---
title: 3D Gaussian Particles
url: https://www.emergentmind.com/topics/3d-gaussian-particles
type: topic
---

# 3D Gaussian Particles

3D Gaussian particles are parametric Gaussian primitives used to represent spatially distributed geometry, radiance, density, or latent state in three dimensions. In contemporary computer graphics, vision, and neural rendering, they usually denote anisotropic kernels with explicit position, covariance, appearance, and opacity, as in 3D Gaussian Splatting and its ray-traced, dynamic, semantic, and scientific-visualization variants [2405.03417] [2407.07090]. In a distinct stochastic-geometric lineage, “Gaussian particles” denote three-dimensional star-shaped random sets whose radial function is obtained by kernel smoothing of Gaussian noise on the sphere [1502.01750]. The modern literature therefore uses a shared term for two related but non-identical objects: explicit scene primitives for differentiable rendering and simulation, and random particles defined through Gaussian random fields.

## 1. Formal definition and parameter spaces

In the graphics and vision literature, a 3D Gaussian particle is typically an anisotropic Gaussian with mean position and covariance, augmented with appearance attributes. A common form is
$$
G(\mathbf{x})=\exp\left(-\frac{1}{2}(\mathbf{x}-\mu)^T\Sigma^{-1}(\mathbf{x}-\mu)\right),
$$
with $\mu \in \mathbb{R}^3$ and $\Sigma \in \mathbb{R}^{3\times3}$; associated attributes include color, opacity, and, in many systems, view-dependent radiance encoded by spherical harmonics coefficients [2405.03417] [2407.07090]. Several works decompose the covariance as $\Sigma = RSS^T R^T$, separating orientation and scale, and dynamic variants often keep color, opacity, and scale fixed while tracking only positions and orientations over time [2410.18912].

Projection to the image plane is commonly handled through a first-order approximation of the projected covariance,
$$
\Sigma' = J R_{cw} \Sigma R_{cw}^T J^T,
$$
or equivalently through viewing and Jacobian matrices in related formulations, after which particles are composited front to back by alpha blending [2405.03417]. A representative rendering expression is
$$
C_i = \sum_{n \leq N} c_n \alpha_n T_n,\qquad T_n=\prod_{m<n}(1-\alpha_m),
$$
with projected Gaussian opacity depending on the image-plane offset and the projected covariance [2405.03417]. This explicit representation is one reason 3D Gaussian methods are described as compatible with real-time rendering and graphics pipelines.

The parameter space is no longer limited to static geometry and color. Dynamics-oriented formulations extend the particle state with latent descriptors or time-dependent deformation parameters. For example, 3DGSim represents a particle as $g_i=(p_i,c_i,r_i,s_i,\sigma_i)$ and augments it with latent features $\tilde g_i=(p_i,f_i)$, while ParticleGS separates standard Gaussian attributes from an $m$-dimensional dynamics latent state vector $z_t$ [2503.24009] [2505.20270]. This suggests that “3D Gaussian particle” has evolved from a purely geometric primitive into a carrier of simulation state, semantics, and task-specific signals.

## 2. Rendering formulations and ray-based generalizations

Most existing methods render particles via rasterization, projecting them to screen space tiles for processing in a sorted order [2407.07090]. This splatting regime is the basis of much of the 3DGS literature, including reconstruction, novel view synthesis, and many downstream editing systems [2405.03417]. Its strengths are explicitness, differentiability, and high throughput on GPUs; its known weaknesses include approximation error in the local 3D-to-2D collapse, edge blurring, and more difficult support for arbitrary camera models or secondary rays [2405.03417] [2505.24053].

Ray-based formulations were introduced precisely to remove those constraints. “3D Gaussian Ray Tracing: Fast Tracing of Particle Scenes” builds a bounding volume hierarchy and casts a ray for each pixel using GPU ray tracing hardware, encapsulating each Gaussian in an anisotropically stretched regular icosahedron and shading batches of intersections in depth order [2407.07090]. The paper also generalizes the kernel family beyond standard Gaussians; degree-2 generalized Gaussians produce over $2\times$ fewer hits with minimal loss in accuracy, and the reported table gives 23.03 PSNR and 143/77 FPS for Gaussian kernels versus 22.68 PSNR and 277/141 FPS for generalized $(n=2)$ kernels on Truck/Deep Blending [2407.07090]. The same work reports that a hits buffer size $k=16$ is empirically optimal and emphasizes support for highly-distorted cameras, stochastic ray sampling, shadows, and reflections.

