Differentiable Partitioning for Directional Appearance

• Differentiable partitioning for directional appearance is a framework that partitions spatial or angular domains to enable efficient gradient-based optimization for view-dependent effects.
• It integrates geometric techniques like self-occluding meshes and adaptive Spherical Voronoi formulations to model both physical occlusion and digital radiance with high fidelity.
• Combining ML optimization, regularization, and explicit fabrication constraints, these approaches offer stable, parameter-efficient control over appearance across diverse rendering pipelines.

Differentiable partitioning for directional appearance encompasses analytic algorithms, optimization techniques, and fabrication strategies for controlling the view-dependent radiance of surfaces and 3D representations. The principal goal is to partition either a spatial or angular domain (e.g., a mesh or the sphere of directions) in a manner that admits efficient gradient computation, enabling learning-based or direct optimization pipelines. Two major approaches are described in recent literature: (1) geometric partitioning of surfaces for fabrication with controlled occlusion and (2) adaptive partition of the spherical domain for appearance modeling in radiance fields. These frameworks offer explicit, stable, and parameter-efficient means for manipulating visual appearance as a function of directional input, achieving demonstrable results in both physically printed objects and photorealistic digital renderers (Perroni-Scharf et al., 2023, Sario et al., 16 Dec 2025).

1. Surface Partitioning through Self-Occluding Meshes

A physical surface with controlled, view-dependent appearance is constructed by discretizing its geometry as a two-dimensional grid of axis-aligned bars. Each bar is assigned a constant color $c_i\in\mathbb{R}^3$ and an extrusion height $h_i\ge0$, forming a 3D heightfield. These ridges are engineered so that their mutual occlusion varies depending on the viewing direction. The analytic visibility of each bar from a given direction allows one to sculpt the effective appearance by optimizing the bar heights and colors. This representation partitions the surface into one-dimensional segments whose collective occlusion profile determines the color visible along each camera ray (Perroni-Scharf et al., 2023).

2. Differentiable Rendering Formulations

The rendering operation for a self-occluding mesh employs a smooth analytical model. Given a set of orthographic camera directions $\mathbf{d}^{(k)}$, each pixel (ray) is mapped to a slice of bars with known widths $w_i$, heights $h_i$, and colors $c_i$. For each slice, boundary heights $y_i$ are computed according to: $y_i = h_i - \frac{d_x}{d_z}\sum_{j=0}^{i-1} w_j$ where $w_{n+1}=0$ and $h_{n+1}=-\infty$.

The pixel color is determined by a smooth Heaviside approximation $g(u)\approx H(u)$: $I = \sum_{i=0}^n c_i [g(o - y_i) - g(o - y_{i+1})]$ where $o$ is the ray’s height at the origin. Gradients with respect to bar color and height are given by: $$\frac{\partial I}{\partial c_j} = g(o-y_j) - g(o-y_{j+1})$$

$$\frac{\partial I}{\partial h_j} = -(c_j - c_{j-1})\,g'(o-y_j)$$

All pipeline components are differentiable, enabling end-to-end gradient-based optimization (Perroni-Scharf et al., 2023).

3. Partitioning the Spherical Domain: Spherical Voronoi Formulation

For explicit radiance field models such as 3D Gaussian Splatting (3DGS), directional appearance is encoded via a differentiable partition of the unit sphere $S^2$. Spherical Voronoi (SV) splits $S^2$ into $K$ learnable cells with centers $\{w_i\}$: $$C_i = \{\,d\in S^2 : i = \arg\min_j\,\arccos(d^T w_j)\,\}$$

To admit differentiation, the SV model replaces hard assignment with softmax-based weights: $$s_i(d) = \frac{\exp(-\tau\,\arccos(d^T w_i))}{\sum_{j=1}^K \exp(-\tau\,\arccos(d^T w_j))}$$ where $\tau$ controls boundary sharpness: small $\tau$ yields overlapping regions, large $\tau$ approaches classical Voronoi. The appearance model is a weighted sum: $R(d) = \sum_{i=1}^K s_i(d)\,a_i(d)$ with per-cell parameters $a_i(d)$ representing diffuse or specular appearance, adapted for view-dependent effects (Sario et al., 16 Dec 2025).

