Vector-Valued Monte Carlo Integration Using Ratio Control Variates

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.