2DGH: 2D Gaussian-Hermite Splatting for High-quality Rendering and Better Geometry Reconstruction

TL;DR

This work addresses the limited expressiveness of 2D Gaussian Splatting for high-quality rendering and precise geometry by introducing Gaussian-Hermite Splatting (2DGH). By embedding Hermite polynomials into the Gaussian kernel and adding a Gaussian-like activation, 2DGH achieves stronger anisotropy and sharper boundaries, enabling better handling of fine geometric details and discontinuities. The method demonstrates state-of-the-art performance in novel-view synthesis and competitive geometry reconstruction across multiple datasets, at the cost of increased parameter count which can be mitigated by adjusting the Hermite rank or primitive count. Overall, 2DGH advances the representational power of splatting primitives and offers a flexible framework for future exploration of alternative basis functions and integration with additional cues such as normals.

Abstract

2D Gaussian Splatting has recently emerged as a significant method in 3D reconstruction, enabling novel view synthesis and geometry reconstruction simultaneously. While the well-known Gaussian kernel is broadly used, its lack of anisotropy and deformation ability leads to dim and vague edges at object silhouettes, limiting the reconstruction quality of current Gaussian splatting methods. To enhance the representation power, we draw inspiration from quantum physics and propose to use the Gaussian-Hermite kernel as the new primitive in Gaussian splatting. The new kernel takes a unified mathematical form and extends the Gaussian function, which serves as the zero-rank term in the updated formulation. Our experiments demonstrate the extraordinary performance of Gaussian-Hermite kernel in both geometry reconstruction and novel-view synthesis tasks. The proposed kernel outperforms traditional Gaussian Splatting kernels, showcasing its potential for high-quality 3D reconstruction and rendering.

Figures (13)

• Figure 1: Compared to the current state-of-the-art 2D Gaussian Splatting (2DGS) huang20242dgs, our proposed method, 2D Gaussian-Hermite Splatting (2DGH), demonstrates superior novel view synthesis performance in highly complex scenes while achieving comparable or even better geometric reconstruction quality under the same number of Gaussians. By modulating the Gaussian functions with Hermite series, the resulting Gaussian-Hermites exhibit a stronger representational capacity than original 2D Gaussians, particularly excelling in the reconstruction of fine complex structures and sharp discontinuous edges.
• Figure 2: Hermite Polynomials. The figure presents Hermite polynomials $H_n(x)$ for $n = 0, 1, 2, \dots, 6$. These orthogonal polynomials exhibit varying degrees of oscillatory behavior as $n$ increases.
• Figure 3: Toy experiment. We use 2 original Gaussians / 2 Gaussian-Hermites to fit the triangle in figure (a). In figure (b), the result produced by fitting with 2 original Gaussians only vaguely represents the general shape of the triangle and fails to clearly capture its edge features. In figure (c), the 2 Gaussian-Hermites more sharply captures the edge features of the triangle.
• Figure 4: Visualization comparing the original Gaussian with the Gaussian after applying GL activation function. Gaussian after GL activation is smoother near the mean and exhibits a steeper decline as it moves away from the mean.
• Figure 5: Visual comparison of rendering quality on Synthetic NeRF dataset. In the drum case, 2DGH provides the best representation of the fine nail structure; in the ficus case, 2DGH most effectively captures the slender leaves viewing from the side.