4. ML Optimization Strategies and Regularization

Surface mesh approaches minimize a composite loss: $$L(H,C) = \frac{1}{K\,m^2}\sum_{k=1}^K\sum_{x,y} \|I_k(x,y;H,C) - D_k(x,y)\|^2 +\lambda_{\rm barrier}\,L_{\rm barrier}(H) +\lambda_{\rm smooth}\,L_{\rm smooth}(H)$$ Barrier and smoothing terms enforce physically meaningful heights, prevent spikes, and guarantee minimum thickness. A coarse-to-fine subdivision scheme initiates with a low-resolution grid (e.g., $8\times8$) and recursively refines to higher resolutions. Block coordinate descent alternates updates between heights and colors. Optional simulated annealing introduces stochastic perturbations to escape local minima. Fabrication constraints, including minimum bar width and smoothness, are differentiably incorporated (Perroni-Scharf et al., 2023).

In the SV/3DGS pipeline, all geometry and appearance parameters—including SV site locations, temperatures, and probe coefficients—are optimized jointly through photometric reconstruction loss: $$\mathcal{L}_{\rm photometric} = \sum_{u,v} \|C_{\rm render}(u,v) - C_{\rm gt}(u,v)\|_2^2$$ Regularization on densities and covariances of Gaussians avoids degenerate solutions. The SV softmax structure guarantees stable gradients and obviates the need for explicit entropy or smoothness penalties on partition weights (Sario et al., 16 Dec 2025).

5. Appearance Modeling: Diffuse and Reflective Effects

In self-occluding mesh fabrication, view-dependent appearance is achieved by modulating bar visibility and color across multiple directions, enabling a surface to display up to five distinct target images with MSE < 0.02 RGB. Heights and colors are chosen such that each direction preferentially reveals a specified subset of bars (Perroni-Scharf et al., 2023).

In the SV radiance pipeline, directional appearance is captured via per-cell diffuse color or reflection probe functions. For reflections, deferred shading is employed: Gaussians are rasterized to G-buffers, after which specular terms are assembled from far-field cubemaps and near-field kNN probes. The roughness parameter modulates the SV temperature, interpolating boundary sharpness: $\tau = (1-R)\,\tau_{\max} + R\,\tau_{\min}$ This strategy allows SV-based models to capture specular highlights, outperforming Spherical Harmonics and Gaussian mixtures, with reported improvements in PSNR of 0.2–0.4 dB across standard datasets (Sario et al., 16 Dec 2025).

6. Fabrication and Real-Time Rendering Workflows

For printable surfaces, optimized heightfields are exported as explicit mesh models—typically one rectangular prism per bar—with assigned heights and colors. UV-curing 3D printers (e.g., Stratasys J55) are employed, achieving bar spacing of 300 dpi. The resultant physical artifacts closely match differentiable render predictions, demonstrating crisp view-dependent effects at sub-millimeter scale; no additional lenses or coatings are required (Perroni-Scharf et al., 2023).

In digital pipelines, SV radiance fields integrate into existing explicit renderers. Rendering and training times for SV models are comparable to baselines (Spherical Harmonics, Gaussians), with reflection-capable SV pipelines operating at approximately 0.45× the speed of vanilla 3DGS while maintaining more efficient training (2.4× faster than MLP-based decoders). Speed-quality trade-offs (e.g., $K=8$ sites per Gaussian) remain favorable for real-time novel view synthesis (Sario et al., 16 Dec 2025).

7. Comparative Analysis and Expressive Capacity

Differentiable partitioning methods contrast with classical basis expansions. Self-occluding mesh techniques exploit spatial partitioning to drive physically grounded occlusion, while Spherical Voronoi provides adaptive directional partitioning with stable, tunable soft boundaries for angular signal modeling. SV avoids issues endemic to Spherical Harmonics (e.g., loss of high-frequency detail, Gibbs artifacts, poor reflection modeling) and achieves robust convergence, stable gradients, and explicit representation. Both approaches enable principled, interpretable control of directional appearance, with demonstrated superiority in representative benchmarks and deployment pipelines (Perroni-Scharf et al., 2023, Sario et al., 16 Dec 2025).

A plausible implication is that differentiable partitioning frameworks, by unifying geometric and directional domain modeling under analytic, optimization-friendly schemes, may serve as foundational tools across both physical fabrication and digital visual synthesis domains, allowing explicit and high-fidelity control over the directionality of appearance.