Gaussian Ray Tracing Techniques

• Gaussian Ray Tracing is a technique that models ray propagation and wave phenomena using 2D/3D Gaussian primitives with precise statistical and mathematical formulations.
• It employs closed-form integrals and optimized acceleration structures to compute ray-primitive intersections, shading, and volumetric effects efficiently.
• The framework underpins state-of-the-art applications in real-time global illumination, differentiable rendering, and advanced computational imaging on modern GPUs.

Gaussian Ray Tracing refers to a class of algorithms and mathematical frameworks involving the propagation of rays through ensembles of Gaussian primitives. These include both high-frequency wave propagation using Gaussian-beam summations in wave equations and explicit modeling/rendering of radiance fields, volumes, or surfaces as mixtures of 2D or 3D Gaussian kernels. Gaussian Ray Tracing unifies the geometric optics of rays with wave-based or particle-based formalisms, and underlies multiple state-of-the-art techniques in computer graphics, computational imaging, and wave physics.

1. Mathematical Foundations and Gaussian Primitives

Gaussian Ray Tracing involves modeling scenes, media, or wavefields with primitives that are either 2D/3D Gaussian density functions or collections of rays associated with Gaussian beams. The most canonical 3D Gaussian primitive is specified by a mean $\mu \in \mathbb{R}^3$ and a positive-definite covariance matrix $\Sigma \in \mathbb{R}^{3 \times 3}$, defining the density:

$$G(x) = \exp\left(-\tfrac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)$$

For ray tracing, such primitives are augmented by per-component color, opacity, surface normal, BRDF parameters, or higher-order coefficients as in 3DGS and related representations (Gao et al., 2023, Moenne-Loccoz et al., 9 Jul 2024, Byrski et al., 31 Jan 2025).

In wave propagation, families of rays parameterized by Gaussian beams or beamlets represent high-frequency solutions to variable-coefficient PDEs, such as the wave, Helmholtz, or elastic equations. The ray trajectory $x(t)$, quadratic phase ansatz, and complex amplitude prefactor are computed via Hamiltonian dynamics and Riccati-type ODEs (Waters, 2015, Huang, 2020).

In volumetric modeling, mixtures of 3D Gaussians define the extinction or emission coefficients of participating media, and their properties yield closed-form integrals for transmittance and path sampling (Condor et al., 24 May 2024).

2. Ray–Primitive Intersection and Accumulation

Given a camera or light ray $r(t) = r_o + t\,r_d$, the intersection with a Gaussian ellipsoid is found by the point of maximal probability density along the ray:

$$t^* = \frac{(\mu - r_o)^T\Sigma r_d}{r_d^T\Sigma r_d}$$

This $t^*$ locates the argmax of $G(r(t))$ along the ray (Gao et al., 2023, Moenne-Loccoz et al., 9 Jul 2024, Byrski et al., 31 Jan 2025). In the case of mesh or polytope proxies, a ray–triangle intersection yields a set of candidate hits later weighted by the Gaussian kernel at those locations.

For volumetric path tracing, the optical depth contribution from a Gaussian along a segment is integrated:

$$\tau_i(t_0, t_1) = \sigma_i\int_{t_0}^{t_1} K_i(r(t))\,dt$$

with $K_i$ the (unnormalized) Gaussian kernel; these integrals are given in closed-form via the error function erf (Condor et al., 24 May 2024). The composited color or radiance along each ray is computed by depth-order alpha compositing with transmittance:

$$C = \sum_{i=1}^N T_i\,\alpha_i\,c_i, \qquad T_i = \prod_{j=1}^{i-1}(1-\alpha_j)$$

where $\alpha_i$ is the effective opacity at the hit and $c_i$ is the color or radiance (Gao et al., 2023, Moenne-Loccoz et al., 9 Jul 2024, Byrski et al., 31 Jan 2025).

For high-density clouds or volumes, stochastic (Monte Carlo) sampling may be employed: rather than process all intersections, a stochastic acceptance per-ray traverses the acceleration structure once, randomly accepting or rejecting each hit as a Bernoulli trial weighted by $\alpha$. This yields an unbiased estimator for the integral and dramatically reduces register pressure (Sun et al., 9 Apr 2025, Hu et al., 23 Mar 2025).

