Importance sampling reduces path-tracer variance by sampling directions proportional to the integrand

In a naive path tracer, scatter directions are chosen uniformly. Importance sampling replaces the uniform distribution with a probability density function (PDF) that concentrates samples where the integrand (the light contribution) is largest — toward bright lights or along the specular lobe. Each sample is weighted by the ratio of the target distribution to the actual sampling distribution. When sampling and integrand align, variance drops dramatically with no bias added. Mixture PDFs combine multiple distributions (e.g., one for lights, one for the BRDF). This is the central technique for making production path tracers practical.

Examples

Instead of uniform hemisphere sampling, sample toward the light source directly: the same number of rays produces a far less noisy image. Weight each sample by 1/pdf(direction) to maintain correctness.

Assessment

Given a uniform PDF and a cosine-weighted PDF over the hemisphere, which gives lower variance for a Lambertian surface? Why?