Bernstein Bounds for Caustics

TL;DR

This work tackles the enduring challenge of rendering sharp caustics on complex geometry by introducing a bound-driven sampling framework that uses Bernstein bounds to conservatively bound the position and irradiance contributed by each triangle tuple. By expressing vertex positions and irradiance as rational functions and handling nonrational components with remainder variables, the method builds piecewise Bernstein bounds that guide efficient, unbiased sampling of triangle tuples and subsequent root-finding within selected tuples. A multi-sample estimator coupled with an optimized sampling probability design reduces variance and concentrates effort on high-contribution paths, enabling fast and reliable caustics rendering. Empirical results on diverse scenes show substantial speedups and variance reductions compared with state-of-the-art methods, with clear pathways for extending the technique to more general geometries and longer path chains in future work.

Abstract

Systematically simulating specular light transport requires an exhaustive search for primitive tuples containing admissible paths. Given the extreme inefficiency of enumerating all combinations, we propose to significantly reduce the search domain by sampling such tuples. The challenge is to design proper sampling probabilities that keep the noise level controllable. Our key insight is that by bounding the range of irradiance contributed by each primitive tuple at a given position, we can sample a subset of primitive tuples with potentially high contributions. Although low-contribution tuples are assigned a negligible probability, the overall variance remains low. Therefore, we derive vertex position and irradiance bounds for each primitive tuple, introducing a bounding property of rational functions on the Bernstein basis. When formulating position and irradiance expressions into rational functions, we handle non-rational components through remainder variables to maintain validity. Finally, we carefully design the sampling probabilities by optimizing the upper bound of the variance, expressed only using the position and irradiance bound. The proposed primitive sampling is intrinsically unbiased. It can be seamlessly combined with various unbiased and biased root-finding techniques within a local primitive domain. Extensive evaluations show that our method enables fast and reliable rendering of complex caustic effects.

Figures (20)

• Figure 1: Overview of the precomputation pass. We trace light beams passing through each triangle tuple and leverage their position and irradiance bounds to sample these triangle tuples. Note that our discussion of irradiance is primarily about its upper bound only.
• Figure 2: Overview of the rendering pass. For a given shading point, the triangle tuples whose position bound covers it form a set $\mathrm{U}$. We aim to sample a subset $\mathrm{S} \subseteq \mathrm{U}$. For each triangle tuple $\mathcal{T} \in \mathrm{S}$, we run existing root-finding methods to solve for admissible paths within it.
• Figure 3: Visualizing the bound of caustics cast by a single triangle reflector. We show the bound and its ratio with reference in the base 10 logarithmic space. Note that the axes are barycentric coordinates $u_k$ and $v_k$ on the receiver. Thus, the position bounds of different pieces may overlap. Top: Our bound is tighter than interval arithmetics wang2020path in both an equal number of pieces and equal time. Ours computes 50 pieces while interval arithmetic computes 4K pieces in equal time. Bottom: Utilizing 500 uniformly sampled paths (in roughly equal time) to fit irradiance distributions (Histogram) like path guiding jensenImportanceDrivenPath1995 or use the range of samples as bounds (Sample-Based) Loubet20 are not conservative and could result in fireflies or bias in rendering.
• Figure 4: With remainder variables, we correctly bound the range of $\sqrt{\beta}$, though looser than the proper range (left). Note that the actual range of $\beta$ is smaller, so the bound could be tighter (right).
• Figure 5: Illustration of the generalized geometric term (GGT). Light emitted in the differential solid angle $\mathrm{d}\Omega_0$ passing through specular surfaces finally hits the differential area $\mathrm{d}A_k$ on the receiver. Note that we use $\mathrm{d}\Omega_0$ instead of the projected solid angle $\mathrm{d}\Omega_0^{\perp}$ because we consider point light sources that emit intensity uniformly in different directions.