Concept: Monte Carlo Integration

• Parent: PathTracer Learning - Phase 1 - Math for Graphics

The Problem

• We need to evaluate: I = ∫_Ω f(x) dx
• For the rendering equation: f(x) = f_r(ω_i, ω_o) * L_i(ω_i) * cos(θ_i)
• This integral has no closed-form solution in general
• Monte Carlo: estimate the integral by random sampling

Monte Carlo Estimator

• Draw N random samples x_1, ..., x_N from distribution p(x)
• Estimator: Î = (1/N) * Σ f(x_i) / p(x_i)
• This is an unbiased estimator: E[Î] = I
• Proof: E[f(x)/p(x)] = ∫ (f(x)/p(x)) * p(x) dx = ∫ f(x) dx = I

Variance and Convergence

• Variance of estimator: Var[Î] = Var[f(x)/p(x)] / N
• Standard deviation (noise): σ = σ_f / √N
• Convergence rate: O(1/√N) — dimension-independent
• To halve noise: need 4× more samples
• Compare to quadrature (trapezoidal rule): O(1/N^(2/d)) — degrades in high dimensions
• For d > 2: Monte Carlo beats quadrature — this is why it’s used for rendering

Uniform Hemisphere Sampling

• Sample direction uniformly over hemisphere
• PDF: p(ω) = 1 / (2π) (constant over hemisphere)
• Sampling: θ = arccos(1 - ξ_1), φ = 2π * ξ_2 where ξ_1, ξ_2 ~ Uniform(0,1)
• Cartesian: x = sin(θ)cos(φ), y = sin(θ)sin(φ), z = cos(θ)
• Estimator for rendering equation:
  • L_o ≈ (2π/N) * Σ f_r(ω_i) * L_i(ω_i) * cos(θ_i)

Cosine-Weighted Hemisphere Sampling

• Sample proportional to cos(θ) — matches the cos(θ) factor in rendering equation
• PDF: p(ω) = cos(θ) / π
• Sampling (Malley’s method): sample disk uniformly, project to hemisphere
  • r = √ξ_1, φ = 2π * ξ_2
  • x = r * cos(φ), y = r * sin(φ), z = √(1 - r²)
• Estimator for Lambertian surface:
  • L_o ≈ (1/N) * Σ (albedo/π) * L_i(ω_i) * cos(θ_i) / (cos(θ_i)/π)
  • = (1/N) * Σ albedo * L_i(ω_i) — the cos and π cancel!
• This is why cosine-weighted sampling is preferred for diffuse surfaces

Variance Reduction

• PathTracer Learning - Concept - Importance Sampling — sample where integrand is large
• Control variates — subtract a known function with same integral
• Stratified sampling — divide domain into strata, sample each
• Quasi-Monte Carlo — use low-discrepancy sequences instead of random
  • Halton sequence, Sobol sequence
  • Converges at O(1/N) instead of O(1/√N) for smooth integrands

Practical Notes

• Always divide by PDF: contribution = f(x) / p(x)
• If p(x) = 0 where f(x) ≠ 0: undefined — ensure support of p covers support of f
• Fireflies: rare samples with very high f(x)/p(x) — clamp or use better sampling