Shared Foundations

Sampling

Where to put the samples, and how the error is distributed when you cannot afford enough of them.

Colour is the family; a dashed line is the second member of it.

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G blue-noise-sampling Blue-Noise Error Distribution monte-carlo-integration Monte Carlo Integration blue-noise-sampling->monte-carlo-integration same error, redistributed to where the eye sees it least cosine-weighted-hemisphere Cosine-Weighted Hemisphere Sampling importance-sampling Importance Sampling cosine-weighted-hemisphere->importance-sampling cancels the cosine factor in the integrand exactly halton-sequence Halton Sequence quasi-monte-carlo Quasi-Monte Carlo halton-sequence->quasi-monte-carlo radical inverse in a different prime per dimension importance-sampling->monte-carlo-integration uniform samples are wasted where the integrand is small owen-scrambling Owen Scrambling sobol-sequence Sobol Sequence owen-scrambling->sobol-sequence a fixed sequence gives structure and no error estimate quasi-monte-carlo->monte-carlo-integration deterministic even coverage converges faster sobol-sequence->quasi-monte-carlo base-2 construction, cheap to evaluate incrementally stratified-sampling Stratified Sampling stratified-sampling->monte-carlo-integration independent random samples clump and leave gaps

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Blue-Noise Error Distribution

Chooses per-pixel sample seeds so that the residual error is high- frequency across the screen. The error is not smaller, but the eye and any subsequ… · 2019

Cosine-Weighted Hemisphere Sampling

Samples directions with density proportional to the cosine of the angle from the normal, cancelling the geometry term in the rendering equation exact…

Halton Sequence

Uses the radical inverse in a different prime base for each dimension. Simple and progressive, but correlations between high dimensions show up as vi…

Importance Sampling

Draw samples proportional to how much they contribute and divide by that density. Same expected value, far less variance, and the foundation every pr…

Monte Carlo Integration

Estimate an integral by averaging the integrand over random samples divided by their density. Converges as the inverse square root of sample count re…

Owen Scrambling

Randomizes a low-discrepancy sequence with a recursive permutation that preserves its stratification, restoring an unbiased estimator with a measurab…

Quasi-Monte Carlo

Replaces random points with deterministic low-discrepancy sequences that cover the domain more evenly, improving the convergence rate on smooth integ…

Sobol Sequence

A base-two low-discrepancy construction with good high-dimensional properties and very fast incremental evaluation, which is why renderers reach for…

Stratified Sampling

Partition the domain and take one sample per cell, so samples cannot clump. Cheap, and effective in low dimensions before the number of strata explod…