Generalized non-exponential Gaussian splatting

Sébastien Speierer; Adrian Jarabo

Figure 1. Non-exponential transmittance for 3D Gaussian Splatting (3DGS). We compare the baseline exponential transmittance model implicit in 3DGS image formation model, against our new non-exponential transmittance models on the Dr Johnson scene, refined from a pretrained 3DGS asset. Top row: Ground-truth reference (left) and reconstructed images for each transmittance model. Bottom row: A plot of the transmittance decay functions (left) and per-pixel overdraw count visualizations. Our non-exponential models achieve comparable image quality to the baseline model, while reducing overdraws by

3

–

4\times

, directly translating to faster rendering.

Figure 2. Physical interpretation of splats. Splats can be thought as disk consisting of small random occluding particles. When light passes through an slab, some light is lost when hitting these particles. For uncorrelated splats (a), light occlusion in each splat is an independent stochastic process. For negatively-correlated splats (b), on the other hand, all light that passes through the first slab is occluded by the second one.

Name	Parameters	$T(\tau_{t})$	$\widebar{p}_{i<k}$
Exponential	N/A	$\exp(-\tau_{t})$	$\alpha_{i}\cdot\prod_{j<i}(1-\alpha_{j})$
Linear	N/A	$\left[1-\tau_{t}\right]_{+}$	$\alpha_{i}$
Quadratic	$c\in[-0.5,\infty)$	$\left[1-\tau_{t}-\frac{c}{2}\cdot\tau_{t}^{2}\right]_{+}$	$\alpha_{i}\cdot(1+c\cdot\widebar{\tau}_{i})$
Blended	$\gamma\in[0,1]$	$\left[\text{lerp}(1-\tau_{t},\exp(-\tau_{t}),\gamma)\right]_{+}$	$\alpha_{i}\cdot(1-\gamma+\gamma\cdot\prod_{j<i}(1-\alpha_{j}))$
( Vicini et al., 2021a )	$\gamma\in[0,1]$	$\text{lerp}(\left[1-\tau_{t}\right]_{+},\exp(-\tau_{t}),\gamma)$	$\text{lerp}(\widebar{p}_{i<k}^{\text{lin}},\widebar{p}_{i<k}^{\text{exp}},\gamma)$
Power-Law	$v\in(-1,\infty]$	$\left[(1+\tau_{t}\cdot v)^{-1/v}\right]_{+}$	$\alpha_{i}\cdot(1+\tau_{i}\cdot v)^{-\frac{1+v}{v}}$

Table 1. Summary of the mother transmittance functions

T(\tau_{t})

and the corresponding discrete probabilities

\bar{p}_{i<k}

for

i<k

used in our experiments. The linear transmittance was proposed by Jarabo et al. (2018) to stochastically model negatively-correlated opaque surfaces. The quadratic is an empirical transmittance, that generalizes the linear transmittance providing sublinear and superlinear decays. The blended transmittance reformulates the blended transmittance proposed by Vicini et al. (2021a) , which is a linear combination of linear and exponential transmittance. Finally, the power-law model was proposed by Davis and Marshak (2004) to model fractal fluctuations in positively-correlated media. The operator

\left[x\right]_{+}=\max(x,0)

is the clamping operator.

Figure 3. Transmittance comparison. We render 100 elliptical Gaussians with random rotation with opacity

\aleph=0.02

, distributed uniformly in depth, for nine different transmittance functions, and compute the inverse transmittance

1-\bar{T}_{N+1}

per pixel (top) and the number of overdraws until transmittance saturates (middle, colormap goes from black to yellow). The bottom plots show the theoretical mother transmittance

T(\tau_{t})

of each model (green) compared with the baseline exponential (dashed red), and the discrete transmittance

\bar{T_{i}}

at the rendered splats (blue dots). Our predicted discrete transmittance matches the theoretical one, which validates that our generalized model for splatting converges to the continuous generalized RTE. The faster is the decay, the less overdraws are required to saturate, and the more opaque is the reconstructed appearance.

Figure 4. Blending comparison. We render 3 Gaussians with different colors and depths (red is closer, then green, and blue at the back) with opacity

\aleph=1

, and (a) without anisotropy and growing in size and (b) anisotropic and with different rotations in the image plane. All splats are parallel to the image plane, and do not intersect. In both (a) and (b), the top row shows the result of blending, the middle row shows the scan-line at the center of the top image (with the black dashed line the sum of the three Gaussians), and the bottom row is the inverse discrete transmittance

1-\bar{T}_{N+1}

. Faster-than-exponential transmittance (third to ninth columns) show sharper blending, which for extreme cases ( power-law with

v=-1

) results in unintuitive blending: This sharper blending behavior is in part due to transparency saturating to one (see middle row), which creates

C_{1}

discontinuities on the blended primitives. Only the exponential transmittance does not show this behavior, due to the statistical uncorrelation between each splat, at the cost of more overdraws.

