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Universal Beta Splatting

  • Conference: ICLR 2026
  • arXiv: 2510.03312
  • Code: Project Page
  • Area: 3D Vision / Neural Rendering
  • Keywords: 3D Gaussian Splatting, Beta Kernel, N-Dimensional, View-Dependent, Dynamic Scene, Real-Time Rendering

TL;DR

This paper proposes Universal Beta Splatting (UBS), which generalizes 3D Gaussian Splatting to an N-dimensional anisotropic Beta kernel. By enabling per-dimension shape control, UBS unifies spatial geometry, view-dependent appearance, and scene dynamics within a single representation, achieving interpretable scene decomposition and state-of-the-art rendering quality.

Background & Motivation

3D Gaussian Splatting (3DGS) achieves real-time rendering via explicit primitives, yet the fixed bell-shaped profile of the Gaussian kernel imposes fundamental limitations:

Spatial dimensions: Sharp boundaries require large numbers of small primitives, resulting in low efficiency.

Angular dimensions: View-dependent effects rely on additional spherical harmonic encoding (48 parameters), leading to a fragmented representation.

Temporal dimensions: Dynamic scenes require auxiliary deformation networks, increasing overall complexity.

Core Insight: Different scene properties demand different kernel behaviors—spatial geometry requires adaptive sharpness, angular appearance ranges from diffuse to specular, and temporal dynamics span from static to fast-moving elements. The Gaussian kernel enforces the same symmetric profile across all dimensions, whereas a Beta kernel can provide per-dimension shape control.

Method

N-Dimensional Beta Kernel

The core density function is defined as:

\[\sigma(\mathbf{x}, \mathbf{q}) = \mathcal{B}(\mathbf{x}, \mathbf{q}; \boldsymbol{\mu}, \boldsymbol{\Sigma}, \mathbf{b}) \cdot o\]

where \(\mathbf{x} \in \mathbb{R}^3\) denotes spatial coordinates, \(\mathbf{q} \in \mathbb{R}^{N-3}\) encodes additional dimensions (view direction / time), and \(\mathbf{b} \in \mathbb{R}^{N-2}\) controls the Beta shape parameter for each dimension. The Beta exponent for each dimension is \(\beta_i = 4\exp(b_i)\): - Negative \(b_i\): flat profile (suitable for smooth surfaces, static elements, diffuse reflection) - Positive \(b_i\): sharp peak (suitable for fine-grained textures, fast motion, specular reflection)

Spatial-Orthogonal Cholesky Parameterization

The covariance matrix is decomposed as:

\[\mathbf{L} = \begin{pmatrix} \mathbf{R}_x \text{diag}(\mathbf{s}_x) & \mathbf{0} \\ \mathbf{L}_{qx} & \mathbf{L}_q \end{pmatrix}\]
  • \(\mathbf{R}_x \in SO(3)\): preserves spatial orthogonal structure (first-order Taylor approximation)
  • \(\mathbf{L}_{qx}\): encodes cross-dimensional correlations
  • Guarantees backward compatibility with the rotation-scale parameterization of 3DGS

Beta-Modulated Conditional Slice

Conditional mean and covariance:

\[\boldsymbol{\mu}_{x|q} = \boldsymbol{\mu}_x + \boldsymbol{\Sigma}_{xq} \boldsymbol{\Sigma}_q^{-1} \text{diag}(\tilde{\boldsymbol{\beta}}_q) (\mathbf{q} - \boldsymbol{\mu}_q)\]
\[\boldsymbol{\Sigma}_{x|q} = \boldsymbol{\Sigma}_x - \boldsymbol{\Sigma}_{xq} \boldsymbol{\Sigma}_q^{-1} \text{diag}(\tilde{\boldsymbol{\beta}}_q) \boldsymbol{\Sigma}_{qx}\]

where \(\text{diag}(\tilde{\boldsymbol{\beta}}_q)\) applies Beta modulation to non-spatial dimensions.

Beta-modulated opacity:

\[o(\mathbf{q}) = o \prod_{i=1}^C (1 - d_i)^{4\beta_{q_i}}\]

Here \(d_i = \tanh(d_i^{raw}) \in [0,1)\) maps the per-dimension Mahalanobis distance to a bounded value.

