Gaussian-Based Representation

Updated 22 December 2025

Gaussian-based representation is a method that expresses signals as a continuous sum of parameterized Gaussians defined by location, covariance, and amplitude.
The approach leverages closed-form operations like differentiation and convolution, ensuring efficient rendering and analysis in high-dimensional spaces.
Its adaptable framework underpins applications in image coding, 3D scene reconstruction, and video representation, bridging classical and neural modeling techniques.

A Gaussian-based representation encodes signals, data, or geometric structure as an explicit, continuous sum (or mixture) of parameterized Gaussian functions. Widely used as an expressive and mathematically tractable modeling substrate, modern Gaussian representations span high-dimensional image, video, 3D, and spatiotemporal domains and are distinguished by their ability to unify dense and continuous representations, enable efficient rendering or analysis, and admit closed-form manipulations of many operators (e.g., differentiation, convolution, or metric distances). This article systematically reviews the formalism, variant formulations, algorithmic construction, and application domains of contemporary Gaussian-based representations, with an emphasis on their mathematical properties and concrete use cases.

1. Mathematical Formalism and Core Parametrization

A Gaussian-based representation of a signal or domain encodes it as a superposition of $N$ Gaussians, each parameterized by location, shape, amplitude, and optionally other attributes. The canonical primitive in $d$ -dimensional Euclidean space is

$G_n(\mathbf{x}) = a_n \cdot \exp\left(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol\mu_n)^\top \boldsymbol\Sigma_n^{-1} (\mathbf{x}-\boldsymbol\mu_n)\right)$

where:

$\boldsymbol\mu_n \in \mathbb{R}^d$ is the center,
$\boldsymbol\Sigma_n \in \mathbb{R}^{d \times d}$ is symmetric positive-definite (covariance/shape),
$a_n$ is the amplitude or attribute vector (e.g., RGB color, opacity, semantic embedding).

The representation aggregates these as a sum (or, in rasterization/conversion settings, a blending sequence):

$f(\mathbf{x}) = \sum_{n=1}^N G_n(\mathbf{x})$

where the physical or semantic meaning of $f$ depends on context (image intensity, field value, feature embedding, etc.).

For example, in 2D image fitting, the GaussianImage framework encodes an image as a sum of 2D Gaussians, each parameterized by an 8-tuple: location (2), covariance (via Cholesky: 3), color (3) (Zhang et al., 13 Mar 2024). In 3D domains, additional attributes such as orientation quaternions, view-dependent radiance (e.g., via spherical harmonics), or semantic vectors are employed, and covariances are often constructed as rotated and scaled matrices to capture anisotropy (Xin et al., 26 Sep 2025, Lee et al., 2023, Chabot et al., 19 Jul 2024).

2. Model Construction, Optimization, and Fitting

Gaussian-based representations are constructed by directly optimizing the locations, shapes, and amplitudes of each primitive. The fitting objective matches the sum-of-Gaussians rendering to a reference signal. In image domains, the primary loss is typically an L2 (MSE) or hybrid L2+D-SSIM over all pixels (Zhang et al., 13 Mar 2024), sometimes supplemented by regularization or structured partitioning. For 3D scene and radiance representations, photometric losses over multiple views or ray samples combined with sparsity, density, or geometric constraints are common (Lee et al., 2023, Huang et al., 8 Jun 2025).

Adaptive deployment is supported by:

Densification and pruning: Algorithms such as SplitNet or confidence-aware splitting create additional Gaussians where residuals concentrate and prune where coverage is redundant or negligible (Fei et al., 24 Oct 2024, Liu et al., 16 Sep 2025).
Hierarchical and multi-level fitting: Two-stage or level-of-Gaussian (LOG) fitting establishes a coarse, low-frequency backbone, followed by high-frequency or residual Gaussians for detail, critical for large or high-entropy signals (Zhu et al., 13 Feb 2025).
Hybrid decomposition: For efficiency, textured mesh models absorb smooth, planar areas, while Gaussians capture high-curvature, detailed, or non-manifold structures (Huang et al., 8 Jun 2025).
Feedforward and pre-trained initialization: Recent methods replace per-instance random or iterative optimizations with predictive networks that infer Gaussian parameters or densities in a feedforward pass, dramatically reducing convergence time (Zeng et al., 30 Jun 2025, Tai et al., 10 Mar 2025, Zhang et al., 20 Mar 2025).

