Supplemental Material for Gradient-Domain Path Tracing

This document supplements the SIGGRAPH 2015 technical paper “Gradient-domain Path Tracing”. We provide additional details and steps in the derivation of a frequency analysis of gradient sampling and reconstruction for Monte Carlo integration, including a numerical experiment that we performed to validate the theory, under its simplifications. We also present full derivations of the Jacobian determinants for our the shift mapping. 1 Frequency Analysis Derivations Here we provide some additional detail and derivation steps for the frequency analysis of gradient-domain Monte Carlo rendering presented in the paper. We follow the same outline as in the paper. 1.1 Frequency Analysis of Gradient Estimation We start by expressing the path difference function as a 1D convolution. For our analysis we use the simple shift mapping Tij(x̄) = (x+ j − i, p̄) that moves the path from pixel i to j without changing the other path parameters. We only compute differences between neighboring pixels in 1D, so we only have one mapping T (x̄) = (x−1, p̄). Clearly, the Jacobian of this mapping is identity and it has a unit determinant. The difference between pixel i and the one next to it is given by integrating the path difference function g(x, p̄) = f(x, p̄)− f(x− 1, p̄),