Code Notes made by Vaibhav Rathod

PathTracer Learning - Concept - BRDF

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Written by Vaibhav Rathod

Concept: BRDF (Bidirectional Reflectance Distribution Function)

Definition

  • f_r(ω_i, ω_o) — ratio of reflected radiance to incident irradiance
  • Units: 1/sr (per steradian)
  • Describes how a surface scatters light
  • ω_i — incoming light direction (toward surface)
  • ω_o — outgoing view direction (away from surface)
  • Both directions are defined in the upper hemisphere (above the surface)

Physical Constraints

  • Non-negativity: f_r(ω_i, ω_o) ≥ 0
  • Reciprocity (Helmholtz): f_r(ω_i, ω_o) = f_r(ω_o, ω_i)
    • Light path can be reversed — important for bidirectional algorithms
  • Energy conservation: ∫_Ω f_r(ω_i, ω_o) cos(θ_i) dω_i ≤ 1 for all ω_o
    • = 1 for perfectly reflective surface
    • < 1 for absorbing surface
    • Violation → surface creates energy → physically impossible

Lambertian BRDF (Diffuse)

  • f_r = albedo / π
  • Constant — scatters equally in all directions
  • The 1/π factor ensures energy conservation
  • Proof: ∫_Ω (albedo/π) * cos(θ) dω = (albedo/π) * π = albedo ≤ 1
  • Rendering equation contribution: L_o = albedo * (1/π) * ∫ L_i * cos(θ) dω
  • With cosine-weighted sampling: simplifies to L_o ≈ albedo * L_i
  • No real surface is perfectly Lambertian — but it’s a good approximation for matte surfaces

GGX Microfacet BRDF (Specular)

  • See PathTracer Learning - Concept - Microfacet Theory for full derivation
  • f_r = D(h) * G(ω_i, ω_o) * F(ω_o, h) / (4 * NdotL * NdotV)
  • Components
    • D(h) — Normal Distribution Function: density of microfacets with normal h
    • G(ω_i, ω_o) — Geometric term: shadowing and masking by microfacets
    • F(ω_o, h) — Fresnel: reflectance as function of angle
  • GGX NDF
    • D(h) = α² / (π * (NdotH² * (α² - 1) + 1)²)
    • α = roughness² (perceptual roughness remapping)
    • α = 0 → perfect mirror, α = 1 → fully rough
  • Smith G term
    • G(ω_i, ω_o) = G1(ω_i) * G1(ω_o) (separable approximation)
    • G1(ω) = NdotV / (NdotV * (1 - k) + k) where k = α/2 (direct lighting)
  • Schlick Fresnel

Disney Principled BRDF

  • Brent Burley (Disney) 2012 — artist-friendly parameterization
  • Parameters: baseColor, metallic, roughness, specular, specularTint, anisotropic, sheen, sheenTint, clearcoat, clearcoatGloss, subsurface, transmission
  • Widely used in production (Unreal, Unity, Godot PBR)
  • Combines diffuse + specular + clearcoat + sheen layers
  • Diffuse term: modified Lambertian with retro-reflection at grazing angles
    • f_diffuse = (baseColor/π) * (1 + (F_D90 - 1)(1-NdotL)^5) * (1 + (F_D90 - 1)(1-NdotV)^5)
    • F_D90 = 0.5 + 2 * roughness * VdotH²

BSDF (Bidirectional Scattering Distribution Function)

  • Generalization of BRDF that includes transmission
  • f_s(ω_i, ω_o) — handles both reflection and refraction
  • BRDF: reflection only (ω_i and ω_o on same side)
  • BTDF: transmission only (ω_i and ω_o on opposite sides)
  • BSDF = BRDF + BTDF
  • Used for glass, water, translucent materials

BRDF Sampling

  • To importance sample a BRDF, need to sample ω_i proportional to f_r * cos(θ_i)
  • Lambertian: cosine-weighted hemisphere sampling
    • θ = arccos(sqrt(1 - ξ₁)), φ = 2π * ξ₂
    • PDF: p(ω) = cos(θ) / π
  • GGX: sample half-vector from NDF, reflect view direction
    • θ_h = arctan(α * √(ξ / (1 - ξ))), φ_h = 2π * ξ_2
    • h = spherical_to_cartesian(θ_h, φ_h) in tangent space
    • ω_i = reflect(-ω_o, h)
    • PDF: p(ω_i) = D(h) * NdotH / (4 * VdotH)
  • VNDF sampling (Heitz 2018) — better for GGX at grazing angles
    • Sample visible NDF: proportional to D(h) * G1(ω_o) * dot(ω_o, h)
    • Fewer wasted samples for rough surfaces at grazing angles

Layered BRDFs

  • Real materials often have multiple layers
  • Clearcoat: thin specular layer over diffuse/specular base
    • f_total = f_clearcoat + (1 - F_clearcoat) * f_base
  • Sheen: retroreflective layer for fabric/velvet
  • Subsurface: light enters and exits at different points (BSSRDF)

Graph View

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