The table below summarizes parameterizations and key operations in common settings:

Domain	Param (per Gaussian)	Rendering/Aggregation Method
2D Image (Zhang et al., 13 Mar 2024, Zhu et al., 13 Feb 2025)	$\mu\in\mathbb{R}^2$ , $\Sigma\in\mathbb{R}^{2\times2}$ , $c'\in\mathbb{R}^3$	Summed Mahalanobis-weighted color
3D Scene (Lee et al., 2023, Chabot et al., 19 Jul 2024)	$\mu\in\mathbb{R}^3$ , $\Sigma\in\mathbb{R}^{3\times3}$ , color SH, opacity	Depth-sorted alpha blending, SH decoding
Spatiotemporal (Pang et al., 8 Jul 2025)	$\mu^t$ , $\Sigma^t$ , $c^t$ , motion fields	Deformed splatting/frame interpolation
High-dim partitioning (Rigas et al., 30 May 2025)	$\mu\in\mathbb{R}^d$ , $\Sigma\in\mathbb{R}^{d\times d}$	Mahalanobis for assignment, search

3. Efficient Rendering, Compression, and Analysis

Gaussian-based representations admit highly efficient, massively parallel rendering and querying, often leveraging GPU-optimized “splatting” kernels. Key properties include:

Permutation-invariant summation in 2D: For single images, additive sum suffices; alpha blending or transmittance accumulation is unnecessary as occlusion is irrelevant (Zhang et al., 13 Mar 2024).
Rasterized ellipsoid splatting in 3D: Each Gaussian is projected, depth-sorted, and alpha-blended front-to-back to yield correct occlusion and volumetric effects (Lee et al., 2023).
Adaptive quantization and entropy coding: Vector quantization, codebook-based compression, and entropy coders (e.g., ANS, Huffman) are crucial for compact storage (Zhang et al., 13 Mar 2024, Lee et al., 2023).
Vectorized initialization from eigenspaces: The EigenGS method maps a principal-component eigenspace to Gaussian space, allowing instant initialization for new signals (Tai et al., 10 Mar 2025).
Differentiable rendering for field analysis: For physics and simulation, the analytic derivatives of Gaussians (gradient, divergence, Laplacian) permit closed-form differential operator evaluation, enabling PDE solvers without grid discretization (Xing et al., 28 May 2024).

Gaussian representations have demonstrated orders-of-magnitude faster decoding than MLP-based INRs or classic codecs, e.g., 2D GaussianImage achieving 1000–2000 FPS vs. typical INR codecs at 10–150 FPS (Zhang et al., 13 Mar 2024). In high-res 3D or video, competitive PSNR/SSIM is achieved at substantial reduction in memory and representation size (Lee et al., 2023, Pang et al., 8 Jul 2025).

4. Extensions: Learning, Graphs, and Unification

Beyond direct parametric encoding, several advancements extend Gaussian representations:

Graph-based relational learning: Gaussian Graph Network (GGN) constructs a multi-view Gaussian graph with message-passing and pooling layers, enabling efficient, generalizable multi-view aggregation, removal of duplicate primitives, and superior view synthesis (Zhang et al., 20 Mar 2025).
Submanifold embedding for neural integration: Mapping each 3D Gaussian to a continuous submanifold field yields an injective, homogeneous feature vector, overcoming non-uniqueness and heterogeneity of parameterization, and supporting robust neural architectures (Xin et al., 26 Sep 2025).
Lie group/lie algebra approaches: Lie algebrized Gaussians (LAG) map GMMs into tangent spaces of the manifold of Gaussian densities, capturing both mean/covariance changes and mixture weights in an inner-product kernel, improving scene classification (Gong et al., 2013).
Unified semantic/geometry encoding and pretraining: 3D Gaussian “anchors” serve as multi-modal volumetric priors in visual pretraining pipelines (Mask-then-render), bridging geometric, textural, and semantic tasks for robust perception (Xu et al., 19 Nov 2024).

