Texel Grid Lighting Model

• Texel grid lighting is a parameterized method for spatially-varying illumination using 2D texture and 3D voxel grids with spherical harmonics or spherical Gaussian lobes.
• It enables efficient, local control over incident radiance through closed-form mathematical models and differentiable inverse rendering techniques.
• The model enhances photorealistic rendering in applications like indoor scene synthesis and facial appearance capture by disentangling lighting from albedo.

The texel grid lighting model is a parameterized representation for spatially-varying illumination, encompassing both physical and non-physical lighting effects, and supporting differentiable inverse rendering for both 2D texture domains and 3D spatial volumes. Modern formulations exploit spherical harmonics (SH) or spherical Gaussian (SG) lobes per grid cell, tessellating texture or volumetric domains to enable efficient, local control over incident radiance, critical for high-fidelity appearance modeling and physically plausible image synthesis in both indoor scenes (Wang et al., 2021) and facial appearance capture (Han et al., 12 Dec 2025). Closed-form mathematical models for texel-wise lighting estimation on distant environment maps provide computational guarantees and accuracy bounds (Alnasser et al., 2017).

1. Foundational Representations

Texel grid lighting is instantiated as either a 2D grid over UV-texture space or a 3D axis-aligned voxel grid covering scene geometry. The 2D case, as in "WildCap" (Han et al., 12 Dec 2025), overlays a regular grid (resolution $(H/g) \times (W/g)$) on UV-unwrapped surfaces, where each grid cell (“texel-light”) stores a vector of SH coefficients (typically order-2, 27 components). In 3D, the volumetric SG model (Wang et al., 2021) partitions space into $X \times Y \times Z$ voxels, storing per-voxel descriptors $L_v = (\alpha_v, c_v, \mu_v, \sigma_v)$, where $\alpha_v$ is density/opacity, $c_v$ is RGB amplitude of a single SG lobe, $\mu_v$ unit direction, and $\sigma_v$ bandwidth. This yields a 4D lighting tensor $L \in \mathbb{R}^{8 \times X \times Y \times Z}$.

Furthermore, environment-map-based closed-form texel grid models (Alnasser et al., 2017) discretize distant lighting onto texel patches of cube-map faces, treating each as a spatially local constant radiance patch. This provides a mathematically precise area-integral interpretation, facilitating rapid per-patch evaluation and analytic error bounds.

2. Lighting Parameterization and Interpolation

Each texel in the grid possesses a set of lighting coefficients, typically corresponding to SH or SG basis functions. In 2D, for a texel at $(u,v)$, SH coefficients are queried via bilinear interpolation of the four surrounding grid nodes: $$\gamma^V(u,v) = (1-\alpha)(1-\beta) V[p,q] + \alpha(1-\beta) V[p+1,q] + (1-\alpha)\beta V[p,q+1] + \alpha\beta V[p+1,q+1]$$ which, combined with a global SH component $\gamma^g$ and a binary mask $M(u,v)$ for shadow regions, gives the local lighting coefficients: $$\gamma(u,v) = \gamma^g + M(u,v) \cdot \gamma^V(u,v)$$ In 3D, volumetric ray-marching is performed: at each sampling point, $(\alpha_k, c_k, \mu_k, \sigma_k)$ are fetched; transmittance $T_k$ accumulates opacity effects; and the composite radiance is

$$L(x, \omega) \approx \sum_{k=1}^N w_k \cdot G(\omega; c_k, \mu_k, \sigma_k)$$

where $G$ is the SG response, and $w_k = T_k \cdot \alpha_k$ (Wang et al., 2021).

3. Integration in Inverse Rendering Workflows

Texel grid lighting models are central to modern inverse rendering pipelines for disentangling albedo, normal, depth, and illumination. In the 3D SG framework (Wang et al., 2021), a 3D CNN predicts all lighting parameters jointly. The renderer applies a physically motivated, energy-conserving image formation process via per-pixel Lambertian integrals: $$\hat{I}_p = \phi \left( \sum_{j=1}^K \frac{A(x_p)}{\pi} L(x_p, l_j) \max(n_p \cdot l_j, 0) \Delta \Omega \right)$$ where $\phi(\cdot)$ enforces HDR-to-LDR mapping. All parameters are optimized end-to-end via differentiable photometric losses including albedo $L_\text{albedo}$, normal $L_\text{norm}$, depth $L_\text{depth}$, lighting $L_\text{light}$, visibility $L_\text{visible}$, entropy $L_\text{reg}$, and physics-based reprojection $L_\text{rerender}$.

