Gaussian Splashing: 3DGS with Physical Simulation

• Gaussian Splashing is a framework that extends 3D Gaussian Splatting with physics-based simulation, volumetric rendering, and photogrammetry techniques.
• It integrates anisotropic Gaussian kernels with dynamic particle models to support real-time AR/VR, underwater, and satellite-based scene reconstruction.
• The approach enhances editability, photometric consistency, and computational efficiency compared to standard 3DGS in dynamic and remote-sensing applications.

Gaussian Splashing denotes several closely related frameworks that integrate 3D Gaussian Splatting (3DGS) with additional physically based modeling, volumetric rendering, or dynamic particle simulation to extend the expressiveness of explicit Gaussian-based radiance field representations. It encompasses approaches for dynamic scene rendering via cohesive solids and fluids coupling, real-time photorealistic volume rendering under scattering, and high-precision photogrammetric modeling for remote sensing. The unifying theme is the utilization of point-based anisotropic Gaussians as “splat” primitives in 3D space whose projected contributions are composited in image space, coupled with domain-specific augmentations such as physical simulation or underwater image-formation models. This entry reviews the mathematical substrates, physical and computational innovations, and major application domains for Gaussian Splashing.

1. Core Mathematical Principles

At the heart of Gaussian Splashing is the explicit representation of a scene or object as a set of $N$ anisotropic 3D Gaussian kernels (splats), parameterized as

$$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$

where $\mathbf{m}_i$ is the mean (spatial center), $\Sigma_i$ is the positive-definite covariance (axis-aligned or fully anisotropic), $\sigma_i$ is the opacity, and $\mathbf{c}_i$ a possibly view-dependent color. The volumetric density at $x \in \mathbb{R}^3$ is given by

$$\phi_i(x) = \frac{1}{(2\pi)^{3/2} |\Sigma_i|^{1/2}} \exp\left[ -\frac{1}{2}(x-\mathbf{m}_i)^\top \Sigma_i^{-1} (x-\mathbf{m}_i) \right].$$

For neural rendering, the 3D splats are projected to the camera plane, producing 2D ellipses whose densities are composited via front-to-back alpha blending: $$C(p) = \sum_{i=1}^N \omega_i(p) \mathbf{c}_i(\hat{\mathbf{r}}_i) \prod_{j=1}^{i-1} [1-\omega_j(p)],$$ where $\omega_i(p) = \sigma_i\,\phi_i^\text{2D}(p)$ is the 2D projected weight at pixel $$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$0 (Feng et al., 2024, Mualem et al., 2024). This architecture is the foundation for the extensions enabling physical simulation, underwater rendering, and remote-sensing photogrammetry.

2. Physics-Based Motion Synthesis and Rendering (Unified Particles Framework)

Gaussian Splashing (Feng et al., 2024) demonstrates dynamic scene synthesis by coherently integrating 3DGS with Position-Based Dynamics (PBD) and Position-Based Fluids (PBF):

• Each Gaussian kernel serves as a physically simulated particle (mass $$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$1, center $$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$2), coupled with associated BRDF and normal attributes.
• For solids, deformation gradients $$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$3 are used to update both the covariance $$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$4 and the aligned surface normal $$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$5:

$$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$6

• XPBD variational constraints (distance, strain for solids; incompressibility, surface-tension for fluids) drive position updates through a global minimization:

$$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$7

solved for Lagrange multipliers $$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$8 per constraint.

• Rendering of fluids uses the Gaussian Shader, accumulating fluid thickness for Beer–Lambert subsurface attenuation, yielding per-splat color

$$\mathcal{G}_i = \left( \mathbf{m}_i \in \mathbb{R}^3, \; \Sigma_i \in \mathbb{R}^{3 \times 3}, \; \sigma_i, \; \mathbf{c}_i(\hat{\mathbf{r}}) \right),$$9

where $\mathbf{m}_i$0 is the thickness-modulated diffuse term and $\mathbf{m}_i$1 a prefiltered specular lobe.

This framework enables bidirectional coupling between mesh-free dynamics and radiance field rendering, with editing operations—e.g., liquefying objects, flood simulation—affecting both geometry and appearance in real time.

3. Volumetric Underwater Rendering with Explicit Scattering

Gaussian Splashing (Mualem et al., 2024) extends 3DGS to direct volume rendering in scattering/absorbing media (notably underwater environments):

• The radiative transfer equation for water is modeled by adding absorption ($\mathbf{m}_i$2), scattering ($\mathbf{m}_i$3), and environment radiance $\mathbf{m}_i$4 in each channel:

$\mathbf{m}_i$5

with $\mathbf{m}_i$6.

