Normalized Gaussian Splatting

• Normalized Gaussian Splatting is a technique that encodes multivariate fields as sums of weighted, normalized Gaussian primitives, ensuring rigorous probabilistic and analytic properties.
• It offers universal approximation and convergence guarantees by leveraging anisotropic, axes-aligned covariance matrices for scalable high-dimensional representations.
• Practical implementations demonstrate improved rendering quality, safer robot trajectory planning, and efficient 4D flow super-resolution with reduced memory usage and faster training.

Normalized Gaussian Splatting is a parametric technique for representing and manipulating multivariate fields using mixtures of normalized Gaussian functions. It forms a theoretical and algorithmic basis for applications ranging from high-fidelity scene rendering and robotics trajectory planning to physics-informed super-resolution of spatiotemporal medical data. The central innovation of normalized Gaussian splatting (NGS) is its principled treatment of normalization for each Gaussian component, enabling rigorous probabilistic interpretation, analytic integral computations, and provable convergence properties.

1. Mathematical Formulation and Normalization

Normalized Gaussian splatting encodes a target function or density as a sum over weighted, normalized Gaussian primitives. For scene or field representation, the density $\sigma:\mathbb{R}^q\to\mathbb{R}$ (for $q$-dimensional space) is parameterized as

$\sigma(x) = \sum_{k=1}^{N} w_k G_k(x),$

where each $w_k > 0$ and $$G_k(x) = (2\pi)^{-\frac{q}{2}}|\Sigma_k|^{-1/2}\exp\left(-\frac{1}{2}(x-\mu_k)^\top \Sigma_k^{-1}(x-\mu_k)\right)$$ is a normalized Gaussian with mean $\mu_k\in\mathbb{R}^q$ and symmetric positive-definite covariance $\Sigma_k\in\mathbb{R}^{q\times q}$ (Michaux et al., 25 Sep 2024, Jo et al., 14 Nov 2025).

Normalization ensures that $\int_{\mathbb{R}^q}G_k(x)\,dx=1$, rendering the mixture interpretable as a probability density or a convex kernel smoother, depending on context. This contrasts with unnormalized splatting approaches, which forgo the normalization factor and thereby lose the probabilistic and analytic properties essential for principled integration and physical modeling.

For vector-valued fields $\mathbf{v}(\mathbf{x})\in \mathbb{R}^p$, the output is expressed as a convex combination:

$$\widehat{\mathbf{v}}(\mathbf{x}) = \sum_{i=1}^N w_i(\mathbf{x}) \mathbf{v}_i, \quad w_i(\mathbf{x}) = \frac{\exp\left(-\frac{1}{2}(\mathbf{x}-\mu_i)^\top \Sigma_i^{-1}(\mathbf{x}-\mu_i)\right)}{\sum_{j=1}^N \exp\left(-\frac{1}{2}(\mathbf{x}-\mu_j)^\top \Sigma_j^{-1}(\mathbf{x}-\mu_j)\right)}.$$

This is the softmax-generated normalized kernel weighting as in the Nadaraya–Watson estimator (Jo et al., 14 Nov 2025).

2. Theoretical Properties and Convergence Guarantees

Normalized Gaussian splatting enjoys rigorous universal approximation and statistical consistency guarantees, extending classical kernel regression theory to high-dimensional, anisotropic, or even axes-aligned mixtures. Under mild regularity assumptions on sampling and covariance scaling,

• For $N$ splats with covariance matrices $\Sigma_i$ satisfying $\operatorname{tr}(\Sigma_i)^{\beta/2}\to 0$ and $N \det(\Sigma_i)^{1/2} \to \infty$ as $N\to\infty$, the estimator

$$\widehat{\mathbf{v}}(\mathbf{x}) \to \mathbf{v}(\mathbf{x}) \quad \text{in probability}.$$

The convergence rate is

$$O_p\left(\frac{1}{N}\sum_{i=1}^N\operatorname{tr}(\Sigma_i)^{\beta/2} + \sqrt{\frac{1}{N^2}\sum_{i=1}^N \det(\Sigma_i)^{-1/2}}\right)$$

for $\beta$-smoothness (Jo et al., 14 Nov 2025). The entirely normalized formulation is necessary—ablation studies report that omitting normalization leads to non-convergence.

In high-dimensional settings, axes-aligned covariances (e.g., $$\Sigma_i = \operatorname{diag}(h_{i1}^2, ..., h_{iq}^2)$$) enable scalable training, and the consistency guarantee remains intact with minimax-optimal rates (Jo et al., 14 Nov 2025).

3. Optimization and Density Control: Steepest Descent Splitting

Densification and point cloud compactness in 3D Gaussian splatting are addressed by a block-splitting optimization that leverages normalization and saddle-point analysis. Given a photometric loss objective over a mixture parameter set $\{\theta^{(i)}\}$, densification replaces a parent Gaussian $\theta^{(i)}$ with $m_i$ offspring $\{\vartheta^{(i)}_j\}$, each weighted by $w^{(i)}_j \ge 0$ with $\sum_j w^{(i)}_j = 1$.

