Normalized Gaussian Splatting (NGS)

• Normalized Gaussian Splatting (NGS) is a method that represents volumetric and spatiotemporal fields as mixtures of normalized Gaussians, ensuring statistical consistency and bounded predictions.
• The technique employs global and local normalization strategies to guarantee a unity-integral density and enable closed-form analytic integration for risk assessment and regression tasks.
• NGS has demonstrated superior performance in robotics trajectory optimization and physics-informed super-resolution, offering efficient, real-time evaluation and improved prediction accuracy.

Normalized Gaussian Splatting (NGS) is a formalism for representing spatial, volumetric, or spatiotemporal fields as mixtures of normalized Gaussian basis functions, where the total mixture either integrates to unity (yielding a valid probability density) or, via local normalization, produces convex-combination predictions at arbitrary query points. This approach enforces statistical and numerical stability, facilitates analytic integration for downstream tasks such as probabilistic collision avoidance or physics-informed regression, and has demonstrated superior performance in robotics trajectory optimization and high-dimensional scientific data super-resolution (Michaux et al., 2024, Jo et al., 14 Nov 2025).

1. Mathematical Formalism and Definitions

NGS represents a target field (for example, a volumetric density or a physical property vector field) as a mixture of Gaussians. For 3D volumetric density, the NGS model is:

$$\sigma(x) = \sum_{i=1}^{n_G} w_i\, N(x; \mu_i, \Sigma_i)$$

where $N(x; \mu, \Sigma)$ is the standard normalized Gaussian:

$$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$

Each component has mean $\mu_i\in\mathbb{R}^q$, symmetric positive-definite covariance $\Sigma_i$, and non-negative weight $w_i$. For regression or field modeling (e.g., velocity), the prediction at a query $x$ is:

$$\hat{\mathbf v}(x) = \sum_{i=1}^N w_i(x)\, \mathbf v_i, \quad w_i(x) = \frac{z_i(x)}{\sum_{j=1}^N z_j(x)}, \qquad z_i(x)=\exp\left(-\frac{1}{2}(x-\mu_i)^\top\Sigma_i^{-1}(x-\mu_i)\right)$$

so that $\sum_i w_i(x)=1$, ensuring bounded, convex predictions (Jo et al., 14 Nov 2025).

2. Normalization Strategies and Justification

NGS may employ either global (weight-sum) or local (component-wise) normalization:

• Global normalization: After training the unnormalized mixture, all weights are divided by the total mass $\sum_j w_j$, and the normalization constant $N(x; \mu, \Sigma)$0 is imposed:

$N(x; \mu, \Sigma)$1

This ensures that the mixture integrates to unity, i.e., $N(x; \mu, \Sigma)$2 (Michaux et al., 2024).

• Local normalization: For regression tasks, at each query $N(x; \mu, \Sigma)$3, components are locally normalized such that their weights sum to one:

$N(x; \mu, \Sigma)$4

which prevents collapse to zero (as would occur for unnormalized sums with query points far from any $N(x; \mu, \Sigma)$5), ensures stability, and bounds the output in the observed value range (Jo et al., 14 Nov 2025).

The normalization guarantees that $N(x; \mu, \Sigma)$6 is a probability density (for density modeling), enabling probabilistic reasoning (e.g., analytic collision risk bounds), and supports numerically robust blending and analytic integration (Michaux et al., 2024, Jo et al., 14 Nov 2025).

3. Analytic Computation for Risk and Constraint Evaluation

NGS admits analytic or tightly bounded evaluation of integrals over geometric primitives, vital for probabilistic robotics and physics-based modeling. For instance, in risk-aware planning, the collision probability of a rigid body (modeled as an $N(x; \mu, \Sigma)$7 ball of radius $N(x; \mu, \Sigma)$8) is bounded as:

$N(x; \mu, \Sigma)$9

with $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$0 a risk threshold and $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$1 the occupancy region. The integral $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$2 can be bounded in closed form using each Gaussian's rotation to axis alignment, allowing a practical and smooth constraint for trajectory optimization:

$$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$3

where $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$4 and $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$5 are axis-aligned parameters (Michaux et al., 2024). This constraint is differentiable and gradient expressions w.r.t. Gaussian parameters can be derived in closed form.

In regression contexts, the NGS model's explicit sum structure admits analytic computation of derivatives for PDE-constrained optimization, such as satisfying Navier–Stokes equations for fluid flows (Jo et al., 14 Nov 2025).