3. Acceleration Structures and Proxy Geometry

Efficient Gaussian ray tracing at scale requires high-performance acceleration structures. Common approaches include:

• BVH over Gaussians: Each Gaussian is bounded by a tight axis-aligned bounding box (AABB) or a small triangle mesh proxy (e.g., an icosahedron at $\sim1\%$ peak density isoline) (Moenne-Loccoz et al., 9 Jul 2024, Byrski et al., 31 Jan 2025).
• OptiX or Embree Ray Tracing: Hardware-accelerated kernels handle ray–triangle intersection and any-hit/closest-hit logic, allowing millions of proxies in real time.
• k-buffer: Per-ray small caches of $k$ closest hits, sorted in depth, batch intersection and compositing to amortize kernel overhead (Moenne-Loccoz et al., 9 Jul 2024, Gu et al., 20 Dec 2024).

Bounding shell optimizations, such as culling Gaussians beyond $\pm3\sigma$ from the mean in their local frame, reduce the number of intersection tests without perceptible error (Condor et al., 24 May 2024). Mesh-based or “flat Gaussian” approximations allow direct manual editing and compatibility with triangle-based graphics pipelines (Byrski et al., 15 Mar 2025).

4. Physically Based and Differentiable Shading

Gaussian Ray Tracing pipelines often integrate full physically based rendering (PBR) formalisms at the primitive level. Each Gaussian is attributed with normals, Disney-style BRDF parameters (base color, roughness, metallic), and view-dependent color in a spherical harmonics basis (Gao et al., 2023). The rendering equation is evaluated per Gaussian:

$$c'_a(\omega_o) = \sum_{i=1}^{N_s} \bigl[f_d + f_s(\omega_o,\omega_i)\bigr] L_i(\omega_i) (\omega_i \cdot n_a) \Delta\omega_i$$

where $f_d$ and $f_s$ denote diffuse/specular terms respectively (Gao et al., 2023, Wu et al., 2 Apr 2025).

Visibility, indirect light, and shadow terms are handled with either explicit, baked visibilities (precomputed via BVH traversal and supervised into learnable SH parameters), or via on-the-fly multi-bounce Monte Carlo ray tracing—either globally (3D ray tracing with compound stochastic acceptance) or in screen-space proxies for real-time effects (Hu et al., 23 Mar 2025, Wu et al., 2 Apr 2025, Gu et al., 20 Dec 2024).

These pipelines are increasingly differentiable end-to-end: all operations on Gaussian parameters (mean, scale, rotation, color, opacity) are analytic with respect to the primitive attributes and rendering expressions, allowing full gradient-based inverse rendering or scene optimization workflows (Gao et al., 2023, Gu et al., 20 Dec 2024, Kohyama et al., 21 Dec 2025).

5. Applications and Advanced Effects

Gaussian Ray Tracing methods underpin numerous applications:

• Novel-view synthesis and scene relighting: Differentiable, point-based representations support photorealistic rendering with dynamic relighting and editing (Gao et al., 2023, Moenne-Loccoz et al., 9 Jul 2024, Wu et al., 2 Apr 2025).
• Soft/Hard Shadows, Reflections, Refraction, Depth of Field, Rolling Shutter, Fisheye: Secondary camera and light effects are handled by tracing arbitrary rays through Gaussian proxies. Time and spatial distortion are incorporated into the spatio-temporal Gaussian definition (4D-GRT) (Liu et al., 13 Sep 2025, Wu et al., 2 Apr 2025).
• Volumetric Scattering and Emission: Mixtures of 3D Gaussians model participating media, where the closed-form attenuation, emission, and scattering integrals allow efficient path tracing and inverse rendering (Condor et al., 24 May 2024, Au et al., 2023).
• Event-based Rendering: Ray tracing through sets of Gaussians at high temporal resolution enables depth and radiance estimation from event camera streams (Kohyama et al., 21 Dec 2025).
• Global Illumination: Two-bounce and multi-bounce stochastic schemes with radiance caching, screen-space tracing, and hybrid rasterization approaches provide real-time global illumination with Gaussians and meshes (Hu et al., 23 Mar 2025, Liu et al., 9 Dec 2025).
• Structured Gaussian Beams in Wave Physics: In wave optics, ray families parameterized as on the Poincaré sphere generate structured Gaussian beams, and their reconstruction bypasses explicit diffraction calculations while capturing caustics and high-frequency propagation (Alonso et al., 2016, Waters, 2015, Huang, 2020).

