Sampling and Variance Analysis for Monte Carlo Integration in Spherical Domain

Abstract

This dissertation introduces a theoretical framework to study different sampling patterns in the spherical domain and their effects in the evaluation of global illumination integrals. Evaluating illumination (light transport) is one of the most essential aspect in image synthesis to achieve realism which involves solving multi-dimensional space integrals. Monte Carlo based numerical integration schemes are heavily employed to solve these high dimensional integrals. One of the most important aspect of any numerical integration method is sampling. The way samples are distributed on an integration domain can greatly affect the final result. For example, in images, the effects of various sampling patterns appears in the form of either structural artifacts or completely unstructured noise. In many cases, we may get completely false (biased) results due to the sampling pattern used in integration.

The distribution of sampling patterns can be characterized using their Fourier power spectra. It is also possible to use the Fourier power spectrum as input, to generate the corresponding sample distribution. This further allows spectral control over the sample distributions.

Since this spectral control allows tailoring new sampling patterns directly from the input Fourier power spectrum, it can be used to improve error in integration. However, a direct relation between the error in Monte Carlo integration and the sampling power spectrum is missing. In this work, we propose a variance formulation, that establishes a direct link between the variance in Monte Carlo integration and the power spectra of both the sampling pattern and the integrand involved.

To derive our closed-form variance formulation, we use the notion of homogeneous sample distributions that allows expression of error in Monte Carlo integration, only in the form of variance. Based on our variance formulation, we develop an analysis tool that can be used to derive theoretical variance convergence rates of various state-of-the-art sampling patterns. Our analysis give insights to design principles that can be used to tailor new sampling patterns based on the integrand.