Vector-Valued Monte Carlo Integration Using Ratio Control Variates

Haolin Lu1,2     Delio Vicini3     Wesley Chang1     Tzu-Mao Li1    
1University of California, San Diego, 2Max Planck Institute for Informatics, 3Google
ACM Transactions on Graphics (Proceedings of SIGGRAPH), 2025
Teaser

Vector-valued integration is ubiquitous in forward and inverse rendering. In forward rendering, the integrated radiance often differs between color channels or wavelength due to chromatic (a) lights (Bunny illuminated by three spherical area lights with RGB radiance values of [3000, 5, 6], [3, 3000, 4], and [4.8, 5.8, 3000]. For better visualization, we underexpose inset images), (b) BSDFs, (c) homogeneous, and (d) heterogeneous media. A common practice to tackle this challenge is sampling the luminance or a random mixture of each wavelength, leading to obvious color noise. In inverse rendering, we estimate (e) the vector-valued scene parameter gradient, or (f) the Hessian matrix (through a derivative-free method) for higher-order optimization. We propose a biased but consistent estimator and its unbiased variants to mitigate the challenge of vector-valued integration. Our methods can significantly reduce the variance with often negligible overhead, and can be easily integrated into existing forward and inverse rendering frameworks.

Abstract

Variance reduction techniques are widely used for reducing the noise of Monte Carlo integration. However, these techniques are typically designed with the assumption that the integrand is scalar-valued. Recognizing that rendering and inverse rendering broadly involve vector-valued integrands, we identify the limitations of classical variance reduction methods in this context. To address this, we introduce ratio control variates, an estimator that leverages a ratio-based approach instead of the conventional difference-based control variates. Our analysis and experiments demonstrate that ratio control variables can significantly reduce the mean squared error of vector-valued integration compared to existing methods and are broadly applicable to various rendering and inverse rendering tasks.

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Haolin Lu, Delio Vicini, Wesley Chang, Tzu-Mao Li. Vector-Valued Monte Carlo Integration Using Ratio Control Variates. ACM Transactions on Graphics (Proceedings of SIGGRAPH), 44(4), August 2025.

BibTex Reference Copy to clipboard

@article{Lu2025VectorValued,
	title        = {Vector-Valued Monte Carlo Integration Using Ratio Control Variates},
	author       = {Haolin Lu and Delio Vicini and Wesley Chang and Tzu-Mao Li},
	year         = 2025,
	month        = aug,
	journal      = {Transactions on Graphics (Proceedings of SIGGRAPH)},
	volume       = 44,
	number       = 4,
}