Cook-Torrance BRDF model

Model that is implemented in Unreal Engine 4. Considered the standard PBR shader for real time applications. Combines diffusive and specular reflection terms.

$$\text{BRDF}(p,{\omega }_o,{\omega }_i)=k_d\cdot f_{\text{Lambert}}+k_s\cdot f_{\text{Torrance-Sparrow}}$$

This model combines the BRDF functions from two other models, the Lambert diffusion model and the Torrance-Sparrow microfacet model. Plugging them in:

$$\text{BRDF}({\omega }_o,{\omega }_i)=k_d\cdot \frac{\text{albedo}}{\pi }+k_s\cdot \frac{DG}{4(w_o\cdot n)(w_i\cdot n)}$$

$k_d$ and $k_s$ are scalars that attenuate the two BRDF functions. They sum up to at most 1 because energy is conserved (cannot gain more energy, cannot be greater than 1).

• $k_s$ is actually just the Fresnel reflectance term from the Torrance-Sparrow model
• Then $k_d=1-k_s=1-F_r$.

The BRDF equation becomes

$$\text{BRDF}({\omega }_o,{\omega }_i)=(1-F_r)\cdot \frac{\text{albedo}}{\pi }+\frac{D{F}_{r}G}{4(w_o\cdot n)(w_i\cdot n)}$$

Where we can use approximations for $D$, $F_r$, and $G$. See the slides for actual formulas.

Splitting up the LTE

This model is made up of two parts, summed together. We can distribute both portions of the BRDF to separate LTE integrals, solve both, and them sum them together.

$$L_o=\int \frac{(1-F_r)\cdot \text{albedo}}{\pi }\text{ (other terms)}d{\omega }_i+\int \frac{D{F}_{r}G}{4(w_o\cdot n)(w_i\cdot n)}\text{(other terms)}d{\omega }_i$$

The “other terms” are from the integral in the Integral.