Voronoi integration of the rendering equation

TL;DR

Monte Carlo rendering suffers from high variance in difficult regions. The authors introduce a Voronoi tessellation reweighting scheme with Poisson point sampling, establishing a variance upper bound that decays faster than standard MC for Hölder continuous integrands. They demonstrate significant variance reduction in rendering tasks, notably near radiance discontinuities and edges, at the cost of tessellation overhead. The results indicate Voronoi-based estimators can improve efficiency when integrand evaluations are costly, suggesting a promising approach for global illumination rendering.

Abstract

In photorealistic image rendering, Monte Carlo methods form the foundation for the integration of the rendering equation in modern approaches. However, despite their effectiveness, traditional Monte Carlo methods often face challenges in controlling variance, resulting in noisy visual artifacts in regions that are difficult to render. In this work, we propose a new approach to the integration of the rendering equation by introducing a Voronoi tessellation reweighting scheme combined with a Poisson point process sampling strategy to address some of the limitations of standard Monte Carlo methods. From a theoretical point of view, we show that the variance induced by a Poisson-Voronoi tessellation is smaller than that of the Monte Carlo method when the intensity of the underlying process is arbitrarily large and when the function to be integrated satisfies a Holder continuity condition.

Figures (9)

• Figure 1: Three Voronoi integration estimators over $W=[-\frac{1}{2}, \frac{1}{2}]^2$. Voronoi cells nuclei inside $W$ are depicted in red, and auxiliary nuclei in blue. Cells used by the estimators are highlighted in yellow.
• Figure 2: Performance of integrators for function not_holder. Each plot compares a reweighting integration method against standard Monte Carlo (labeled mc). Solid lines represent the average estimate over 10,000 runs, with sample count $n$ ranging from $2^5$ to $2^{14}$. Shaded regions indicate the dispersion of these estimates within $\pm 1$ standard deviation. The estimators $E_V$, $E_C$ and $E_F$ (labeled vor, cvor and fvor respectively) demonstrate significant variance reduction. As expected, the clipped Voronoi reweighting method cvor exhibits estimation bias, while still reducing the variance.
• Figure 3: Comparative efficiency of three algorithms for evaluating the integral of the function not_holder.
• Figure 4: Simulating numerous potential light paths connecting a virtual camera $C$ to light sources leads to the accumulation of light contributions $(L_i)$ over a pixel area $W$, defining its final color. In a standard Monte Carlo framework, these contributions are averaged without accounting for the geometry of their distribution.
• Figure 5: Illustration of Monte Carlo rendering convergence with YAPT. The estimate gradually approaches the exact scene radiance as the sample count increases, resulting in a progressively more accurate estimation of the scene's true radiance.

Theorems & Definitions (2)

• Proposition 1
• proof