Variance and Convergence Analysis of Monte Carlo Line and Segment Sampling

Abstract

Recently researchers have started employing Monte Carlo-like line sample estimators in rendering, demonstrating dramatic reductions in variance (visible noise) for effects such as soft shadows, defocus blur, and participating media. Unfortunately, there is currently no formal theoretical framework to predict and analyze Monte Carlo variance using line and segment samples which have inherently anisotropic Fourier power spectra. In this work, we propose a theoretical formulation for lines and finite-length segment samples in the frequency domain that allows analyzing their anisotropic power spectra using previous isotropic variance and convergence tools. Our analysis shows that judiciously oriented line samples not only reduce the dimensionality but also pre-filter C0 discontinuities, resulting in further improvement in variance and convergence rates. Our theoretical insights also explain how finite-length segment samples impact variance and convergence rates only by pre-filtering discontinuities. We further extend our analysis to consider (uncorrelated) multi-directional line (segment) sampling, showing that such schemes can increase variance compared to unidirectional sampling. We validate our theoretical results with a set of experiments including direct lighting, ambient occlusion, and volumetric caustics using points, lines, and segment samples.