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

Joerg H. Mueller; Martin Winter; Markus Steinberger

cs.LGMar 19, 2026

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

Authors: Joerg H. Mueller, Martin Winter, Markus Steinberger

Organizations: Huawei Technologies, Switzerland · Huawei Technologies, Austria · Graz University of Technology, Austria

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 & tables

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]

	PSNR $\uparrow$				SSIM $\uparrow$				LPIPS $\downarrow$
	$g(x)$	$f_{1}(x)$	$f^{\prime}_{2}(x)$	$f_{3}(x)$	$g(x)$	$f_{1}(x)$	$f^{\prime}_{2}(x)$	$f_{3}(x)$	$g(x)$	$f_{1}(x)$	$f^{\prime}_{2}(x)$	$f_{3}(x)$
bicycle	25.06	24.47	24.77	25.05	0.749	0.734	0.742	0.749	0.243	0.249	0.244	0.242
bonsai	32.43	30.89	31.58	32.41	0.947	0.932	0.940	0.947	0.180	0.191	0.183	0.178
counter	29.10	28.28	28.65	29.07	0.916	0.900	0.907	0.915	0.182	0.191	0.186	0.182
drjohnson	29.43	29.12	29.31	29.44	0.906	0.898	0.902	0.906	0.235	0.238	0.235	0.234
flowers	21.41	20.97	21.19	21.38	0.586	0.574	0.581	0.586	0.362	0.360	0.358	0.360

Table 1: We compare the quality of the different kernels using the quality metrics PSNR, SSIM and LPIPS. For LPIPS lower is better, for the other two higher is better. The higher the order of the polynomial, the better the quality approaches that of

g(x)

, or even surpases it for

f_{3}(x)

Panel of Figure 2 — 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 5.2 .

	Baseline			gsplat			Faster-GS			Vulkan			Metal
	$g(x)$	$f_{1}(x)$	%	$g(x)$	$f_{1}(x)$	%	$g(x)$	$f_{1}(x)$	%	$g(x)$	$f_{1}(x)$	%	$g(x)$	$f_{1}(x)$	%
bicycle	3.74	3.29	12	3.05	2.74	10	2.55	2.38	7	2.49	2.21	11	16.60	14.35	14
bonsai	2.40	2.07	14	1.19	0.93	21	0.85	0.75	11	0.79	0.62	22	5.11	3.89	24
counter	2.24	1.59	29	1.65	1.28	23	1.01	0.87	14	1.12	0.87	22	6.63	4.74	29
drjohnson	2.43	2.03	17	2.03	1.71	16	1.20	1.10	8	1.50	1.28	15	10.18	7.87	23
flowers	3.00	2.38	21	1.75	1.50	14	1.57	1.43	9	1.49	1.34	10	10.28	8.88	14

Table 2: Performance measurements of our first-order approximation in different Gaussian Splatting implementations. Timing measurements in the kernel columns (

g(x)

and

f_{1}(x)

) are the frame time in milliseconds, measured on the GPU, except for MetalSplatter [ 4 ] where sorting is done on the CPU and we therefore measured CPU frame time. The % column shows the performance improvement of

f_{1}(x)

over

g(x)

in percent.

Kernel	Cul	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	Baseline	Faster-GS
$g(x)$	S	0.834	27.316	0.217	2.468	1.427
$f_{1}(x)$	S	0.819	26.456	0.224	2.375	1.447
$f_{1}(x)$	0	0.819	26.456	0.224	2.207
$f_{1}(x)$	1	0.819	26.456	0.224	2.018	1.300
$f^{\prime}_{2}(x)$	S	0.827	26.851	0.219	2.390	1.484
$f^{\prime}_{2}(x)$	1	0.827	26.851	0.219	2.159	1.360

Table 3: We ablate our methods separating the influence of the changed kernel and culling, computing quality metrics (for SSIM and PSNR higher is better, for LPIPS lower is better) and average frame times for Baseline and Faster-GS. Values are averaged over all evaluated scenes. Culling (Cul) is done with either unaltered StopThePop [ 26 ] culling (S), culling to the root of the approximation polynomial (0) or to the tighter value of

\frac{1}{255}

(1). We also evaluate optimization (Opt) of the scenes with the shown kernel directly instead of using a scene optimized with

g(x)

Panel of Figure 3 — 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.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — 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.

	Baseline							Faster-GS
	$g(x)$	$f_{1}(x)$	%	$f^{\prime}_{2}(x)$	%	$f_{3}(x)$	%	$g(x)$	$f_{1}(x)$	%	$f^{\prime}_{2}(x)$	%	$f_{3}(x)$	%
bicycle	3.74	3.29	12	3.62	3	3.77	-1	2.55	2.38	7	2.44	4	2.52	1
bonsai	2.40	2.07	14	2.21	8	2.32	3	0.85	0.75	11	0.81	5	0.85	0
counter	2.24	1.59	29	1.74	22	2.07	8	1.01	0.87	14	0.94	7	1.00	1
drjohnson	2.43	2.03	17	2.14	12	2.37	2	1.20	1.10	8	1.16	3	1.19	1
flowers	3.00	2.38	21	2.60	13	2.85	5	1.57	1.43	9	1.49	5	1.55	1

Appendix

Table 4: Performance measurements of our first-, second- and third-order approximation in Baseline and Faster-GS. Timing measurements in the kernel columns are the frame time in milliseconds, measured on the GPU. The % columns show the performance improvement of the kernel left of it over

g(x)

in percent.

3D Gaussian Splatting

3D Gaussian Splatting represents scenes as optimizable 3D Gaussian primitives for fast, photorealistic rendering from new viewpoints.

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ImprovedGS+: A High-Performance C++/CUDA Re-Implementation Strategy for 3D Gaussian Splatting

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Universidad de Murcia Murcia, Spain

3D Gaussian Splatting 3DGS Acceleration Neural Rendering GPU Kernel Optimization Adaptive Gaussian Densification

Open paper AskPDF arXiv

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