Bernstein Bounds for Caustics

2025-04-27

AI Summary

This paper addresses the high noise and slow convergence in caustic rendering under specular transport. We propose an unbiased path integral method based on dual-bound importance sampling. Our key innovation is the first application of tight rational-function bounds under the Bernstein basis to light transport analysis, jointly modeling vertex positions and irradiance upper bounds over primitive tuples to construct a variance-controllable sampling distribution. The method integrates Bernstein polynomial bound estimation, analytical remainder-variable derivation, local root solving, and variance upper-bound optimization to efficiently identify high-contribution paths. Experiments demonstrate that, while strictly preserving unbiasedness, our approach significantly reduces noise and accelerates rendering by 3–8×, enabling stable, high-fidelity caustic generation in complex dynamic scenes.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• Reducing search domain for specular light transport paths
• Designing sampling probabilities to control noise levels
• Bounding irradiance contributions for efficient primitive tuple sampling

Innovation

• Sampling primitive tuples to reduce search domain
• Bounding irradiance range using Bernstein basis
• Optimizing sampling probabilities for low variance