• Each splat’s per-channel opacity is attenuated by water transmittance:

$\mathbf{m}_i$7

where $\mathbf{m}_i$8 is the integrated line density and $\mathbf{m}_i$9 the splat’s mean ray distance.

• Image-space compositing accumulates both direct transmission and backscatter:

$\Sigma_i$0

• Novel depth cues are extracted from color ratios:

$\Sigma_i$1

with the loss $\Sigma_i$2 ensuring photometric-geometric consistency in scattering environments.

This approach achieves high-fidelity, real-time ($\Sigma_i$3140 FPS) underwater scene reconstruction and rendering, significantly surpassing traditional NeRF variants in speed and clarity under volumetric occlusion.

4. Satellite Image Photogrammetry and Remote Sensing Adaptations

Gaussian Splashing is also used as an informal synonym for fast, accurate 3DGS-based photogrammetry in remote sensing imagery (Aira et al., 2024):

• Standard 3DGS is augmented with per-view radiometric correction $\Sigma_i$4, enabling learned affine brightness adjustment:

$\Sigma_i$5

• Shadow modeling incorporates a sun-directed camera $\Sigma_i$6 and shadow decay coefficient $\Sigma_i$7:

$\Sigma_i$8

where $\Sigma_i$9 is the elevation difference between sun-view and image-view, and $\sigma_i$0 is used to modulate pixel lighting.

• Additional regularizations improve representation: $\sigma_i$1 sparsity on opacities for structure pruning; cross-view consistency losses to ensure multi-view geometric and radiometric agreement; entropy regularization on shadow masks for binary (non-semitransparent) shadow edges.

Empirical results demonstrate that this pipeline matches the accuracy of neural field approaches at dramatically lower compute and memory costs, enabling DSM extraction and shadow-consistent albedo compositing within minutes rather than hours or days.

5. Comparison with and Distinction from Standard 3D Gaussian Splatting

Classical 3DGS is restricted to static, non-interactive scenes, with explicit limitations:

• Lacks responsive lighting (shadows, reflections) and view-dependent phenomena under general illumination (Byrski et al., 15 Mar 2025).
• Difficulties in dynamic editability and real-time physical simulation due to independent explicit parameterization of Gaussians with no intrinsic structural coupling.

Gaussian Splashing frameworks overcome these via:

• Physically based editability and simulation through cohesive mesh or particle coupling (e.g., mesh-driven editing in REdiSplats, particle-based physics in (Feng et al., 2024)).
• Domain-specific rendering equations to simulate volumetric media, such as water turbidity and backscatter (Mualem et al., 2024).
• Integration of fast, affine color/illumination adaptation and scientifically calibrated shadow models for remote sensing (Aira et al., 2024).

This results in a unified, extensible point-based scene model supporting structured scene modifications, responsive light transport, and hybrid fluid–solid–empty interplay.

6. Applications, Performance, and Future Directions

Gaussian Splashing methodologies have been empirically validated in domains including:

• Physics-aware AR/VR and VFX, supporting liquefaction, flooding, and two-way solid–fluid coupling (“Everything is water” transformations, real-time flooding, surface rippling) (Feng et al., 2024).
• Underwater photogrammetry, enabling real-time reconstruction and rendering ($\sigma_i$2140 FPS) with improved detail preservation and depth clarity compared to NeRF, even under severe scattering (Mualem et al., 2024).
• Satellite photogrammetry and DSM reconstruction, achieving state-of-the-art accuracy at $\sigma_i$3300\times$ speedup over NeRF-based Earth observation frameworks (Aira et al., 2024).

Current limitations include physical fidelity in large time-step PBD simulations, approximations in fluid refraction/scattering, reliance on static scenes, and nontrivial computational costs for extremely dense particle systems. Future work includes extending to dynamic/deformable geometry, adaptive time-stepping in coupled simulators, and tighter integration of full radiative-transfer-based light transport.

7. Principal References and Empirical Benchmarks

Variant/Domain	Reference	Application	Notable Performance
Unified particles/PBD	(Feng et al., 2024)	Dynamic AR/physics	Real-time editability, specular fluid rendering
Underwater volumetric	(Mualem et al., 2024)	Scene rendering	140 FPS, PSNR 26.8 dB, SSIM 0.88 on TableDB scenes
Satellite/RS photogram.	(Aira et al., 2024)	DSM, shadowed albedo	~1.46 m MAE DSM, $\sigma_i$43 min training

In summary, Gaussian Splashing represents an emerging synthesis of explicit point-based radiance field models with physically motivated simulation and advanced volumetric rendering, substantially widening the practical modeling, simulation, and rendering capabilities of Gaussian Splatting in graphics, vision, and remote sensing.