Skew-Normal Splatting (SNS) for Real-Time Rendering

• Skew-Normal Splatting (SNS) is a scene representation method that replaces standard Gaussian primitives with learnable skew-normal kernels to better capture asymmetric structures.
• SNS maintains analytical tractability under affine transformation and marginalization, ensuring closed-form gradients and seamless integration within existing Gaussian splatting pipelines.
• Decoupled parameterization and block-wise optimization enhance training stability, yielding quantifiable improvements in PSNR, SSIM, LPIPS, and a reduction of up to 10% in primitive count.

Skew-Normal Splatting (SNS) is a splatting-based scene representation for real-time novel view synthesis in which the standard Gaussian primitive of 3D Gaussian Splatting (3DGS) is replaced by an Azzalini Skew-Normal primitive. In "3D Skew-Normal Splatting," the method is introduced to improve representation compactness under a finite primitive budget, especially near asymmetric structures such as object boundaries and one-sided surfaces, while preserving analytical tractability under affine transformations and marginalization so that it can be integrated into existing Gaussian Splatting rasterization pipelines (Wu et al., 14 May 2026).

1. Position within splatting-based scene representations

3DGS is described as a leading representation for real-time novel view synthesis, with its core strength attributed to an efficient kernel-based scene representation in which Gaussian primitives provide favorable mathematical and computational properties. SNS is motivated by the observation that, under a finite primitive budget, the symmetric shape of each primitive directly affects representation compactness, especially near asymmetric structures such as object boundaries and one-sided surfaces. Recent works are characterized as having explored more complex kernel distributions, but as either remaining within the elliptical family or relying on hard truncation, which limits continuous shape control and introduces distributional discontinuities. SNS addresses this by introducing a learnable and bounded skewness parameter that can continuously interpolate between symmetric Gaussians and Half-Gaussian-like shapes (Wu et al., 14 May 2026).

A related formulation, "3D Skew Gaussian Splatting with Any Camera Trajectory Visualization Engine," presents 3D Skew Gaussian Splatting (3DSGS) as a framework that extends the standard primitive to a general Skew Gaussian counterpart, couples it with an enhanced opacity representation, adds a depth-aware densification strategy, and re-derives the CUDA rasterization pipeline to universally support both symmetric and skew Gaussians in a decoupled, free-camera interactive visualization engine. This suggests a closely related line of work in which skewed splats are used not only for reconstruction quality but also for spatial data exploration and interactive visualization (Zhao et al., 18 May 2026).

2. Primitive definition and limiting behavior

In SNS, the fundamental primitive is Azzalini’s $d$-dimensional Skew-Normal density,

$$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$

where

$$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$

$\mu\in\mathbb R^d$ is the location, $\Sigma\in\mathbb R^{d\times d}$ is the covariance, $\alpha\in\mathbb R^d$ is the skewness vector, and $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$ is the CDF of a standard normal. The SNS paper explicitly identifies two limiting cases. If $\alpha=0$, then $\Phi(0)=\tfrac12$ and the primitive reduces to a standard symmetric Gaussian. As $\|\alpha\|\to\infty$, the CDF term approaches an indicator and the density tends to $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$0, i.e. a half-Gaussian (Wu et al., 14 May 2026).

The related 3DSGS presentation uses the notation

$$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$1

with reduction to the symmetric case when $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$2. It further notes that, by choosing $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$3, one can stretch the density more on the side where $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$4. Across both formulations, the defining property is intrinsic asymmetry controlled by a skewness or slant vector rather than by hard truncation (Zhao et al., 18 May 2026).

3. Affine projection, marginalization, and rasterization

A central property of SNS is tractability under affine camera projection. Let $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$5 and approximate the camera locally by the $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$6 matrix $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$7. The induced 2D primitive at pixel coordinate $$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$8 is again skew-normal,

$$p(x;\mu,\Sigma,\alpha) \;=\;2\,\varphi_d(x;\mu,\Sigma)\;\Phi\!\bigl(\alpha^\top\Sigma^{-1/2}(x-\mu)\bigr),$$9

with

$$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$0

This projected form is described as admitting a closed form and exact gradients. More generally, if $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$1 and $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$2, then

$$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$3

so moments and density remain in the same family under affine maps and marginalization. The paper also stresses a point that is frequently mishandled in asymmetric distributions: the Skew-Normal’s “center” $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$4 is not its true mean, so SNS derives an analytic offset so that tile-based bounds use the true projected mean (Wu et al., 14 May 2026).