In facial appearance capture (Han et al., 12 Dec 2025), texel grid lighting enables robust disentanglement of network-baked shadow artifacts from clean skin albedo, when integrated into a hybrid data-driven/model-based pipeline that incorporates a diffusion prior for real skin reflectance (via DDPM) and joint optimization of lighting coefficients. The photometric loss

$$L_\text{pho}(A, \theta) = \| I_{UV} - \Gamma_\theta[A, N_c] \|_2^2$$

is minimized alongside regularization terms ($L_{TV}, L_{neg}$) to encourage spatially smooth, non-negative local shading offsets restricted to predetermined mask regions.

4. Mathematical Principles of Texel Lighting Evaluation

Physically based texel grid lighting models harness closed-form solutions for the rendering equation, particularly under distant lighting (Alnasser et al., 2017). For Lambertian BRDFs, the shaded radiance at $p$ is expressed as

$$L_o(p) = k_d I/d^2 \int_A \max(0, n \cdot \omega_i) (N_a \cdot \omega_i) dA$$

where the area integral over each texel is computed via analytic polynomial monomial integration, yielding $O(1)$ cost. Phong-like BRDFs introduce $O(\text{sh}^2)$ complexity per patch. Summation over all texel patches for cube maps yields efficient, noise-free lighting synthesis, with accuracy governed by texel size and spatial frequency of incident illumination. Texel area-integrals inherently recover the true solid angle subtended by each texel, unlike discrete SH or Monte Carlo schemes.

5. Handling Non-Physical Effects and Disentanglement

Texel grid models explicitly support explanation of non-physical effects such as shadow baking. In applications like "WildCap" (Han et al., 12 Dec 2025), network outputs often contain high-frequency dark patches (“baked-in shadows”) which cannot be physically reproduced by global SH or environment maps. By restricting local SH offsets to regions flagged in mask $M(u,v)$ and regularizing to only darken, texel grid lighting attributes such artifacts to illumination rather than reflectance, preserving clean albedo estimates. Diffusion-guided posterior sampling enforces realistic reflectance distributions and effectively resolves the scale ambiguity between lighting and albedo.

A plausible implication is that local texel grid lighting increases the robustness and physical correctness of appearance modeling, even in unconstrained data-driven pipelines.

6. Comparative Analysis vs. Alternative Lighting Strategies

The texel grid approach occupies a middle ground between 2D lighting summaries and fully general neural radiance fields:

Representation	Spatial Variation	Angular Frequency	Occlusion/Parallax	HDR/Reflectance Decoupling	Memory/Cost
Environment Map (EM)	None	Low–Varied	No	No	Minimal/O(HW)
Spherical Harmonics (SH)	Per-pixel/patch	Low–Limited	No	No	Moderate/O(HWL²)
2D Spatially–Varying SG Lobes	Plane-tied	Moderate	No	Partial	Per-pixel SG
NeRF-style RGB+α Volumes	Volumetric	View-dependent	Yes	Hard to recover HDR	High (MLPs, features)
Volumetric Spherical Gaussians (SG)	Volumetric (3D grid)	High (Sharp SGs)	Yes	Decoupled SG amplitude	Large/O(XYZ8)

Texel grid lighting provides full 3D spatial variation, high-frequency angular detail, and physically motivated rendering, but at increased memory and computational cost (O(XYZ8) floats; N≈128 ray samples per pixel; differentiable 3D CNNs). Environment maps and SH models are compact but suffer from limitations in spatial variation, occlusion handling, and directional resolution.

7. Computational Complexity and Error Analysis

Closed-form area-integral texel lighting, as detailed by Alnasser & Foroosh (Alnasser et al., 2017), avoids Monte Carlo noise and outperforms spherical harmonics with respect to high-frequency reproduction for Lambertian surfaces. For Lambertian texels, evaluation cost is $O(1)$ per patch. For Phong-like BRDFs, cost increases to $O(\text{sh}^2)$ due to polynomial expansion. Empirical results report peak signal-to-noise ratios exceeding $33$ dB at practical texel size-to-distance ratios, often surpassing $40$ dB under typical parameterization.

Spatial resolution ($g$ in 2D grids; voxel density in 3D) directly trades off between local artifact removal and smoothing. In (Han et al., 12 Dec 2025), a grid size $g=96$ (approximately $10 \times 10$) gives optimal balance in facial appearance pipelines.

8. Significance and Applications

The texel grid lighting model is integral for end-to-end differentiable rendering pipelines that demand joint estimation of reflectance, geometry, and spatially-varying illumination from images or video. Key applications include inverse rendering in complex indoor scenes (Wang et al., 2021) and unconstrained facial appearance capture (Han et al., 12 Dec 2025), enabling photorealistic object insertion and robust recovery of skin reflectance in uncontrolled settings. The methodology also provides exact area-based lighting evaluation for distant illumination on environment maps (Alnasser et al., 2017).

This suggests that texel grid lighting, via volumetric SG or local SH parametrizations, provides an extensible and physically informed substrate for future research across neural rendering, appearance capture, and graphics.