Cook Torrance BRDF Review

Cook-Torrance reflectance equation

r = ka + sum over lights of [ lc * (n.l) * (drd + srs) ], where the diffuse and specular terms are scaled by energy-conserving weights d and s (s + d = 1)

Cook-Torrance specular term rs

rs = (D * G * F) / (4 * (n.l) * (n.v)) -- the three pluggable microfacet functions divided by a normalization denominator

Why divide by 4, not pi, in rs

The original 1982 Cook-Torrance paper had a typo using pi in the denominator; the correct derivation of the microfacet specular term uses 4(n.l)(n.v)

D term (Normal Distribution Function)

Describes the fraction of microfacets whose normal aligns with the half-vector h, i.e. the facets oriented to reflect light straight at the viewer. Defines the shape/size of the specular highlight

G term (Geometric Attenuation Function)

Accounts for microfacets shadowing or masking each other; returns the fraction of facets that are neither occluded from the light nor from the viewer

F term (Fresnel Function)

Accounts for the Fresnel effect: reflectance increases at grazing (high) angles of incidence

Microfacet model

Treats a surface as a huge collection of tiny, randomly oriented mirror-like facets rather than one smooth surface; roughness controls how scattered the facet normals are

Roughness parameter alpha

Cook-Torrance's roughness control, 0 = perfectly smooth, 1 = maximally rough; conventionally alpha = roughness^2

Why square roughness into alpha

Perceptually linear roughness values map more naturally to visual results when squared before being fed into the D and G functions (artist-friendly remap)

GGX / Trowbridge-Reitz distribution

D(h) = alpha^2 / (pi * ((n.h)^2 * (alpha^2 - 1) + 1)^2) -- an alternative NDF to Blinn-Phong/Beckmann with longer, softer highlight tails

How GGX differs visually from Blinn-Phong

GGX has a narrower highlight peak but much longer falloff tails, producing a tight bright core with a soft glow around it, closer to measured real-world materials

Blinn-Phong NDF (Cook-Torrance form)

Dblinn = 1/(pi*alpha^2) * (n.h)^(2/alpha^2 - 2) -- the classic Blinn-Phong exponent rewritten in terms of alpha and normalized for energy conservation

NDF normalization condition

Integral over the hemisphere of D(h)*(h.n) dwi must equal 1, ensuring the distribution doesn't add or remove energy

Cook-Torrance's original G formula

G = min(1, 2(h.n)(n.v)/(v.h), 2(h.n)(n.l)/(v.h)) -- takes the minimum of the view-masking and light-shadowing terms

Smith geometry function

G(n,v,l) = G1(n.v) * G1(n.l), splitting occlusion into a separate view term and light term multiplied together; the standard pairing for GGX

Schlick-GGX approximation

G1(x,k) = x / (x*(1-k) + k), a fast single-term approximation used inside the Smith geometry function

Karis' k remap for direct lighting

k = (roughness + 1)^2 / 8 -- the roughness-to-k mapping recommended for analytic (non-IBL) light sources in Smith-Schlick-GGX

Schlick's Fresnel approximation

F = F0 + (1 - F0) * (1 - (v.h))^5 -- a fast polynomial approximation to the full Fresnel equations

F0 (reflectance at normal incidence)

The fraction of light reflected when viewed straight-on (0 degree incidence); low (~0.02-0.05) for dielectrics, high and colored for metals

F0 from index of refraction

F0 = ((n - 1) / (n + 1))^2, where n is the material's index of refraction; used for dielectrics

Metallic workflow F0 formula

F0 = mix(vec3(0.04), albedo, metallic) -- dielectrics get a flat ~4% reflectance, metals reflect (and tint) their own albedo color

Why metals have no diffuse term

In a metal, light that enters the surface is absorbed or re-emitted as reflection rather than scattering back out diffusely, so metals only exhibit specular (tinted) reflection

Energy-conserving diffuse weight kd

kd = (1 - F) * (1 - metallic) -- diffuse light is whatever fraction wasn't reflected specularly (1-F), and metals get none of it

Normalized Lambertian diffuse term

diffuse = kd * albedo / pi -- Cook-Torrance divides diffuse by pi (unlike un-normalized Blinn-Phong) to keep the BRDF physically energy-balanced

Half vector h

h = normalize(l + v) -- the vector halfway between the light direction and the view direction; microfacets aligned with h are the ones that reflect light toward the viewer

Denominator epsilon guard

A tiny constant (e.g. 1e-4) added to 4NdotVNdotL in the specular denominator to prevent divide-by-zero at grazing angles

Roughness clamping

Roughness is clamped to a small minimum (e.g. 0.045) instead of allowing exactly 0, since alpha=0 collapses the GGX denominator and causes divide-by-zero/NaN highlights

glTF metallic-roughness workflow

The standard PBR material convention: an albedo (base color) map, a combined or separate metallic value, a roughness value, plus normal, ambient occlusion, and emissive maps

albedoMap / albedoTint

Base color texture and tint multiplier; replaces the old diffuseMap in a PBR material, feeding both the diffuse term and (via metallic) the specular F0

metallicMap / metallicFactor

Texture and scalar controlling how conductor-like a surface is (0 = dielectric, 1 = metal); the factor scales the texture or acts as the value directly if no map is bound

roughnessMap / roughnessFactor

Texture and scalar controlling microfacet roughness fed into both the D and G functions; same texture/scale-or-standalone pattern as metallic

aoMap (ambient occlusion map)

Texture storing how exposed each point is to ambient/indirect light; multiplies the ambient term so crevices and contact points appear darker

emissiveMap / emissiveTint

Texture and tint for self-illumination that is added to the final lit color independent of any light source, unaffected by the BRDF

normalMap usage (unchanged by PBR conversion)

Sampled and remapped from [0,1] to [-1,1] to perturb the surface normal; behaves identically whether the shading model is Blinn-Phong or Cook-Torrance

Ambient term as an IBL placeholder

ambient = ambientLight * albedo * ao -- a flat constant standing in for real image-based lighting; gives the AO map something to modulate until proper IBL (irradiance + prefiltered specular + BRDF LUT) is added

Point light attenuation (from the shader)

attenuation = clamp(1 - distance^2/radius^2, 0, 1)^2 -- a smooth, physically-inspired falloff bounded by a light radius, independent of the BRDF used

Spot light cone falloff

fallout = clamp((theta - outerCutOff) / (innerCutOff - outerCutOff), 0, 1), where theta is the cosine angle between the light-to-fragment direction and the spot's forward direction

Directional light characteristics

Uses a constant direction (no position) and no distance attenuation, representing an infinitely distant light source like the sun

NdotL guard in the light loop

Specular and diffuse contributions are only computed when n.l > 0, since a surface facing away from the light receives no illumination and the BRDF math is undefined/wasted otherwise

Why old Blinn-Phong wasn't physically based

It uses an arbitrary fixed exponent tuned by eye, no energy conservation between diffuse and specular, and no real microfacet or Fresnel behavior -- effective for plastics but not general materials

Rendering equation role of DGF/4NdotVNdotL

This whole fraction is the specular BRDF term inside the reflectance sum; it replaces the single pow(NdotH, shininess) term from Blinn-Phong