Anti-Aliased Low Discrepancy Samplers for Monte Carlo Estimators in Physically Based Rendering

Abstract

When you display a 3D object on a computer screen, we transform this 3D scene into a 2D image, which is a set of organized colored pixels. We call Rendering all the process that aims at finding the correct color to give those pixels. This is done by integrating all the light rays coming for every directions that the object’s surface reflects back to the pixel, the whole being ponderated by a visibility function. Unfortunately, a computer can not compute an integrand. We therefore have two possibilities to solve this issue: We find an analytical expression to remove the integrand (statistic based strategy). Numerically approximate the equation by taking random samples in the integration domain and approximating the integrand value using Monte Carlo methods. Here we focused on numerical integration and sampling theory. Sampling is a fundamental part of numerical integration. A good sampler should generate points that cover the domain uniformly to prevent bias in the integration and, when used in Computer Graphics, the point set should not present any visible structure, otherwise this structure will appear as artifacts in the resulting image. Furthermore, a stochastic sampler should minimize the variance in integration to converge to a correct approximation using as few samples as possible. There exists many different samplers that we will regroup into two families: Blue Noise samplers, that have a low integration variance while generating unstructured point sets. The issue with those samplers is that they are often slow to generate a pointset. Low Discrepancy samplers, that minimize the variance in integration and are able to generate and enrich a point set very quickly. However, they present a lot of structural artifacts when used in Rendering. Our work aimed at developing hybriod samplers, that are both Blue Noise and Low Discrepancy.