< Back to the program



Can't Invert the CDF? The Triangle-Cut Parameterization of the Region under the Curve

Eric Heitz

We present an exact, analytic and deterministic method for sampling densities whose Cumulative Distribution Functions (CDFs) cannot be inverted analytically. Indeed, the inverse-CDF method is often considered the way to go for sampling non-uniform densities. If the CDF is not analytically invertible, the typical fallback solutions are either approximate, numerical, or non-deterministic such as acceptance-rejection. To overcome this problem, we show how to compute an analytic area-preserving parameterization of the region under the curve of the target density. We use it to generate random points uniformly distributed under the curve of the target density and their abscissae are thus distributed with the target density. Technically, our idea is to use an approximate analytic parameterization whose error can be represented geometrically as a triangle that is simple to cut out. This triangle-cut parameterization yields exact and analytic solutions to sampling problems that were presumably not analytically resolvable.

A Comprehensive Theory and Variational Framework for Anti-aliasing Sampling Patterns

Cengiz Oztireli

In this paper, we provide a comprehensive theory of anti-aliasing sampling patterns that explains and revises known results, and show how patterns as predicted by the theory can be generated via a variational optimization framework. We start by deriving the exact spectral expression for expected error in reconstructing an image in terms of power spectra of sampling patterns, and analyzing how the shape of power spectra is related to anti-aliasing properties. Based on this analysis, we then formulate the problem of generating anti-aliasing sampling patterns as constrained variational optimization on power spectra. This allows us to not rely on any parametric form, and thus explore the whole space of realizable spectra. We show that the resulting optimized sampling patterns lead to reconstructions with less visible aliasing artifacts, while keeping low frequencies as clean as possible.

Practical Product Sampling by Fitting and Composing Warps

David Hart, Matt Pharr, Thomas Müller, Ward Lopes, Morgan McGuire, Peter Shirley

We introduce a Monte Carlo importance sampling method for integrands composed of products and show its application to rendering where direct sampling of the product is often difficult. Our method is based on warp functions that operate on the primary samples in [0,1)^n, where each warp approximates sampling a single factor of the product distribution. Our key insight is that individual factors are often well-behaved and inexpensive to fit and sample in primary sample space, which leads to a practical, efficient sampling algorithm. Our sampling approach is unbiased, easy to implement, and compatible with multiple importance sampling. We show the results of applying our warps to projected solid angle sampling of spherical triangles, to sampling bilinear patch light sources, and to sampling glossy BSDFs and area light sources, with efficiency improvements of over 1.6× on real-world scenes.