Torrance-Sparrow microfacet model

Developed in 1967 as a way to represent metallic surfaces.

Surfaces described by this model specifically assume that at the microfacet level, the material behaves like small tiny mirrors that reflect rays across its surface normal. In other words, they are perfectly specular.

BRDF equation

Using this model, the BRDF is given by

$$\text{BRDF}(p,{\omega }_o,{\omega }_i)=f(p,{\omega }_o,{\omega }_i)=\frac{D({\omega }_h)\cdot F_r({\omega }_o)\cdot G({\omega }_o,{\omega }_i)}{4\cos ({\theta }_o)\cos ({\theta }_i)}$$

where ${\theta }_o$ and ${\theta }_i$ are the angles between the microfacet surface normal and ${\omega }_o$ and ${\omega }_i$, respectively.

• $D({\omega }_h)$ is the distribution function that gives the probability a microfacet is perpendicular to the surface normal ${\omega }_h$, the half-angle vector between ${\omega }_o$ and ${\omega }_i$. It is the normal of the microfacet that reflected ${\omega }_o$ into ${\omega }_i$.

$${\omega }_h=\text{AVG}(\text{view},\text{light})=\text{AVG}({\omega }_o,{\omega }_i)=\frac{{\omega }_o+{\omega }_i}{2}$$

• $F_r$ is the Fresnel reflectance term: we assume microfacets individually reflect light according to Fresnel’s equations, which models the reflectivity at the surface.
• $G$ is a geometric attenuation term that accounts for masking and shadowing. See Lighting effects.

Calculation terms

We can pick and choose how to calculate $D$, $F_r$, and $G$ to balance physical accuracy with performance.

• For $D$, we can use the Trowbridge-Reitz distribution for approximation.
• For $F_r$, we can use Schlick’s approximation for Fresnel’s reflectance term.
• For $G$, we can use Smith’s Schlick-GGX approximation.

See the slides for the actual formulas.

Blinn-Phong distribution function

Proposed by Jim Blinn in 1977. $D$ is approximated by an exponential falloff. We have

$$D({\omega }_h)=\frac{\alpha +2}{2\pi }\cdot \text{dot}({\omega }_h,N{)}^{\alpha }$$

where $N$ is the surface normal and $\alpha$ is the exponential power that we want.

• The greater the exponent, the greater the falloff occurs, which we know from above corresponds to a smoother surface.
• The orientation with the greatest probability is in the actual surface normal direction, falling off to no microfacets oriented to the normal. In smoother surfaces, the falloff occurs quickly, but it’s more gradual for rougher surfaces.