A second line of work seeks exactness rather than merely a ray-compatible proxy. 3DGEER derives a closed-form density integral along a ray traversing a 3D Gaussian and replaces screen-space association with a Particle Bounding Frustum and Bipolar Equiangular Projection. In its formulation,
$$
T(\mathbf{o},\mathbf{d})=\sigma \exp\left(-\frac{1}{2}D^2\right),\qquad
D^2=\frac{(\mathbf{o}_u \times \mathbf{d}_u)^\top(\mathbf{o}_u \times \mathbf{d}_u)}{\mathbf{d}_u^\top \mathbf{d}_u},
$$
which removes the projective approximation underlying conventional 3DGS and is reported to reach 327 FPS on RTX 4090 while improving reconstruction quality, especially under wide FoV and fisheye inputs [2505.24053]. In tomography, an analogous argument appears in “Radioactive 3D Gaussian Ray Tracing for Tomographic Reconstruction,” where the line integral through an anisotropic Gaussian is evaluated analytically for arbitrary rays rather than by affine collapse to a detector-plane Gaussian, yielding a more physically consistent forward model and explicit support for nonlinear geometric corrections such as arc-correction in PET [2602.01057].

A recurring misconception is that 3D Gaussian particles are inherently tied to splatting. The ray-tracing and exact volumetric literature shows that the same primitive can support rasterization, BVH-based ray tracing, line integration, and differentiable inverse problems, with the principal disagreement lying in how the particle contributes to a ray, not in the underlying representation itself [2407.07090] [2505.24053].

## 3. Dynamics, simulation, and temporal extrapolation

Dynamic uses of 3D Gaussian particles replace per-frame static reconstruction with explicit temporal evolution of particle attributes. Gaussian-Flow models a dynamic scene as a set of 3D Gaussian particles whose position, rotation, and color vary through a Dual-Domain Deformation Model,
$$
S(t)=S_0+D(t),\qquad
D(t)=\sum_{n=0}^{N} a_n t^n + \sum_{l=1}^{L}\left(f^l_{\sin}\sin(lt)+f^l_{\cos}\cos(lt)\right),
$$
combining polynomial fitting in the time domain with Fourier fitting in the frequency domain [2312.03431]. The reported result is a $5\times$ faster training speed compared to per-frame 3DGS modeling, while retaining real-time rendering and strong novel-view quality.

Other formulations explicitly reinterpret Gaussians as particles for learned physics. Dynamic 3D Gaussian Tracking for Graph-Based Neural Dynamics Modeling trains a particle-based dynamics model on sparse control particles downsampled from densely tracked Gaussian reconstructions, predicts their motion with a graph neural network, and interpolates dense Gaussian motion through linear blend skinning [2410.18912]. 3DGSim, by contrast, learns a simulator end to end from multi-view RGB videos: it encodes images into a 3D Gaussian particle representation, propagates dynamics via a transformer, and renders frames using 3D Gaussian splatting, with reported ability to capture rigid, elastic, and cloth-like interactions together with realistic lighting effects [2503.24009].

ParticleGS moves the dynamics into a latent ODE. It augments the standard Gaussian kernel with a dynamics latent state and evolves the global component by
$$
\frac{dz(t)}{dt}=f_\theta(z_t,t),
$$
then decodes the latent state into affine deformation parameters for each Gaussian [2505.20270]. According to the paper, this prior-free formulation achieves comparable reconstruction quality to existing approaches and significantly outperforms them in future frame extrapolation. The central claim is not just better frame fitting but more effective modeling of the Gaussian particle dynamics system itself.

Physics-based simulation has also been coupled directly to Gaussian particles. “Let it Snow!” converts static 3D Gaussians to a particle-based representation, simulates dynamic particles with the Material Point Method, and maps them back to time-dependent Gaussians,
$$
\mathcal{G}_p(t)=\{\mathbf{x}_p(t),\sigma_p(t),\mathbf{S}_p(t),\mathbf{R}_p(t),\mathbf{C}_p(t)\},
$$
with specialized collision handling for snowfall, rainfall, sandstorm, and fog [2504.05296]. “Gaussian Splashing” integrates Gaussian particles with Position-Based Dynamics for both solids and fluids, adds normals and physically-based shading parameters to each kernel, and updates covariance and normals under deformation [2401.15318]. Across these works, the particle is simultaneously a rendering primitive and a simulation carrier, rather than a mere display proxy.

## 4. Semantic labeling, editing, and robustness to weather

Once 3D Gaussian particles are explicit, they can be assigned labels and manipulated at object level. LabelGS augments each Gaussian with an object label and introduces cross-view consistent semantic masks, an Occlusion Analysis Model, a Main Gaussian Labeling model, and a Gaussian Projection Filter to lift 2D labels to 3D while avoiding label conflict [2508.19699]. Its label-aware loss operates on sets of Gaussians sharing the same label, and the paper reports a 22X speedup in training compared to Feature-3DGS at a resolution of 1440X1080, together with mIoU of 0.92–0.93 and PSNR up to 34.26 on 3D-OVS [2508.19699]. The formulation shows that scene understanding can be attached directly to the primitive level rather than only to rendered images.

