RadiantFoam: Rendering & Metal Foam Advances

• RadiantFoam (RF) is a dual-domain concept combining a piecewise-constant volumetric scene representation for real-time differentiable rendering and engineered metal foams for accelerator applications.
• In rendering, RF employs a Voronoi-based mesh ray tracing algorithm with explicit geometry and closed-form gradients, enabling high FPS without rasterization overhead.
• For accelerator use, RF metal foams exhibit high porosity, effective conductivity, and optimized gas evacuation, ensuring robust performance in high-heat, vacuum environments.

RadiantFoam (RF) refers to distinct yet conceptually related domains, denoting: (1) an explicit, piecewise-constant volumetric scene representation for real-time differentiable rendering via mesh-based ray tracing in computer vision and graphics; and (2) open-cell or closed-cell conducting metal foams engineered for demanding accelerator and vacuum applications. In both contexts, the name “RadiantFoam” highlights the structuring of space or material into a highly porous, polyhedral partition—whether physically realized or abstracted as a computational model.

1. Scene Representation: Voronoi Foam in Differentiable Rendering

RadiantFoam represents a scene as a closed-cell foam—a partition of $\mathbb{R}^3$ into convex polyhedra defined by a set of Voronoi sites $\mathbf{p}_1,\dots,\mathbf{p}_N\in\mathbb{R}^3$ (Govindarajan et al., 3 Feb 2025). Each cell is

$$c_i = \{\mathbf{x} \in \mathbb{R}^3 : \|\mathbf{x}-\mathbf{p}_i\| \le \|\mathbf{x}-\mathbf{p}_j\|,\, \forall j\ne i \}.$$

Every cell $c_i$ is parameterized by:

• A constant density $\sigma_i\ge0$,
• A view-dependent color model $c_i(\omega)$, implemented via low-order spherical harmonics.

Unlike 3D Gaussian Splatting, which uses overlapping anisotropic Gaussians and rasterization, RF employs disjoint, injective, and explicit geometry, enabling exact ray-cell intersections. The dual Delaunay triangulation varies combinatorially, but the Voronoi cells deform smoothly with site positions, conferring full differentiability (except on zero-measure flip events).

This structure is highly distinct from MLPs or grid-based NeRFs, offering explicit geometric boundaries and partitioning space without ambiguity or overlap. The closed-form nature of the Voronoi cell boundaries is crucial for differentiable rendering.

2. Volumetric Mesh Ray Tracing Algorithm

At the core of RF is a per-ray mesh traversal algorithm adapted from tetrahedral mesh traversal [Weiler et al. 2003], but generalized to arbitrary convex Voronoi cells (Govindarajan et al., 3 Feb 2025, Rech et al., 18 Dec 2025). The rendering proceeds as follows:

• The initial cell is determined via a nearest-site lookup.
• For each cell $i$ along the ray, the outgoing intersection with each neighbor $j\in N(i)$ is found by solving the bisector-plane equation: $$(\mathbf{p}_j-\mathbf{p}_i)\cdot (\mathbf{o} + t\mathbf{d}) = ( \|\mathbf{p}_j\|^2 - \|\mathbf{p}_i\|^2 )/2$$ yielding

$$t = \frac{ ( \|\mathbf{p}_j\|^2 - \|\mathbf{p}_i\|^2 )/2 - (\mathbf{p}_j-\mathbf{p}_i)\cdot \mathbf{o} }{ (\mathbf{p}_j-\mathbf{p}_i)\cdot \mathbf{d} }$$

for the smallest front-facing intersection ($t>0$).

• The traversal accumulates color and transmittance for the segment $[t_0, t_1)$ inside cell $i$: $$\alpha = 1 - \exp(-\sigma_i \delta), \quad C \gets C + T\cdot \alpha \cdot c_i(\omega), \quad T \gets T\cdot (1-\alpha),$$ where $\delta = t_1 - t_0$.
• Since each cell's neighborhood has small constant degree ($\sim$12 for 3D Voronoi), no acceleration structure is needed: the method scales as $O(k)$ per-ray step, yielding real-time throughput.

This piecewise-constant, segment-wise rendering admits an exact implementation of the volumetric rendering integral.

3. Mathematical Foundations and Differentiability

RF relies on analytic formulations for color, geometry, and gradient flow (Govindarajan et al., 3 Feb 2025, Rech et al., 18 Dec 2025):

• Rendering Integral: $$c_r = \int_{t_\mathrm{min}}^{t_\mathrm{max}} T(t)\, \sigma(r(t))\, c(r(t), \omega)\, dt, \quad T(t) = \exp(-\int_{t_\mathrm{min}}^{t} \sigma(r(u))\,du)$$
• Piecewise-Constant Discretization: For $N$ segments,

$$c_r = \sum_{n=1}^N T_n\cdot(1-e^{-\sigma_n \delta_n})\, c_n, \qquad T_n = \prod_{j=1}^{n-1} e^{-\sigma_j \delta_j}$$

• Gradient Flow: Because segment lengths $\delta_n$, densities $\sigma_n$, and colors $c_n$ all depend on learnable parameters, backpropagation follows the chain rule: $$\frac{\partial c_r}{\partial p_i} = \sum_n \frac{\partial c_r}{\partial \delta_n} \frac{\partial \delta_n}{\partial t_n} \frac{\partial t_n}{\partial p_i}$$ The intersection time $t_n$ depends analytically on site positions via the bisector plane, permitting closed-form gradients for differentiable optimization.
• Loss Functions: The objective combines photometric error $L_\mathrm{rgb}$ and a quantile loss $L_\mathrm{quantile}$ to focus density on surfaces: $$L = L_\mathrm{rgb} + \lambda\cdot L_\mathrm{quantile}$$
• Differentiability: No discrete discontinuities arise at Voronoi-flip events, as they occur on zero-measure sets.