Universal Compatibility

\(\mathbf{b}\) Setting Equivalent Method
\(N=3\), \(b_x=0\) ≈ 3DGS
\(N=3\), \(b_x \neq 0\) ≈ DBS
\(N=6\), \(\mathbf{b}=\mathbf{0}\) ≈ 6DGS
\(N=7\), \(\mathbf{b}=\mathbf{0}\) ≈ 7DGS

Interpretable Scene Decomposition

The learned Beta parameters naturally enable unsupervised scene decomposition: - Spatial \(b_x\): negative → smooth surfaces; positive → fine-grained textures - Angular \(b_d\): negative → diffuse; positive → specular - Temporal \(b_t\): negative → static elements; positive → dynamic elements

Loss & Training

\[\mathcal{L} = (1-\lambda_{SSIM})\mathcal{L}_1 + \lambda_{SSIM}\mathcal{L}_{SSIM} + \lambda_o \sum_i |o_i| + \lambda_\Sigma \sum_i \|\mathbf{s}_i\|_1\]

Opacity regularization ensures effective MCMC densification, while the scale penalty encourages primitive relocation.

Parameter Efficiency

  • Static scenes: 41% fewer parameters than 3DGS (no 48-parameter spherical harmonics required)
  • Dynamic scenes: 73% fewer parameters than 4DGS

Key Experimental Results

Static Scenes

NeRF Synthetic (UBS-6D vs. 3DGS vs. 6DGS):

Scene 3DGS PSNR 6DGS PSNR UBS-6D PSNR
chair 35.60 35.55 36.72
ficus 35.49 34.62 36.90
materials 30.50 30.63 32.90
lego 36.06 35.22 36.95

PSNR improvements reach up to +8.27 dB on the 6DGS-PBR dataset.

Dynamic Scenes

D-NeRF and 7DGS-PBR (UBS-7D vs. 4DGS vs. 7DGS): - PSNR improvements reach up to +2.78 dB - Particularly pronounced advantages on complex spatio-temporal-angular correlation scenes (cardiac motion, daylight variation, translucent deformation)

Key Findings

  1. Initializing Beta parameters to zero guarantees convergence starting from the Gaussian limit.
  2. The first-order approximation of spatial-orthogonal Cholesky achieves accuracy comparable to exact rotation, with lower computational cost.
  3. The MCMC optimization strategy remains effective for Beta kernels.
  4. Real-time rendering performance is on par with 3DGS.

Training Efficiency

  • 30K iterations
  • Static: single RTX 4090, approximately 8–10 minutes
  • Dynamic: single V100, consistent with 4DGS/7DGS baselines

Highlights & Insights

  1. Unified framework: A single Beta kernel primitive jointly handles spatial, angular, and temporal dimensions, replacing multiple specialized encodings.
  2. Backward compatibility: Setting Beta=0 degrades to the Gaussian case, providing a guaranteed performance lower bound.
  3. Unsupervised decomposition: Learned Beta parameters naturally disentangle geometry, appearance, and motion.
  4. Substantial parameter reduction: 41–73% fewer parameters with simultaneous performance gains.
  5. Real-time rendering: Full CUDA-accelerated implementation.

Limitations & Future Work

  1. The number of parameters in the N-dimensional Cholesky parameterization grows with dimensionality.
  2. The bounded support of the Beta kernel may require more primitives in far-field regions.
  3. Validation is currently limited to 7 dimensions; effectiveness at higher dimensionalities remains to be investigated.
  4. Dynamic scene training requires a batch size of 4, resulting in relatively high GPU memory consumption.
  5. The first-order rotation approximation may be insufficiently accurate under extreme rotation angles.
  • Alternative kernel designs: GES, TNT-GS, Half-GS, and Disc-GS modify the 3D Gaussian; DBS introduces a 3D Beta kernel.
  • High-dimensional Gaussians: 6DGS (spatial + view) and 7DGS (spatial + view + time) model conditional distributions over extended dimensions.
  • Dynamic scenes: D-NeRF employs deformation fields; 4DGS directly extends the temporal dimension.
  • Alternative primitives: 3D Convex Splatting, Triangle Splatting, and others.

Rating

  • Novelty: ⭐⭐⭐⭐⭐ — The N-dimensional Beta kernel unified framework constitutes an elegant theoretical contribution.
  • Practicality: ⭐⭐⭐⭐⭐ — Plug-and-play compatibility combined with real-time rendering.
  • Clarity: ⭐⭐⭐⭐ — Mathematical derivations are clear, though notation is dense.
  • Significance: ⭐⭐⭐⭐⭐ — Establishes a universal primitive framework for radiance field rendering.