Environment Importance Sampling: Match Directions, Probabilities and Lighting Weights

Spend samples where the integral matters

Environment lighting sums contributions arriving from many directions. A small bright source can dominate the result while occupying little angular area, so uniformly chosen directions may rarely hit it. Importance sampling selects promising directions more often and compensates for that selection in the estimator. A suitable distribution reduces variance; a poorly chosen one can increase it. [1]

The relevant contribution includes incident radiance, the surface reflection function and the angular weighting. Environment brightness alone is therefore a useful proposal, not always the best match for every material and view.

Keep the probability measure explicit

Each sampled contribution must be divided by its sampling probability density in the same measure as the integral. For lighting directions, that is commonly solid angle. A probability defined over texture coordinates is not automatically a directional density.

Latitude-longitude environment texels cover different solid angles near the poles and equator. PBRT's environment-light construction weights the image distribution with the corresponding sine factor. [2] Other parameterizations require their own conversion. Ensure the distribution remains capable of sampling every region with a nonzero contribution.

Worked example: verify the compensation

Consider a discrete toy integration problem with two bin contributions, 8 and 2. The target sum is 10. Choose the bins with probabilities 0.8 and 0.2, respectively.

For the first bin the weighted estimate is 8 / 0.8 = 10; for the second it is 2 / 0.2 = 10. Either selection returns the correct total in this deliberately ideal example. Averaging the raw selected contributions instead has expectation 0.8 * 8 + 0.2 * 2 = 6.8, which is not the target sum.

These are discrete probabilities, not solid-angle densities. The example isolates the correction rule; a real environment integral also requires the correct angular mapping and reflection weighting.

Validate sampling before prefiltering

Start with a constant environment and a fixed material. Compare estimated brightness as the sample count changes. Then add a narrow bright region and rotate the environment across seams and poles. Inspect the sampled-direction distribution, returned densities and convergence against a high-sample reference.

If sampling a microfacet normal before converting it to a reflected direction, use the density for that final direction. Karis's course notes show this relationship in an importance-sampled lighting implementation. They also describe storing roughness-dependent environment filtering and a separate reflection-function integral for runtime use. [3]

Separate noise reduction from approximation error

A prefiltered environment lookup can be fast while still approximating the full lighting integral. Assumptions about view direction and filtering shape can lose stretched reflections at grazing angles. [3] More preprocessing samples reduce sampling noise but do not remove those assumptions. Evaluate different roughnesses and bright-source sizes, and distinguish distribution construction cost, filtering cost and runtime lookup cost when comparing implementations.

Sources

• [1] Improving Efficiency: Importance Sampling — Matt Pharr, Wenzel Jakob and Greg Humphreys, Physically Based Rendering.
• [2] Infinite Area Lights — Matt Pharr, Wenzel Jakob and Greg Humphreys, Physically Based Rendering.
• [3] Real Shading in Unreal Engine 4 — Brian Karis, Epic Games, original SIGGRAPH course notes.