These analytic properties are what permit direct insertion into existing Gaussian Splatting rasterizers. In SNS, each 2D Gaussian kernel is replaced by a 2D Skew-Normal kernel in the existing tile-based rasterizer, with affine matrices $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$5 precomputed per tile and corrected operational centers used for bounding boxes. In the related 3DSGS system, the forward pass evaluates the projected 2D Gaussian term, computes $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$6, evaluates $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$7, and composites front-to-back. When $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$8, the CDF term becomes constant and the code path reduces exactly to the standard Gaussian case, so one CUDA kernel can handle both with a single branch on $$\varphi_d(x;\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr),$$9; the backward pass remains analytic by differentiating through the $\mu\in\mathbb R^d$0, $\mu\in\mathbb R^d$1, and $\mu\in\mathbb R^d$2-blending operations (Zhao et al., 18 May 2026).

4. Decoupled parameterization and optimization

SNS identifies a strong coupling between scale, rotation, and skewness parameters and responds with a decoupled parameterization and a block-wise optimization strategy designed to enhance training stability and accuracy. The covariance is factorized as

$\mu\in\mathbb R^d$3

where $\mu\in\mathbb R^d$4 and $\mu\in\mathbb R^d$5. The skew vector is parameterized through an unconstrained latent $\mu\in\mathbb R^d$6 and a unit direction $\mu\in\mathbb R^d$7:

$\mu\in\mathbb R^d$8

This enforces bounded skewness, expressed in the paper as $\mu\in\mathbb R^d$9 or bounded as desired (Wu et al., 14 May 2026).

Optimization proceeds in two stages. In the warm-up stage, over the first $\Sigma\in\mathbb R^{d\times d}$0 iterations, all parameters $\Sigma\in\mathbb R^{d\times d}$1 are jointly updated. The subsequent alternating stage uses Block Coordinate Descent in cycles of length $\Sigma\in\mathbb R^{d\times d}$2: for the first $\Sigma\in\mathbb R^{d\times d}$3 steps, $\Sigma\in\mathbb R^{d\times d}$4 are frozen while $\Sigma\in\mathbb R^{d\times d}$5 are updated; for the remaining $\Sigma\in\mathbb R^{d\times d}$6 steps, $\Sigma\in\mathbb R^{d\times d}$7 are frozen while $\Sigma\in\mathbb R^{d\times d}$8 are updated. Adam is used for $\Sigma\in\mathbb R^{d\times d}$9, SGHMC for positions $\alpha\in\mathbb R^d$0, and the loss is a photometric $\alpha\in\mathbb R^d$1 reconstruction term plus regularization on opacity and skew-magnitude. The ablation study reports a progression from Vanilla SNS at 30.06 dB PSNR on Mip-NeRF360 to 30.12 dB with $\alpha\in\mathbb R^d$2-decomposition, 30.14 dB with alternating optimization, and 30.17 dB for Full SNS, with the intermediate variants also improving SSIM and LPIPS (Wu et al., 14 May 2026).

5. Opacity modeling, densification, and engine-level variants

A major variant introduced in 3DSGS concerns opacity and color modeling. Each splat has a learned base opacity $\alpha\in\mathbb R^d$3 and contributes at a screen-space sample $\alpha\in\mathbb R^d$4 through

$\alpha\in\mathbb R^d$5

Compared to symmetric splatting, the extra CDF term sharpens one side of the footprint and enables directional opacity transitions. The same work also introduces spatially varying internal opacity through two base opacities $\alpha\in\mathbb R^d$6 and a segmentation direction $\alpha\in\mathbb R^d$7:

$\alpha\in\mathbb R^d$8

Color accumulation remains front-to-back compositing with view-dependent color $\alpha\in\mathbb R^d$9 via spherical harmonics (Zhao et al., 18 May 2026).