Theoretical analysis yields these results (Wang et al., 8 May 2025):

• Opacity normalization: For two offspring, $w^{(i)}_1 = w^{(i)}_2 = 1/2$; thus, each inherits half the parent's opacity, preserving local density: $o^{(i)}_1 = o^{(i)}_2 = o^{(i)} \cdot 1/2$.
• Saddle-escape condition: Only Gaussians at negative-curvature points (i.e., minimal eigenvalue $\lambda_{\min}(S^{(i)}(\theta))<0$ of the splitting matrix) benefit from splitting. Exactly two children are sufficient.
• Split direction: Children are placed at $\pm\epsilon v_{\min}(S^{(i)})$ (the eigenvector of the most negative curvature), restoring descent even as first-order gradients vanish.

This block-normalized splitting is foundational in the SteepGS algorithm, yielding $\sim50\%$ reduction in Gaussians, 20–40% less memory, and maintained or improved reconstruction quality (Wang et al., 8 May 2025).

4. Practical Implementations and Algorithmic Strategies

Implementation of normalized Gaussian splatting hinges on proper initialization, density control, and memory management, particularly for high-dimensional or large-scale data.

Key strategies include (Jo et al., 14 Nov 2025):

• Initialization: Uniform Gaussian grid placement with initial field values sampled from low-resolution data.
• Axes-aligned splats: To manage computational complexity in high $q$, covariances are constrained to diagonal, reducing optimization cost while preserving convergence.
• Gaussian merging: To prevent redundant splats (degeneracies), a cosine similarity graph over unnormalized influence vectors $\mathbf{z}_i$ is constructed. Connected clusters (e.g., similarity $>0.9$) are periodically merged, averaging means, bandwidths, and values, and re-predicting the field at merged centers. This merging is critical for efficiency and to avoid out-of-memory failures.
• Differentiable integration: For robotics and planning, analytic bounds (via error function) are used for integrals of normalized mixtures over geometric volumes, which is only tractable due to normalization (Michaux et al., 25 Sep 2024).

5. Applications in Scene Synthesis, Robotics, and Scientific Computing

Normalized Gaussian splatting underpins state-of-the-art systems in several application domains:

• Real-time rendering and compact representation: 3DGS with normalization supports efficient novel view synthesis, enabling GPU-accelerated high-resolution rendering with reduced point count (Wang et al., 8 May 2025).
• Risk-aware motion planning: Normalized splats allow analytic upper bounds on robot–scene collision probabilities. SPLANNING, a trajectory optimization system, employs these bounds to deliver differentiable, real-time, risk-constrained planning in dense, photorealistic scenes. Empirically, it yields a higher area under the precision–recall curve and more collision-free successes versus NeRF and deterministic baselines (Michaux et al., 25 Sep 2024).
• Physics-informed super-resolution: PINGS-X models high-resolution, spatiotemporal fields (e.g., 4D flow MRI) using normalized, axes-aligned Gaussians and periodic merging. Formal convergence guarantees, closed-form PDE residuals, and ablation results confirm the necessity of normalization and merging for both accuracy and scaling. PINGS-X achieves substantial reductions in walltime and memory use, with 2–5$\times$ faster training and lower $L^2$ error versus physics-informed neural networks (Jo et al., 14 Nov 2025).

6. Empirical Outcomes and Comparative Evaluations

Evaluation of normalized Gaussian splatting reveals:

• Rendering quality: Normalization preserves PSNR/SSIM in scene synthesis benchmarks, with negligible loss compared to unnormalized baselines (Wang et al., 8 May 2025, Michaux et al., 25 Sep 2024).
• Trajectory safety and efficiency: SPLANNING with normalized splats outperforms state-of-the-art planners on collision-free success rates, with real-time cycle times ($\sim0.2$–$0.3$ s) and analytic, gradient-based constraints (Michaux et al., 25 Sep 2024).
• Scientific field modeling: In 4D flow MRI, normalized, merged splatting delivers lower error and dramatically shorter training times. Ablations confirm normalization is essential for convergence, and merging prevents out-of-memory and accuracy degradation (Jo et al., 14 Nov 2025).

Domain	Key Algorithm	Impact of Normalization
Scene Rendering	SteepGS (Wang et al., 8 May 2025)	50% fewer points, real-time, no quality loss
Robotics Planning	SPLANNING (Michaux et al., 25 Sep 2024)	Analytic collision bounds, higher success rates
4D Flow Super-Res	PINGS-X (Jo et al., 14 Nov 2025)	Faster, more stable, guaranteed convergence

7. Limitations and Open Directions

While normalized Gaussian splatting confers analytic and computational advantages, it introduces trade-offs:

• The necessity of normalization can complicate rasterization and may slightly reduce efficiency in hardware-optimized pipelines originally developed for unnormalized splats (Michaux et al., 25 Sep 2024).
• Merging strategies require careful thresholding to balance compactness and field fidelity (Jo et al., 14 Nov 2025).
• In high-dimensional applications, axes alignment may compromise anisotropic structure capture, though it substantially accelerates training and preserves minimax rates in practice.

Further research is warranted into adaptive, structure-aware normalization, online merging, and extensions to non-Gaussian or multimodal parametric mixtures. Empirical evidence suggests normalization is indispensable for theoretical soundness and practical scalability across emerging fields of real-time scene synthesis, robotics, and scientific imaging (Jo et al., 14 Nov 2025, Wang et al., 8 May 2025, Michaux et al., 25 Sep 2024).