Concept: Importance Sampling
Core Idea
- Sample more where the integrand is large → lower variance
- Optimal PDF:
p*(x) = |f(x)| / ∫|f(x)|dx — proportional to integrand
- With optimal PDF: variance = 0 (but requires knowing the integral — circular)
- In practice: approximate
p(x) ≈ f(x) using known distributions
Why It Works
- Estimator:
Î = (1/N) * Σ f(x_i) / p(x_i)
- If
p(x) ∝ f(x), then f(x)/p(x) ≈ constant → variance ≈ 0
- If
p(x) is uniform, f(x)/p(x) varies a lot → high variance
- Key: the ratio
f(x)/p(x) should be as flat as possible
BRDF Importance Sampling
- For Lambertian BRDF:
f_r = albedo/π, cos(θ) factor
- Sample proportional to
cos(θ) → cosine-weighted hemisphere sampling
- PDF:
p(ω) = cos(θ)/π
- Weight:
f_r * cos(θ) / p(ω) = (albedo/π) * cos(θ) / (cos(θ)/π) = albedo
- For GGX specular BRDF
- Sample the half-vector
h from GGX NDF
p(h) = D(h) * dot(N, h) — proportional to NDF- Convert to incident direction:
ω_i = reflect(-ω_o, h)
- PDF for
ω_i: p(ω_i) = p(h) / (4 * dot(ω_o, h))
- GGX sampling:
θ_h = arctan(α * √(ξ_1 / (1 - ξ_1))), φ_h = 2π * ξ_2
Light Source Importance Sampling
- Sample a point on a light source directly
- PDF:
p(x) = 1 / area_of_light (uniform over light surface)
- Convert to solid angle PDF:
p(ω) = p(x) * r² / cos(θ_light)
r = distance to light, θ_light = angle at light surface
- This is Next Event Estimation (NEE) — see PathTracer Learning - Concept - Next Event Estimation
Multiple Importance Sampling (MIS)
- Combine multiple sampling strategies optimally
- Problem: BRDF sampling is good for specular, light sampling is good for diffuse
- MIS combines both without double-counting
- Balance heuristic:
w_i(x) = p_i(x) / Σ_j p_j(x)
- Power heuristic (β=2):
w_i(x) = p_i(x)² / Σ_j p_j(x)² — usually better
- MIS estimator:
Î = Σ_i (1/N_i) * Σ_j w_i(x_ij) * f(x_ij) / p_i(x_ij)
Environment Map Sampling
- Sample directions proportional to environment map luminance
- Build 2D CDF from luminance values
- Sample row (θ) then column (φ) using inverse CDF
- PDF:
p(ω) = L(ω) / ∫L(ω)dω — proportional to luminance
- Dramatically reduces variance for scenes lit by HDR environment maps