The same 3DSGS framework adds a depth-aware densification strategy. Standard 3DGS densification is described as tracking the 2D positional gradient $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$0. The added depth-gradient term is

$\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$1

and a splat is split or cloned whenever either $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$2 or $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$3 exceeds a threshold. By including $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$4, the method states that it avoids “floaters” near occlusion edges. Splitting is performed along the principal axis of greatest variance of the 3D scale matrix $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$5, with new covariances and skew-vectors reassigned to fit the original shape (Zhao et al., 18 May 2026).

At the implementation level, SNS integrates into a Gaussian Splatting engine by replacing each 2D Gaussian kernel with the 2D Skew-Normal kernel, precomputing affine matrices $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$6 per tile, projecting $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$7, caching $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$8 and $\Phi(z)=\tfrac12[1+\erf(z/\sqrt2)]$9 per primitive, batching primitives by screen region for GPU tiling, applying frustum culling via tightened bounds using the mean shift, and using algebraic simplifications in the backward pass to avoid repeated matrix inverses. The 3DSGS implementation stores parameters in Structure-of-Arrays form on device, including $\alpha=0$0, $\alpha=0$1, rotation quaternions $\alpha=0$2, skew $\alpha=0$3, opacity $\alpha=0$4, and SH coefficients, and combines tile-based rasterization, shared-memory buffering, a fast GPU-friendly $\alpha=0$5 approximation, branch minimization, and CPU–GPU decoupling in VisEngine; camera trajectories and resolution changes are serialized out-of-band so that the CUDA renderer runs asynchronously to maintain $\alpha=0$6 FPS interactive rates in Blender (Wu et al., 14 May 2026).

6. Quantitative results, qualitative behavior, and projected extensions

On three standard benchmarks—Mip-NeRF360, Tanks&Temples, and Deep Blending—SNS is reported to outperform 3DGS and other non-Gaussian splatting methods. The dataset-average PSNR values reported for SNS are 30.17 dB versus 29.90 dB (SSS) on Mip-NeRF360, 25.08 dB versus 24.87 dB on Tanks&Temples, and 30.30 dB versus 30.07 dB on Deep Blending. SSIM improves by up to +0.004 over the best baselines, and LPIPS drops by approximately 0.006 on Tanks&Temples. The qualitative gains listed in the paper are sharper object boundaries such as window frames and train edges, better recovery of thin and one-sided surfaces such as curtain folds and tower rails, and fewer “holes” near depth discontinuities. A further primitive-budget study reports that skewed kernels achieve the same fidelity with approximately 10% fewer components (Wu et al., 14 May 2026).

The related 3DSGS results on the average of seven Mip-NeRF 360 scenes report 28.88 PSNR, 0.870 SSIM, 0.182 LPIPS, 99 FPS, and 3.22 M kernels for standard 3DGS, versus 29.78 PSNR, 0.888 SSIM, 0.145 LPIPS, 91 FPS, and 3.12 M kernels for 3DSGS. On Tanks & Temples and Deep Blending, the reported trend is +0.3–0.5 dB PSNR, +0.01 SSIM, substantially lower perceptual error, and often fewer total primitives. Error-heatmaps are stated to show up to 30% lower absolute error along high-frequency boundaries, and the generalized CUDA backend sustains approximately 90 FPS at 1080p on an RTX A6000, enabling fully unconstrained free-camera exploration in Blender (Zhao et al., 18 May 2026).

The future directions named for SNS are explicit. They include replacing the exact erf-based CDF with faster polynomial approximations to further accelerate rendering, hierarchical or hybrid schemes that only deploy skewed kernels near edges and fall back to simple Gaussians elsewhere, and joint specular BRDF or neural warping in the splatting framework for highly glossy scenes. These directions follow directly from the central SNS premise: a continuous skewness parameter can bridge symmetric Gaussians and half-Gaussians while keeping affine projection, marginalization, and alpha-blend in closed form, thereby targeting higher compactness on asymmetric geometry without abandoning the computational structure that made Gaussian splatting effective (Wu et al., 14 May 2026).