From ex(p) to poly: Gaussian Splatting with Polynomial Kernels

Abstract

Recent advancements in Gaussian Splatting (3DGS) have introduced various modifications to the original kernel, resulting in significant performance improvements. However, many of these kernel changes are incompatible with existing datasets optimized for the original Gaussian kernel, presenting a challenge for widespread adoption. In this work, we address this challenge by proposing an alternative kernel that maintains compatibility with existing datasets while improving computational efficiency. Specifically, we replace the original exponential kernel with a polynomial approximation combined with a ReLU function. This modification allows for more aggressive culling of Gaussians, leading to enhanced performance across different 3DGS implementations. Our results show a notable performance improvement of 4 to 15% with negligible impact on image quality. We also provide a detailed mathematical analysis of the new kernel and discuss its potential benefits for 3DGS implementations on NPU hardware.

Figures (4)

• Figure 1: Different kernel functions for Gaussian Splatting depending on the quadric $x$. $g(x)$ is the usual exponential function and $f_N(x)$ are polynomial approximations of order $N$ approximated with gradient descent on the $L_1$ loss in comparison to $g(x)$. The plot shows exactly the relevant range of the kernel function for $x \sim \left[0, -2\ln\left(\frac{1}{255}\right)\right]$.
• Figure 2: Sample render cut-outs of the test dataset taken from some of the test scenes, rendered using the different kernels. In most cases, the different kernels are indistinguishable. Only scenes with white backgrounds (3rd and 4th row) show noticeable artifacts for $f_1(x)$ and $f_2(x)$ that we discuss in section \ref{['sec:limitations']}.
• Figure 3: Artifacts caused by different mechanisms. (a) shows the truck scene rendered with $g(x)$ but with the opacity-aware culling of $f_1(x)$, culling to $\frac{1}{255}$. In contrast to using $f_1(x)$, using $g(x)$ introduces noticeable blocky artifacts especially for bigger splats as the culling is too tight and the opacity at the culling edge not low enough. (b) shows what happens when the treehill scene is color clamped before blending as happens in vk_gaussian_splatting. This also happens in vanilla Gaussian Splatting when an upper limit on the color is introduced.
• Figure 4: To determine the range within which the kernel is evaluated we consider the worst case based on a tile based culling approach. The kernel is only evaluated for tiles which the splat touches. The worst case is when the splat just barely touches a tile in which case the kernel is still evaluated for the point within the tile furthest away from the splat center. In the one-dimensional case (a) this maximum distance to the cut-off $p_c$ plus the tile size $p_t$. In the two-dimensional case (b) $p_c$ is computed from the smaller splat scale and the diagonal of the tile needs to be added instead of just the tile size.