4. Performance and System Characteristics

RadiantFoam is implemented in PyTorch, with custom CUDA kernels for Voronoi updates and ray traversal (Govindarajan et al., 3 Feb 2025):

• Hardware: Runs on any programmable GPU/WebGL target, with no hardware ray-tracing cores or OptiX dependency.
• Storage: $O(N)$ for sites (3 floats), densities (1 float), SH coefficients ($\sim$10 floats), neighbor lists ($\sim$12 per cell); for $N\approx 200$k, total $\lesssim 50$MB.
• Throughput: 512×512 images at 300 FPS on an RTX 4090; each ray traverses $\sim$100 cells (scene-dependent); per-step costs are a handful of dot products.
• Efficiency: By avoiding both dense per-ray sampling and MLP network queries, RF achieves real-time speeds on par with 3D Gaussian Splatting.

5. Empirical Comparison with Splatting and Ray-Traced Models

Extensive comparative evaluation on datasets like Mip-NeRF 360 and Deep Blending demonstrates the following (Govindarajan et al., 3 Feb 2025, Rech et al., 18 Dec 2025):

Method	FPS	PSNR (dB)	SSIM	LPIPS
3D Gaussian Splatting	260–300	29.3	0.88	0.20
3DGS+OptiX (ray tracing)	78–119	≈29.3	≈0.88	≈0.20
RadiantFoam	200–300	28.5	0.83	0.21

RF achieves PSNR within $0.5$ dB of splatting, with $\sim$2× higher FPS than hardware-accelerated ray tracing, and matches or exceeds ray-based Gaussian Splatting in speed and fidelity.

Qualitatively, RF supports advanced light transport (reflection, refraction), arbitrary projections, temporal phenomena (rolling shutter, motion blur), and relighting. It exhibits sharp silhouette and consistent occlusion, free of the popping artifacts typical for splatting.

6. Limitations and Evolving Directions

Major limitations of the foundational RF model include (Govindarajan et al., 3 Feb 2025, Rech et al., 18 Dec 2025):

• Voronoi equidistance constraints mean large smooth surfaces become expensive to represent, requiring numerous low-density (“empty”) cells.
• Surface extraction via post-hoc density thresholding can yield topological holes and floaters.
• Only static scenes under fixed illumination are supported in the basic model.

Addressing these, future research directions include:

• Generalizing beyond strict Voronoi structures (free-form convex cell optimization).
• Dynamic, time-varying foam parameters for non-static scenes.
• Fast foam composition for layering/editing.
• Integrating global illumination, generative priors, and interactive scene editing tools.

Recent work with SDFoam introduces a hybrid explicit-implicit approach, coupling global SDFs to Voronoi cells for improved surface regularity, mesh watertightness, and extraction speed, at comparable rendering rates and photometric fidelity (Rech et al., 18 Dec 2025).

7. RadiantFoam in Physical and Accelerator Contexts

RadiantFoam also refers to engineered metal foams used as vacuum beam-liner patches in synchrotron environments (Petracca et al., 2014). These foams exhibit:

• Porosity $\varphi_\mathrm{RF}=0.92$, pore diameter $D_\mathrm{RF}=0.8$ mm, ligament $d_\mathrm{RF}=0.12$ mm, surface area $A_s\sim 1.8\times 10^4$ m$^2$/m$^3$.
• Effective electrical conductivity $\sigma_\text{eff,RF}\approx 8.4\times 10^4$ S/m, with Drude-type frequency response and plasmonic behavior.
• Skin depth $\delta_f\approx 4\times 10^{-4}$ m, frequency-independent sheet resistance $R_f\sim 10^{-5}\Omega$.
• High gas escape probability ($f_\mathrm{RF}\sim 0.07$–$0.10$) compared to LHC slotted patches, enabling order-of-magnitude lower equilibrium $\mathrm{H}_2$ densities.
• Specific strength (1–2 MPa), low mass, and high thermal conductivity ($k_\text{eff,RF}\sim 5$ W/m·K), making them viable for high heat loads.

Drawbacks include increased inductive reactance ($X_f \propto \omega/\omega_p$), reduced bulk conductivity compared to pure copper, and sensitivity to oxide growth.

Practical recommendations target optimal placement, patch thickness, oxide minimization (e.g., Ag/Au plating), and coverage fraction to balance impedance and evacuation performance.

8. Integrative Assessment and Research Trajectory

RadiantFoam, as a computational model, revives volumetric mesh ray tracing for real-time differentiable rendering, avoiding the constraints of rasterization while achieving explicit geometry, analytic gradients, and high throughput. Its extension to hybrid models bridges the gap to reliable explicit surface reconstruction. As a material structure, it serves critical accelerator vacuum and thermal roles owing to its microstructural and multiscale conductive properties.

Unified by the spatial logic of polyhedral partitioning and optimized transmission of energy—be it radiance or gas molecules—RadiantFoam exemplifies the convergence of geometric modeling, computational physics, and high-throughput simulation, with ongoing research poised to expand its adaptability, efficiency, and application domains (Govindarajan et al., 3 Feb 2025, Petracca et al., 2014, Rech et al., 18 Dec 2025).