Gaussian Splatting Primitives

• Gaussian splatting primitives are explicit, parameterized volumetric elements defined by multivariate Gaussians that encode both geometry and appearance.
• They employ a hybrid structure with anchor primitives and coupled primitives, using on-the-fly prediction to dramatically reduce storage needs.
• Rate-distortion optimization and entropy modeling yield compact, photorealistic scene reconstructions applicable to real-time AR/VR and efficient streaming.

Gaussian splatting primitives are explicit, parameterized volumetric elements used as the core representation in state-of-the-art 3D scene modeling and rendering pipelines. Each primitive models a local region of 3D space via a multivariate Gaussian function, encoding both geometry and appearance. Recent research has focused on improving the compactness and efficiency of these primitive sets to facilitate practical deployment in real-world applications.

1. Definition and Principles of Gaussian Splatting Primitives

A Gaussian splatting primitive in the 3DGS (3D Gaussian Splatting) paradigm consists of:

• Location: $\mu \in \mathbb{R}^3$ (mean of the Gaussian, its center in 3D).
• Covariance: $\Sigma \in \mathbb{R}^{3 \times 3}$ (shape and orientation, typically anisotropic).
• Appearance attributes: color $c \in \mathbb{R}^3$, opacity $\alpha \in [0,1]$, and possibly higher-order parameters (e.g., spherical harmonics for view-dependence).

For rendering, the scene is projected via "splatting": each Gaussian primitive contributes to the color and transmittance of pixels it overlaps on the view plane, and the contributions are composited using alpha blending.

This approach combines the explicitness and parallelism of raster graphics with the smooth, continuous support of a probabilistic volumetric representation, enabling high-quality scene reconstruction with real-time rendering performance.

2. Hybrid and Predictive Representation Schemes

A principal challenge with conventional 3DGS is the need to store millions of primitives for photorealistic results—raising storage and transmission demands. To address this, the CompGS framework introduces a hybrid primitive structure:

• Anchor Primitives ($\boldsymbol{\Omega}$): A sparse set of fully detailed Gaussians. Each anchor contains all attributes (geometry, appearance, embedding) and acts as a reference for a local neighborhood.
• Coupled Primitives ($\boldsymbol{\Gamma}$): The majority of Gaussians. Rather than storing their full attributes, each coupled primitive is predicted on-the-fly from its associated anchor via a learnable residual embedding. The coupled primitive's parameters are derived from the anchor and the residual using learned affine transformations and neural networks.

This design leverages the strong local correlations among nearby primitives (i.e., spatial redundancy) and encodes only the deviations—drastically compressing the data size required for high-quality reconstruction.

Mathematically, for a coupled primitive $\gamma_k$ associated with anchor $\omega$:

• Geometry: $$[\mu_k, \Sigma_k] = \mathcal{A}(\mu_\omega, \Sigma_\omega \mid \beta_k)$$, with $\beta_k$ predicted from the fused anchor and residual embeddings.
• Appearance: Predicted from both anchor embedding and residual via differentiable neural networks, possibly including view-dependent components.

3. Rate-Constrained Optimization for Compact Representation

The CompGS framework employs a rate-distortion optimization framework to balance storage size with rendering quality: $\mathcal{L} = \lambda R + D$ where:

• $R$ is the expected code length (bits) needed to encode the anchor and coupled embeddings, explicitly modeled via learned entropy (probabilistic) models over quantized parameters,
• $D$ quantifies rendering distortion (e.g., mean squared error between rendered and target images),
• $\lambda$ controls the bitrate-quality trade-off.

Entropy models for the anchor and residuals are parameterized by conditional Gaussians, capturing statistical dependencies for efficient compression. This approach minimizes redundancy both between primitives (via prediction) and within parameters (via entropy modeling), producing highly compact scene representations.

4. Empirical Compression and Quality Results

Empirical validation on challenging datasets (e.g., Tanks & Temples) demonstrates that this methodology results in:

• Up to 110× compression ratios relative to uncompressed 3DGS.
• Single-digit MB model sizes (vs. hundreds of MB), with rendering quality (PSNR, SSIM, LPIPS) matching or exceeding the state-of-the-art, even at high compression.
• Improved compactness and fidelity over prior methods, including Compact3D, Compressed3DGS, and EAGLES.

Ablation studies confirm the necessity of both components: removing either the hybrid structure or rate-constrained optimization significantly degrades performance, and the best trade-off occurs at around $K=10$ coupled primitives per anchor.

Quantitative Table (Tanks & Temples, selected results):

Method	PSNR	Size (MB)
Kerbl et al. (uncompressed)	23.72	434.38
Niedermayr et al. (prior SOTA)	23.58	17.65
CompGS (proposed, best quality)	23.70	9.60
CompGS (lowest bitrate)	23.11	5.89

5. Mathematical Summary

Formally, for anchors $\Omega$ and coupled primitives $\Gamma$, the optimization is

$$\Omega^*, \Gamma^* = \underset{\Omega, \Gamma}{\arg\min}~ \lambda R + D$$

where:

• Coupled geometry prediction: $$\mu_k, \Sigma_k = \mathcal{A}(\mu_\omega, \Sigma_\omega \mid \beta_k)$$, with transformation and offsets $\beta_k$ derived from anchor/residual fusion.
• Appearance prediction: $\alpha_k = \kappa(\epsilon \oplus h_k)$, $c_k = \zeta(\epsilon \oplus h_k)$.
• Entropy models predict rate with $$R_{f_{\omega}} = E_\omega\left[-\log p(\tilde{f}_{\omega})-\log p(\eta_f)\right]$$ per anchor.

Proper quantization and entropy modeling for all encoded attributes enable efficient representation with negligible quality loss.

6. Practical Implications and Future Research

Compressed representations of Gaussian splatting primitives, as advanced by CompGS, make high-fidelity neural scene rendering feasible in bandwidth-constrained and resource-sensitive contexts. Successful areas of application include:

• Real-time 3D reconstruction and rendering for AR/VR and web-based visualization.
• Efficient transmission and streaming of large-scale photorealistic scenes.
• Integration with augmented/dynamic content pipelines (e.g., for video, AR, or telepresence).

Identified future research avenues include:

• Data-driven, adaptive anchor selection to further reduce redundancy.
• Extension of correlation modeling to global (non-local) dependencies, possibly via graph neural networks or attention.
• Joint compression of geometry and color attributes for further efficiency.
• Progressively refinable bitstreams and scalable transmission schemes.
• Generalization to dynamic scenes and temporally coherent video.

7. Summary Table

Aspect	CompGS Approach
Compression method	Hybrid: anchors + residual-based coupled primitives
Redundancy reduction	Predictive coding; only residuals encoded for most Gaussians
Optimization	Joint end-to-end rate-distortion minimization with entropy modeling
Performance	Best compactness, competitive/better image quality than all prior
Extensibility	Amenable to future advances in context modeling, perceptual loss, etc.

CompGS establishes a new standard in compressed explicit scene modeling, demonstrating that carefully structured predictive coding of Gaussian splatting primitives, coupled with a principled bitrate-quality optimization, can deliver compact, practical representations without sacrificing photorealistic rendering quality.