Radiance Meshes in GPU-Accelerated Rendering

• Radiance meshes are volumetric representations built via Delaunay tetrahedralization that enable closed-form volume rendering.
• They employ dynamic mesh maintenance and neural field queries to continuously optimize geometry and appearance.
• The approach ensures efficient GPU rendering with superior memory scaling compared to Voronoi-based or fixed-grid schemes.

Radiance meshes are volumetric representations in which space is partitioned into constant-density tetrahedral cells using a 3D Delaunay tetrahedralization. This construction enables a closed-form evaluation of the volume rendering equation and high-performance rendering via standard GPU pipelines. The Delaunay framework offers advantages over Voronoi-based or fixed-grid schemes in terms of memory scaling and hardware compatibility, and forms the core of recent differentiable radiance field approaches (Mai et al., 3 Dec 2025).

1. Mathematical Foundations of Delaunay Tetrahedralization

A radiance mesh is derived from the Delaunay tetrahedralization of a finite 3D point set $V = \{v_1, \ldots, v_N\} \subset \mathbb{R}^3$, producing a partition $T$ of $\operatorname{conv}(V)$ into non-overlapping tetrahedra with all vertices in $V$. The Delaunay tetrahedralization is the unique such partition (under general position) satisfying the empty circumsphere condition: for every simplex $t = (v_i,v_j,v_k,v_l) \in T$, no other site $v_m \in V \setminus \{v_i,v_j,v_k,v_l\}$ satisfies $\|v_m - O(t)\|_2 < R(t)$, where $O(t)$ and $R(t)$ are the center and radius of the circumsphere through $\{v_i,v_j,v_k,v_l\}$. The Delaunay tetrahedralization is dual to the Voronoi diagram and ensures each cell is a tetrahedron with four exactly triangular faces, which are ideal primitives for GPU rasterization (Mai et al., 3 Dec 2025).

2. Construction and Dynamic Maintenance

Radiance meshes are dynamically maintained by recomputing the Delaunay tetrahedralization of an evolving point set every fixed number of optimization steps. This is typically performed using a robust incremental insertion algorithm (Bowyer–Watson) or flipping-based approaches (Pachner moves/3D edge-flips), as implemented in libraries such as CGAL or TetGen. Small perturbations of vertex positions can induce local mesh updates (flips) to restore the empty-circumsphere criterion. The tetrahedralization is periodically rebuilt to match the optimized vertex configuration while preserving valid mesh topology (Mai et al., 3 Dec 2025, Binninger et al., 7 May 2025).

3. Differentiable Field Representation and Optimization

A fundamental challenge in radiance mesh optimization is preserving the continuity of the radiance field as vertex positions change and the intrinsic mesh connectivity may flip. Rather than directly attaching radiance (density, color) parameters to vertices—since such assignments can exhibit discontinuities under edge flips—the radiance mesh paradigm attaches these attributes to the tetrahedra themselves. Each cell’s appearance parameters are queried at the centroid of each tetrahedron via a neural network (e.g., hash-grid neural field, as in Instant-NGP).

To further mitigate topological discontinuities at flip events, a circumcenter smoothing strategy is employed: densities and colors are interpolated via a Zip-NeRF-style Gaussian downweighting, using the circumsphere radius to ensure smooth transitions across mesh updates. Gradients flow through the neural field query location (generally the centroid) to the vertex positions, enabling end-to-end differentiable optimization of both geometry and appearance (Mai et al., 3 Dec 2025).

4. Hardware-Accelerated Closed-Form Volume Rendering

Radiance meshes provide a pathway for GPU-accelerated, exact volume rendering. For each tetrahedron $t$, its intersection with a camera ray $r(t) = o + t d$ is determined via Cyrus–Beck plane clipping, yielding entry $t_\mathrm{in}$ and exit $t_\mathrm{out}$. Given a constant cell density $\sigma_t$ and a linearly interpolated color $c_t(r(t))$, the emission-only volume rendering integral is

$$\Delta C_t = \int_{t_\mathrm{in}}^{t_\mathrm{out}} c_t(r(t)) \sigma_t \exp(-\int_{t_\mathrm{in}}^t \sigma_t ds) dt,$$

which, under constant $\sigma_t$, admits a closed-form solution involving only the color at entry/exit, $d_t = \sigma_t (t_\mathrm{out} - t_\mathrm{in})$, and $\alpha_t = 1 - e^{-d_t}$. The final color contribution is

$$\Delta C_t = (1-\alpha_t/d_t)\,c_t^\mathrm{in} + (\alpha_t/d_t - e^{-d_t})\,c_t^\mathrm{out}.$$

This compact evaluation is implemented entirely in fragment shaders after tetrahedron data is emitted via mesh shaders, enabling real-time view synthesis at $>200$ FPS for million-cell meshes on standard GPUs (Mai et al., 3 Dec 2025).

5. Vertex Optimization, Topological Events, and Regularization

The set of radiance mesh vertices $\{v_i\}$ are direct optimization parameters. As these are perturbed during learning, the Delaunay structure is periodically recomputed, resulting in discrete connectivity changes (Pachner flips) when the empty circumsphere property is violated. Because field parameters are attached and interpolated at the tetrahedron level, continuity in appearance is preserved even as topology changes.

The location at which the field network $G_\theta$ is queried is important: querying at circumcenters ensures value continuity across flips (as shared circumcenters coincide at flip events) but is numerically ill-conditioned for skinny tets. The solution in (Mai et al., 3 Dec 2025) is to query at centroids for stability, downweighting poor-quality cells via a Zip-NeRF Gaussian, with signal regularization implemented via the circumsphere radius. This enables robust autodiff optimization and prevents instability as the mesh adapts.

6. Performance, Scalability, and Hardware Suitability

The radiance mesh approach achieves low memory overhead (linear in the number of points/cells) and supports large-scale meshes (tested at $1.2$ million tets with $\approx$19 ms render passes). Tetrahedral meshes are particularly well suited for rasterization and ray tracing engines: each cell requires only four triangles and can be sorted in front-to-back order via circumsphere-based “power” values, enabling efficient compositing.

Compared with alternative methods—such as Voronoi-based schemes, which require significantly greater tessellation to produce triangle faces, or fixed voxel grids, which have cubic memory scaling—the tetrahedral decomposition achieves superior memory efficiency and exact hardware compatibility with mesh shading pipelines (Mai et al., 3 Dec 2025, Binninger et al., 7 May 2025).

7. Broader Applicability, Extensions, and Limitations

Radiance meshes have been used for real-time volumetric rendering, differentiable graphics, view synthesis, and simulation where exact integral evaluation, mesh extraction, and efficient memory usage are critical. Their compatibility with mesh extraction, editing, and hardware rasterization supports a range of physics-based and vision applications.

However, Delaunay-based schemes are typically limited by the potential for mesh degeneracies (skinny, sliver tetrahedra), especially in non-uniform point sets, motivating regularization and filtering strategies. Furthermore, the static partitioning yields limited adaptivity, and the recomputation cost for large point sets may be nontrivial, although amortized by remeshing every set number of iterations. State-of-the-art libraries and GPU kernels (TetGen, HXT, CGAL) implement these algorithms at scales ranging from tens of thousands to billions of tetrahedra, enabling applications in geometry processing, scientific simulation, and differentiable rendering (Mai et al., 3 Dec 2025, Binninger et al., 7 May 2025, Marot et al., 2020, Marot et al., 2018).