6. Performance, Variants, and Limitations

Recent works have demonstrated real-time to interactive framerates for complex 3D Gaussian scenes using RTX/GPU hardware, with scaling to millions of primitives. For example, 3D Gaussian ray tracing at $800 \times 800$ achieves 55–190 FPS depending on kernel and scene (Moenne-Loccoz et al., 9 Jul 2024). Stochastic single-pass algorithms are 4–8× faster than all-hit approaches and avoid memory blowup, albeit with controllable Monte Carlo noise (Sun et al., 9 Apr 2025).

Hybrid approaches such as “reflection-baked” tracing combine per-primitive ray-baked coefficients with fast 2D splatting, offering $\times$7 speedups with little quality loss in highly reflective scenes (Liu et al., 9 Dec 2025). Alternative kernels, e.g., generalized Gaussian or Epanechnikov, offer speed/memory tradeoffs at the cost of smoothness at boundaries (Condor et al., 24 May 2024).

Some limitations remain: (i) For very dense or highly overlapping scenes, the number of contributing Gaussians per ray can grow; k-buffer and stochastic methods mitigate but do not eliminate this. (ii) In wave beam applications, caustics may necessitate special treatment or higher-order reconstructions (Waters, 2015, Alonso et al., 2016). (iii) On resource-limited hardware, acceleration via mesh proxies or single-pass traversal is required (Sun et al., 9 Apr 2025).

7. Historical and Theoretical Context

The theoretical origins of Gaussian Ray Tracing trace to the development of Gaussian-beam approximations in wave and quantum mechanics, ray-based beam summation in optics (e.g., structured beams on the Poincaré sphere (Alonso et al., 2016)), and multi-scale volumetric transport in radiative transfer. In practical graphics, 3D Gaussian Splatting and its ray-traced extensions (3DGRT, RaySplats, 4D-GRT) have established point-based, differentiable pipelines as viable alternatives to mesh and voxel-based approaches (Moenne-Loccoz et al., 9 Jul 2024, Liu et al., 13 Sep 2025, Byrski et al., 31 Jan 2025). Hybrid rendering systems now combine mesh-like editability, real-time ray-tracing effects, and wave-based physical fidelity in a unified, mathematically principled framework.

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References:

• "Relightable 3D Gaussians: Realistic Point Cloud Relighting with BRDF Decomposition and Ray Tracing" (Gao et al., 2023)
• "3D Gaussian Ray Tracing: Fast Tracing of Particle Scenes" (Moenne-Loccoz et al., 9 Jul 2024)
• "3D Gaussian Inverse Rendering with Approximated Global Illumination" (Wu et al., 2 Apr 2025)
• "Real-time Global Illumination for Dynamic 3D Gaussian Scenes" (Hu et al., 23 Mar 2025)
• "Stochastic Ray Tracing of Transparent 3D Gaussians" (Sun et al., 9 Apr 2025)
• "RaySplats: Ray Tracing based Gaussian Splatting" (Byrski et al., 31 Jan 2025)
• "Don't Splat your Gaussians: Volumetric Ray-Traced Primitives for Modeling and Rendering Scattering and Emissive Media" (Condor et al., 24 May 2024)
• "A ray-optical Poincaré sphere for structured Gaussian beams" (Alonso et al., 2016)
• "Low Regularity Ray Tracing for Wave Equations with Gaussian beams" (Waters, 2015)
• "Eulerian Gaussian beams for high frequency wave propagation in inhomogeneous media of arbitrary anisotropy" (Huang, 2020)
• "HybridSplat: Fast Reflection-baked Gaussian Tracing using Hybrid Splatting" (Liu et al., 9 Dec 2025)
• "IRGS: Inter-Reflective Gaussian Splatting with 2D Gaussian Ray Tracing" (Gu et al., 20 Dec 2024)
• "Geometric-Photometric Event-based 3D Gaussian Ray Tracing" (Kohyama et al., 21 Dec 2025)
• "Every Camera Effect, Every Time, All at Once: 4D Gaussian Ray Tracing for Physics-based Camera Effect Data Generation" (Liu et al., 13 Sep 2025)
• "REdiSplats: Ray Tracing for Editable Gaussian Splatting" (Byrski et al., 15 Mar 2025)