Concept: Multiple Importance Sampling (MIS)

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

The Problem

• No single sampling strategy is optimal for all situations
• BRDF sampling: great for specular surfaces, terrible for small lights
• Light sampling (NEE): great for diffuse surfaces, terrible for specular
• Naively combining both: double-counts contributions → biased result
• MIS: combine multiple strategies optimally without bias

The Core Idea

• Veach & Guibas 1995 — one of the most important papers in rendering
• Each sample gets a weight based on how likely each strategy was to produce it
• Strategies that are “better suited” to a sample get higher weight
• Result: low variance everywhere, no double-counting

MIS Estimator

• Have n sampling strategies with PDFs p_1, ..., p_n
• Take n_i samples from strategy i
• MIS estimator: Î = Σ_i (1/n_i) * Σ_j w_i(x_ij) * f(x_ij) / p_i(x_ij)
• w_i(x) — MIS weight for strategy i at sample x
• Must satisfy: Σ_i w_i(x) = 1 for all x where f(x) ≠ 0

Balance Heuristic

• w_i(x) = (n_i * p_i(x)) / Σ_j (n_j * p_j(x))
• Provably optimal in a certain sense (Veach 1997)
• Simple to implement
• For equal sample counts (n_i = 1): w_i(x) = p_i(x) / Σ_j p_j(x)

Power Heuristic (β=2)

• w_i(x) = (n_i * p_i(x))^β / Σ_j (n_j * p_j(x))^β
• With β = 2: w_i(x) = p_i(x)² / Σ_j p_j(x)² (for equal sample counts)
• Usually better than balance heuristic in practice
• Reduces variance by suppressing contributions from poorly-suited strategies
• Veach showed β=2 is a good practical choice

MIS for NEE + BRDF Sampling

• The most common use case in path tracing
• Strategy 1: sample light source (NEE)
  • PDF: p_light(ω) — probability of sampling direction ω via light sampling
• Strategy 2: sample BRDF
  • PDF: p_brdf(ω) — probability of sampling direction ω via BRDF sampling
• MIS weights (power heuristic, β=2):
  • w_light = p_light² / (p_light² + p_brdf²)
  • w_brdf = p_brdf² / (p_light² + p_brdf²)
• Combined estimator:

  // NEE sample
  vec3 L_nee = f_r * L_e * G * V / p_light;
  float w_nee = pow2(p_light) / (pow2(p_light) + pow2(p_brdf_at_light_dir));
   
  // BRDF sample
  vec3 L_brdf = f_r * L_incoming * cos_theta / p_brdf;
  float w_brdf = pow2(p_brdf) / (pow2(p_brdf) + pow2(p_light_at_brdf_dir));
   
  vec3 L_total = w_nee * L_nee + w_brdf * L_brdf;

When MIS Helps Most

• Specular surfaces near area lights
  • BRDF sampling: good (samples specular lobe)
  • Light sampling: bad (light is large, but BRDF is narrow)
  • MIS: correctly weights BRDF samples higher
• Diffuse surfaces near small lights
  • BRDF sampling: bad (random hemisphere, rarely hits small light)
  • Light sampling: good (directly samples the light)
  • MIS: correctly weights light samples higher
• Environment map lighting
  • BRDF sampling: good for specular
  • Environment map sampling: good for bright regions
  • MIS: combines both optimally

MIS for Multiple Light Sources

• With N lights: sample one light, weight by p_light_i / Σ_j p_light_j
• Or: sample all lights and apply MIS weights
• PathTracer Learning - ReSTIR — handles many lights efficiently with MIS

Practical Notes

• Always evaluate p_brdf at the NEE direction and p_light at the BRDF direction
• If p_light = 0 for a BRDF sample (e.g., light is a point): w_brdf = 1
• If p_brdf = 0 for a NEE sample (e.g., delta BRDF): w_nee = 1
• Handle degenerate cases: avoid division by zero

Related

• PathTracer Learning - Concept - Importance Sampling
• PathTracer Learning - Concept - Next Event Estimation
• PathTracer Learning - Concept - BRDF