4. Algorithmic Workflow and Efficiency

The NGS pipeline consists of the following main stages:

Offline (Model Construction):

• Acquire multi-view RGB-D or images plus poses, or low-resolution field samples.
• Initialize $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$6 Gaussians with means, covariances, and weights (and property vectors as needed).
• Optimize photometric or data-matching loss (plus L2 penalties for mass normalization or merge/split criteria).
• Enforce normalization by re-scaling weights and reintroducing Gaussian normalizers.
• For axes-aligned scenarios (e.g., in high-dimensional scientific data), confine all $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$7 to be diagonal, reducing parameter count from $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$8 to $$N(x; \mu, \Sigma) = (2\pi)^{-3/2}|\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)$$9 per Gaussian (Jo et al., 14 Nov 2025).

Online (Query/Planning/Regression):

• For robotics: compute occupancy balls for robot configurations, evaluate closed-form risk bounds, enforce as hard constraints within a trajectory optimization problem.
• For regression/physics-inference: perform batch updates on Gaussian parameters, enforce physical PDE constraints, and apply periodic split/merge operations based on influence or error metrics to manage model capacity and redundancy (Michaux et al., 2024, Jo et al., 14 Nov 2025).

NGS’s differentiable constraints allow GPU-accelerated, real-time execution for both robotics and high-dimensional regression.

5. Empirical Performance and Properties

Empirical results across robotics and scientific data tasks highlight the performance and stability advantages of NGS:

• Trajectory Optimization: On cluttered manipulation benchmarks (Kinova Gen3, various obstacle counts), NGS-based planners achieve 69/52/28 successes (with 1/5/11 collisions) for 10/20/40 obstacles, outperforming most baselines except for those using full ground-truth geometry (Michaux et al., 2024).
• Reconstruction Fidelity: Normalized 3DGS matches standard (unnormalized) 3DGS on PSNR/SSIM and depth RMSE after 30k training iterations.
• Computation Time: Planning with NGS requires 0.19–0.30 seconds per plan (<0.5 s budget) and 6–8 ms per risk constraint evaluation.
• Physics-Informed Super-Resolution (PINGS-X): NGS regression models enable rapid training (2.6 h on 4D flow MRI, vs 30 h for PINN), with errors approximately three times lower. Axes-alignment and merging constrain memory and runtime while preserving convergence (Jo et al., 14 Nov 2025).
• Precision/Recall Trade-offs: NGS-based collision constraints yield simultaneously high precision and recall, with tunable risk threshold $\mu_i\in\mathbb{R}^q$0 governing the trade-off; other methods tend to be either overly conservative or under-recall.
• Convergence and Approximation Guarantees: NGS is proved to be a dynamic, learnable analog of the Nadaraya–Watson estimator, with formal consistency and convergence rates under mild smoothness and bandwidth conditions. Axes-aligned (diagonal) covariance reduces parameterization while preserving statistical consistency (Jo et al., 14 Nov 2025).

6. Model Variants and Adaptive Scalability

NGS can be adapted for efficiency and scalability via:

• Axes-Aligned Covariance: By restricting to diagonal covariances, parameter count and computational complexity are reduced while preserving approximation properties and analytic tractability, particularly important in 4D or higher-dimensional settings.
• Gaussian Merging and Splitting: To mitigate degenerate clustering or redundancy (especially in PDE-constrained or regression scenarios), periodic merging is performed. Gaussians with high similarity (measured by cosine similarity of their influence vectors) are merged, stabilizing $\mu_i\in\mathbb{R}^q$1 and improving computational efficiency (Jo et al., 14 Nov 2025).
• Comparison to Standard 3DGS: Standard 3DGS uses unnormalized $\mu_i\in\mathbb{R}^q$2-compositing weights and full anisotropic rotations in parameterization, potentially leading to instability and inefficiency; NGS corrects these issues via normalization and parameter discipline.

7. Applications and Outlook

NGS is employed in multiple domains:

• Risk-Aware Robot Trajectory Optimization: Provides a rigorous, differentiable, and analytically integrated risk constraint compatible with continuous real-time planning (Michaux et al., 2024).
• Physics-Informed Super-Resolution and Field Regression: Underpins highly efficient, stable models for tasks such as 4D flow MRI super-resolution, enabling training time and memory savings while maintaining or improving error rates (Jo et al., 14 Nov 2025).

A plausible implication is that NGS forms a foundation for future developments in continuous probabilistic environmental mapping and physics-regularized scientific modeling, especially for applications demanding analytic integration, bounded outputs, and computational tractability in high dimensions.

---

Key references:

• "Let's Make a Splan: Risk-Aware Trajectory Optimization in a Normalized Gaussian Splat" (Michaux et al., 2024)
• "PINGS-X: Physics-Informed Normalized Gaussian Splatting with Axes Alignment for Efficient Super-Resolution of 4D Flow MRI" (Jo et al., 14 Nov 2025)