally, Gaussian splatting tends to contain a large number of outliers that, while not affecting the rendering quality, need to be handled correctly to not produce noisy results in geometry processing applications. In this work, we present a novel framework that operates directly on Gaussian splatting representations for geometry processing tasks. Our work introduces a graph-based outlier removal designed for Gaussian distributions as well as a formulation to compute the Laplace-Beltrami operator, a widely used tool in geometry processing, directly on Gaussian splatting. Both use the Mahalanobis distance to account for the anisotropic nature of Gaussians. Our experiments show superior performance to the point cloud Laplacian operator and competitive performance to the traditional Laplacian operator computed on a mesh, while avoiding the need for intermediate representation conversion.