Moment-Based 3D Gaussian Splatting

• The paper introduces MB3DGS, an advanced volumetric rendering method that leverages statistical moment theory to achieve order-independent, physically-plausible transmittance reconstruction.
• It computes per-pixel moment estimations using closed-form 3D Gaussian projections and reconstructs optical depth via Hankel matrix techniques, eliminating the need for per-pixel sorting.
• Empirical evaluations demonstrate improved PSNR, SSIM, and reduced artifacts compared to traditional methods, while maintaining efficient performance on standard GPU rasterization pipelines.

Moment-Based 3D Gaussian Splatting (MB3DGS) is an advanced volumetric rendering framework addressing the principal physical fidelity limitations of rasterization-based 3D Gaussian Splatting (3DGS). It provides order-independent, physically-plausible transmittance computation for scenes modeled as mixtures of semi-transparent 3D Gaussians, all within standard GPU rasterization pipelines and without per-pixel sorting or ray tracing. MB3DGS leverages statistical moment theory to compactly summarize volumetric density along camera rays, enabling accurate analytic reconstruction of transmittance and emission/absorption effects in complex, overlapping media (Müller et al., 12 Dec 2025).

1. Challenges in Traditional 3D Gaussian Splatting

Standard 3D Gaussian Splatting replaces the physically-accurate volumetric emission–absorption model with alpha-blended “splat” rendering, which enforces a simplistic, screen-space blending operation. This approach inherently assumes that Gaussian splats do not overlap or are perfectly front-to-back sorted, and it typically employs coarse point-sampling of density (usually at the Gaussian centroid) along the ray. These assumptions result in significant artifacts:

• Incorrect color mixing: Overlapping semi-transparent particles are not composited using the true volumetric attenuation integral, resulting in unphysical color blends.
• Order dependence: The rendered output directly depends on splat draw order, yielding popping artifacts and instability under viewpoint changes.
• Limitations of prior volumetric extensions: Volumetric variants such as Vol3DGS and ray-traced Gaussians either reintroduce inefficient per-pixel sample sorting (defeating the speed advantage of rasterization) or require explicit ray marching at frame cost.

MB3DGS is designed to overcome these limitations by enabling physical light attenuation modeling with order-independent accumulation, fully within a rasterization-based pipeline.

2. Modeling Ray Density with Statistical Moments

The foundational assumption is a scene modeled as a weighted mixture of 3D Gaussians: $$\rho(x) = \sum_{i} w_i \mathcal{N}(x \mid \mu_i, \Sigma_i), \quad w_i \geq 0$$ For a given pixel, the camera ray is parameterized as $r(t) = o + t d$. Per-ray, each 3D Gaussian reduces analytically to a 1D Gaussian along the ray: $$\rho_i(t) = w_i e^{-(t - \mu_{in})^2/(2\sigma_i^2)}$$ where $\mu_{in}$ and $\sigma_i$ are the ray-projected mean and standard deviation.

MB3DGS characterizes the total per-pixel density $\rho(t) = \sum_i \rho_i(t)$ using a finite set of statistical moments:

• Raw (power) moments:

$m_k = \int_{t_n}^{t_f} t^k \rho(t) dt$

• Power-warped moments (for large $k$ stability):

$m_k^g = \int_{t_n}^{t_f} [g(t)]^k \rho(t) dt$

with $g$ a monotonic transform, typically power-based.

• Trigonometric (Fourier) moments:

$$m_k^F = \int_{t_n}^{t_f} \exp(i a g(t)) \rho(t) dt, \quad a = k(2\pi - \theta)$$

For each Gaussian, $m_0$ and $m_1$ can be computed in closed form, and higher-order moments follow from a three-term recurrence anchored by linearization and integration by parts. This per-pixel, per-ray moment estimation is the critical preparatory step, enabling global, compact representation of density information without storing or sorting all samples.

3. Moment-Based Analytic Transmittance Reconstruction

Rendering requires the computation of transmittance $T(a \to b) = \exp\left(-\int_a^b \rho(s) ds\right)$ along camera rays — traditionally challenging under unsorted, overlapping volumetric primitives.

MB3DGS solves this using the classical moment problem. Rather than explicitly recovering the entire density function, it reconstructs a continuous approximation of the optical depth $A(t) = -\ln T(t_n \to t)$ from the finite set of moments $\{ m_k \}$. The principal steps, adapting Münstermann et al.'s order-independent transparency approach to continuous densities, are:

1. Construct the Hankel matrix $H_{ij} = m_{i+j}$.
2. Compute kernel polynomial roots (from the Hankel eigenproblem) to obtain discrete nodes $\{ x_j \}$, with one node fixed at $t$.
3. Solve a Vandermonde system for weights $\{ w_j \}$ such that their moments match $\{ m_k \}$.
4. Estimate opacity from nodal weights:

$$A_L(t) = \sum_{x_j \leq t} w_j, \quad A_U(t) = \sum_{x_j < t} w_j$$

$$\hat{T}(t) = \exp\left( - [ (1-B)A_L(t) + B A_U(t) ] \right),\quad B = 0.25$$

This procedure yields an analytic, order-independent estimate $\hat{T}(t)$ for the transmittance along each ray — entirely without ray tracing or per-pixel sorting.