Parameter	Recon.	Refine.
Max iterations	15 000	1 000
Wall-clock budget	5 000 s	1 800 s
Max primitives	200 000	(inherited)
Centers LR	0.13	0.015
Scales LR	0.08	0.03
Quaternions LR	0.45	0.1

Table 2. Hyperparameters used for the reconstruction and refinement experiments.

Figure 5. Reconstruction results on NeRF synthetic scenes. Each row shows one scene ( Chair , Hotdog , Lego and Materials ), with the ground-truth reference on the left and the reconstructions using each of the four image formation models to the right. The rightmost column plots PSNR convergence over wall-clock time (top) and iteration count (bottom). Under a fixed time budget, the non-exponential models run significantly more iterations and converge to a higher PSNR, while per-iteration convergence remains comparable across all models.

Figure 6. Refinement results on real scenes. Each row shows one scene ( Dr Johnson , Playroom , Train and Truck ), with the ground-truth reference on the left and the reconstructions using each of the four image formation models to the right. Insets show the per-pixel overdraw count. The non-exponential models maintain comparable visual quality comparable to the exponential baseline while substantially reducing overdraw, which results into

3

–

4\times

rendering speedups.

Reconstruction
	Superlinear (Ours)			Linear (Ours)			Sublinear (Ours)			Exponential
Scene	PSNR $\uparrow$	MSE $\downarrow$	SSIM $\uparrow$	PSNR $\uparrow$	MSE $\downarrow$	SSIM $\uparrow$	PSNR $\uparrow$	MSE $\downarrow$	SSIM $\uparrow$	PSNR $\uparrow$	MSE $\downarrow$	SSIM $\uparrow$
Chair	29.16	0.0012	0.9721	28.85	0.0013	0.9654	28.13	0.0015	0.9530	27.72	0.0017	0.9471
Hotdog	32.77	0.0005	0.9777	32.01	0.0006	0.9735	30.52	0.0009	0.9669	32.39	0.0006	0.9731
Lego	30.26	0.0009	0.9748	30.19	0.0010	0.9731	29.44	0.0011	0.9622	28.61	0.0014	0.9536
Materials	28.33	0.0015	0.9573	28.47	0.0014	0.9589	28.04	0.0016	0.9532	27.26	0.0019	0.9448

Table 3. Image quality metrics across reconstruction and refinement experiments.

Reconstruction
	Superlinear (Ours)			Linear (Ours)			Sublinear (Ours)			Exponential
Scene	FPS $\uparrow$	Speedup $\uparrow$	Avg. OD $\downarrow$	FPS $\uparrow$	Speedup $\uparrow$	Avg. OD $\downarrow$	FPS $\uparrow$	Speedup $\uparrow$	Avg. OD $\downarrow$	FPS $\uparrow$	Speedup $\uparrow$	Avg. OD $\downarrow$
Chair	38.4	4.63 $\times$	17.8	24.0	2.89 $\times$	20.8	8.9	1.07 $\times$	34.2	8.3	1.00 $\times$	79.0
Hotdog	36.4	4.39 $\times$	18.7	23.5	2.83 $\times$	21.0	9.2	1.11 $\times$	35.4	8.3	1.00 $\times$	73.2
Lego	56.5	6.73 $\times$	13.9	38.9	4.63 $\times$	18.2	19.3	2.30 $\times$	28.0	8.4	1.00 $\times$	70.3
Materials	69.9	6.19 $\times$	14.6	60.7	5.37 $\times$	16.8	24.4	2.16 $\times$	26.6	11.3	1.00 $\times$	65.9

Table 4. Rendering performance across reconstruction and refinement experiments.

Generalized non-exponential Gaussian splatting

Abstract

Figures & tables

3D Gaussian Splatting

Exploration Matters for Escaping the Blur Trap in 3D Gaussian Splatting

Stochastic Ray Tracing for the Reconstruction of 3D Gaussian Splatting

QIRF Quantum-Inspired Non-Orthogonal Function-Space Compression for 3D Gaussian Splatting

Abstract

3D Gaussian Splatting

Explore similar work

Exploration Matters for Escaping the Blur Trap in 3D Gaussian Splatting

Stochastic Ray Tracing for the Reconstruction of 3D Gaussian Splatting

QIRF Quantum-Inspired Non-Orthogonal Function-Space Compression for 3D Gaussian Splatting