Concept: Multiple Importance Sampling (MIS)
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