4. Quadrature and Additive Rasterization Workflow

After moment collection and transmittance reconstruction, MB3DGS evaluates the radiance integral for each contributing Gaussian: $$L_i = \int_{t_n}^{t_f} \hat{T}(t_n \to t) \rho_i(t) L_e^i(d) dt$$ The method proceeds as follows:

• Sampling intervals: For each Gaussian, $N$ intervals along its main axis are determined via the inverse CDF of the 1D projected Gaussian:

$$t_j = \mu_{in} + \sigma_i \Phi^{-1}(u_j), \quad u_j = (j - 1/2)/N$$

• Sub-interval quadrature: The $[t_n, t_f]$ range is split into intervals $[t_j, t_{j+1}]$; the local density is approximated as constant within each.
• Interval contribution: Radiance is then

$$L_i \approx \sum_{j=1}^N \Delta T_j \cdot \rho_i(t_j) \cdot L_e^i(d)$$

with $$\Delta T_j = \hat{T}(t_n \to t_j) - \hat{T}(t_n \to t_{j+1})$$.

• Additive accumulation: All $L_i$ values are summed for the pixel in any order, since all order dependence has been eliminated in the transmittance computation.

The core rasterization algorithm in MB3DGS is summarized as:

for each pixel p: initialize moments m[0…K]=0 for each Gaussian G covering p: compute per-G moments miᵢ[0…K] via closed-form m += miᵢ reconstruct T̂(·) from m[0…K] for each G covering p: sample intervals {tⱼ} in [tₙ,t𝒻] Li=0 for j in 1…N: ΔT = T̂(tₙ→tⱼ) − T̂(tₙ→tⱼ₊₁) Li += ΔT * ρᵢ(tⱼ) * L_e^i(d) L[p] += Li

No per-pixel linked lists, explicit sorting, or extra memory is required beyond the moment buffers and standard additive blending.

5. Empirical Evaluation: Datasets, Metrics, and Quality

MB3DGS has been benchmarked across standard datasets, including Mip-NeRF-360 (nine unbounded scenes), Tanks & Temples (“train” and “truck”), and DeepBlending (“drjohnson”, “playroom”). The principal evaluation metrics are PSNR (higher is better), SSIM (higher is better), and LPIPS (lower is better), comparing with baselines 3DGS, StopThePop, Vol3DGS, EVER, and Don’t Splat.

Method	PSNR (↑)	SSIM (↑)	LPIPS (↓)
3DGS	23.72	0.846	0.178
StopThePop	27.31	0.814	0.213
Vol3DGS	23.67	0.851	0.174
MB3DGS	24.25	0.869	0.145

Particle counts range from 1–4 million, and training times (end-to-end) are 3–15 hours on a single high-end GPU.

Qualitative improvements with MB3DGS include:

• Physically-based blending of semi-transparent effects such as reflections on glass and smoke.
• Accurate volumetric color mixing at complex intersections (e.g., synthetic six-splat tests).
• Elimination of popping under viewpoint changes due to draw-order independence.

Ablation studies reveal that higher-order trigonometric moments (e.g., $N=5$) closely recover ground truth occlusion and color mixing, while lower-order or purely power moments lead to under-occlusion and overblending. Real-scene ablations show that removing the consistency regularizer or replacing the confidence-interval proxy with EWA leads to measurable PSNR declines and runtime increases. Substituting the default ADC for volumetric ADC sharply increases point-count and runtime with substantial PSNR loss.

6. Implementation Aspects and Order-Independence

MB3DGS comprises two primary rasterization passes:

• Moment pass: Each splat additively accumulates its per-pixel moments in floating-point framebuffers. The summation is commutative, ensuring draw-order independence.
• Quadrature pass: Each splat computes its radiance contribution based on the analytic transmittance field reconstructed from the global moments. This is again commutative and supports out-of-order accumulation.

This design eliminates the need for per-pixel linked lists, screen-space ordering, or ray tracing, and is implementable on standard GPU rasterization hardware.

7. Limitations and Prospective Directions

MB3DGS, while resolving major compositing and physicality issues of prior methods, inherits key limitations:

• ADC-based densification: Reliance on heuristic adaptive density control splitting/pruning can yield under-reconstruction or blurring in finely-detailed or extreme-parallax geometry.
• Calibration sensitivity: Mis-calibrated camera intrinsics or extrinsics may result in local over-fits and opacity artifacts.
• Moment truncation: The moment-based approach imposes approximation errors for density variations at frequencies higher than the retained moment order, leading to potential under/over-estimation of occlusion.

Suggested avenues for future improvement include:

• Integration of robust, possibly learned, densification strategies harmonizing ADC and moment stability.
• Utilization of higher-order or adaptive moment bases (e.g., wavelets, splines).
• Joint optimization of camera calibration in the MB3DGS framework.
• Extension to modeling scattering media, secondary volumetric phenomena, and global illumination.

MB3DGS thus represents an overview of statistical-moment theory and efficient rasterization, bridging the gap between fast, real-time rendering and physically-accurate, order-independent neural scene representation (Müller et al., 12 Dec 2025).