Approximate Program Smoothing Using Mean-Variance Statistics, with Application to Procedural Shader Bandlimiting

Abstract

We introduce a general method to approximate the convolution of a program with a Gaussian kernel. This results in the program being smoothed. Our compiler framework models intermediate values in the program as random variables, by using mean and variance statistics. We decompose the input program into atomic parts and relate the statistics of the different parts of the smoothed program. We give several approximate smoothing rules that can be used for the parts of the program. These include an improved variant of Dorn et al. [DBLW15], a novel adaptive Gaussian approximation, Monte Carlo sampling, and compactly supported kernels. Our adaptive Gaussian approximation handles multivariate Gaussian distributed inputs, gives exact results for a larger class of programs than previous work, and is accurate to the second order in the standard deviation of the kernel for programs with certain analytic properties. Because each expression in the program can have multiple approximation choices, we use a genetic search to automatically select the best approximations. We apply this framework to the problem of automatically bandlimiting procedural shader programs. We evaluate our method on a variety of geometries and complex shaders, including shaders with parallax mapping, animation, and spatially varying statistics. The resulting smoothed shader programs outperform previous approaches both numerically and aesthetically.