5. Applications Across Domains

Gaussian-based representations have found utility in a broad array of domains:

Image coding, compression, and representation: Explicit mixtures track high-frequency details with competitive rate–distortion, fast (real-time) decoding, and practical deployment on resource-constrained devices (Zhang et al., 13 Mar 2024, Zhu et al., 13 Feb 2025, Zeng et al., 30 Jun 2025).
Scene reconstruction and view synthesis: 3D Gaussian Splatting offers an explicit, memory-efficient alternative to volumetric NeRFs, supporting rapid novel-view rendering and hybridization with mesh for improved efficiency (Lee et al., 2023, Huang et al., 8 Jun 2025).
Object detection and geometric regression: G-Rep recasts arbitrary-oriented bounding representations (OBB, QBB, PointSet) as Gaussians, directly optimizing statistical distances for robust, unified detector heads (Hou et al., 2022).
Video representation: Dynamic 2D Gaussian fields, deformed by hybrid motion models, deliver high-fidelity video with sub-second per-frame training and 10x–30x faster decoding (Pang et al., 8 Jul 2025).
Simulation and PDEs: Fluid solvers formulated over sums of Gaussians (grid-free) connect Lagrangian element motion and Eulerian constraints for memory-efficient, vorticity-preserving continuous fields (Xing et al., 28 May 2024).
High-dimensional indexing: Adaptive mixtures of $N$ -dimensional Gaussians (GARLIC) learn space partitions for vector search and $k$ -NN classification with fast indexing, progressive refinement, and strong generalization (Rigas et al., 30 May 2025).
Phase and holographic field representation: Complex-valued 2D Gaussians for holography reduce parameter count and memory, outperforming pixelwise approaches in scalability and fidelity (Zhan et al., 19 Nov 2025).

6. Quantitative Performance and Model Efficiency

Recent empirical studies demonstrate the practical impact of Gaussian-based approaches:

Method	FPS	Memory (GB)	PSNR (dB)	Task	Reference
GaussianImage (2D)	2,000	0.4	44.1	Image comp.	(Zhang et al., 13 Mar 2024)
LIG (9K×9K image)	20	16.7–20.3	37.5–42.2	Large image	(Zhu et al., 13 Feb 2025)
3DGS (Mip-NeRF 360)	120	746 MB	27.46	Scene synth.	(Lee et al., 2023)
Compact 3DGS (+PP)	128	29 MB	27.03	Scene synth.	(Lee et al., 2023)
Hybrid mesh–GS	231	0.74M splats	24.28	Indoor 3D	(Huang et al., 8 Jun 2025)
GARLIC (SIFT1M 128D)	—	—	Recall@1=0.69	K-NN	(Rigas et al., 30 May 2025)

Further, model compression techniques (VQ, codebooks, entropy coding), dynamic SH orders, pruning, and unified pre-training contribute to state-of-the-art accuracy with reduced computational or storage cost (Lee et al., 2023, Liu et al., 16 Sep 2025, Xu et al., 19 Nov 2024).

7. Unique Properties, Limitations, and Open Directions

Gaussian-based approaches offer a unique blend of mathematical tractability, analytic differentiability, and adaptivity:

Permutation invariance and closed-form operators: Summation and differentiation are analytic, supporting efficient optimization and field analysis (Xing et al., 28 May 2024).
Flexibility in statistical interpretation: Mixtures can be interpreted probabilistically (for uncertainty) or deterministically (as explicit renderers/fields).
Hybridization with classical and neural methods: Smoothly integrates with mesh, deep, and graph neural frameworks; admits unification with eigenspace and Lie group representations (Huang et al., 8 Jun 2025, Tai et al., 10 Mar 2025, Gong et al., 2013, Xin et al., 26 Sep 2025).

Limitations include trade-offs between parameter count and signal complexity (very high-frequency structures may require many primitives or specialized learning schedules), domain-specific rendering optimizations, and the potential for parameter redundancy or non-uniqueness if the base space is not managed (e.g., ambiguous quaternions or covariances in 3DGS (Xin et al., 26 Sep 2025)).

Open research directions include learning robust and universal priors over Gaussian spaces, advancing submanifold or graph-based encodings for neural integration, and extending adaptive dense-to-sparse transitioning in hybrid representations to time-varying or cross-modal data.

Gaussian-based representations now form a core modeling and computational primitive across machine learning and graphics, combining analytic structure, empirical efficiency, and extensibility across modalities and scales (Zhang et al., 13 Mar 2024, Zhu et al., 13 Feb 2025, Lee et al., 2023, Liu et al., 16 Sep 2025, Zhang et al., 20 Mar 2025, Zeng et al., 30 Jun 2025, Pang et al., 8 Jul 2025, Rigas et al., 30 May 2025, Tai et al., 10 Mar 2025, Xin et al., 26 Sep 2025, Zhan et al., 19 Nov 2025).