Editing-oriented systems extend the same idea from labels to unseen appearance. FruitNinja fills object interiors with densely distributed opaque Gaussians and uses a pre-trained diffusion model to progressively inpaint cross-sectional views, together with voxel-grid-based smoothing, so that slicing a reconstructed object reveals synthesized interior texture in real time without additional optimization [2411.12089]. The OpaqueAtom GS strategy enforces dense, highly opaque particles to avoid the bias toward larger particles that destabilizes training and to preserve sharp color transitions. This is a different use of the Gaussian primitive from classical novel-view synthesis: the particle field is treated as a volumetric substrate for later topological change.

Adverse-weather reconstruction supplies a complementary case in which unwanted particles must be removed rather than added. WeatherGS explicitly categorizes weather artifacts into dense particles and lens occlusions, removes the former with an Atmospheric Effect Filter, extracts masks for the latter with a Lens Effect Detector, and then trains 3D Gaussians only on unoccluded regions using masked $L_1$ and D-SSIM losses [2412.18862]. The paper reports that this procedure consistently produces high-quality, clean scenes across various weather scenarios. A plausible implication is that, in outdoor reconstruction, the phrase “3D Gaussian particles” may refer both to the desired scene representation and to nuisance image phenomena that must be prevented from becoming part of that representation.

## 5. Compression, distribution, and scientific visualization

Because a 3D scene may require very many explicit particles, a major line of work treats the Gaussian set itself as the optimization target. Optimized Minimal 3D Gaussian Splatting prunes redundancy through a combination of rendering importance and local distinctiveness, then compresses geometry and appearance with hybrid neural features and sub-vector quantization [2503.16924]. The reported results are nearly 50% less storage than the previous state of the art, scene sizes of about 4 MB in the XS variant, roughly 0.4M Gaussians, and 600+ FPS rendering while maintaining high rendering quality [2503.16924]. ProtoGS instead learns Gaussian prototypes derived by K-means within structure-from-motion anchor groups and renders directly from those prototypes; the reported NeRF-Synthetic comparison gives 45K primitives and PSNR 33.45 for ProtoGS versus 261K and 33.21 for 3DGS, with average rendering speed of 225 FPS [2503.17486].

Scalability at system level has been addressed by distributing particle optimization across HPC nodes. Distributed 3D Gaussian Splatting for High-Resolution Isosurface Visualization partitions data across nodes, adds ghost cells at partition boundaries, applies background masks, trains local Gaussian models on multi-node multi-GPU systems, and merges splats for global rendering [2509.12138]. On Richtmyer–Meshkov datasets of about 106.7M Gaussians, the paper reports up to 3X speedup across 8 nodes on Polaris while preserving image quality. This work makes explicit that the “particle scene” may be far larger than a single GPU can hold.

Scientific-visualization studies generalize 3D Gaussian particles beyond view-dependent scene radiance to sparse volumetric data. One OpenVDB-based line constructs Gaussians from leaf nodes, blocks, or tiles of sparse grids and ray marches them with analytic line integration for optical-depth accumulation in OptiX [2509.11377]. Another studies level-of-detail approximations of VDB datasets using non-overlapping Gaussian particles, reporting low-, medium-, and high-LOD conversions, PSNRs in the 18–28 dB range, and rendering above 150 FPS even for multi-million Gaussian scenes [2504.04857]. These works stress transfer-function-agnostic rendering and full volumetric coverage rather than photorealistic surface appearance. They also show that Gaussian particles are not confined to image-based reconstruction; they can serve as compact surrogates for regular grids, AMR volumes, and point clouds.

## 6. Stochastic-geometry usage and conceptual boundaries

In stochastic geometry, the term has a different but mathematically precise meaning. A Gaussian random particle is a compact star-shaped set
$$
Y=\bigcup_{u\in S^2}\{o+ru: 0\le r \le X(u)\},
$$
where the radial function is generated by kernel smoothing of a Gaussian random measure on the sphere,
$$
X(u)=\int_{S^2} K(v,u)\,L(dv),\qquad u\in S^2
$$
[1502.01750]. The surface roughness of the resulting particle is controlled by the correlation decay of the field on the sphere; if the fractal index is $\alpha$, then the Hausdorff dimension of the surface is
$$
D=3-\frac{\alpha}{2}.
$$
For power kernels, the construction yields particles with boundaries of any Hausdorff dimension between 2 and 3 [1502.01750].

This stochastic-geometric usage should not be conflated with the explicit anisotropic splats of contemporary rendering systems. The former is a model of random star-shaped sets on $S^2$; the latter is a representation of scenes, volumes, or dynamic states by finite collections of learnable Gaussian kernels. What the two usages share is the idea that Gaussian structure provides a controllable balance between locality, smoothness, and analytic tractability. In current arXiv literature, that shared structure has made 3D Gaussian particles a unifying primitive for real-time rendering, inverse problems, physical simulation, semantic scene understanding, and large-scale scientific visualization [2405.03417] [2407.07090].

Source: https://www.emergentmind.com/topics/3d-gaussian-particles