Physically Based Rendering

Blinn-Phong shading model:

$L=C_{diff} N \cdot L + C_{spec} (N \cdot H)^m$

$C_{diff}$: Diffuse color
$C_{spec}$: Specular color
$N$: Normal vector
$L$: Light direction vector
$H$: Half-angle vector (sight direction vector & light direction vector)
Which is also $Normalized(L + V)$

BRDF:

$L=f_{BRDF} \cos \theta_{l}$

$f_{BRDF} = k_{diff} \frac{C_{diff}}{\pi} + k_{spec}VDF$

$D_{TR} = \frac{Roughness^2}{\pi (\cos^2\theta_h(Roughness^2 - 1) + 1)^2}$

$F = F_0 + (1 - F_0)(1 - \cos \theta_h)^5$

$F_0 = Metalic \times Albedo + (1 - Metalic) \times [0.04, 0.04, 0.04]$

$V = \frac{G_{S-GGX}(\theta_l)G_{S-GGX}(\theta_v)}{4 \cos \theta_l \cos \theta_v}$

$G_{S-GGX}(\theta) = \frac{\cos \theta}{k + (1 - k) \cos \theta}$

$k = \frac{(Roughness + 1)^2}{8}$

$k_{spec} = F_0$

$k_{diff} = (1 - F_0)(1 - Metalic)$

For Roughness:

$L=f_{BRDF} N \cdot L$

$L = (k_{diff} \frac{C_{diff}}{\pi} + k_{spec}VDF) N \cdot L$

$L = (k_{diff} \frac{C_{diff}}{\pi} + k_{spec} (\frac{G_{S-GGX}(\theta_l)G_{S-GGX}(\theta_v)}{4 \cos \theta_l \cos \theta_v}) (\frac{Roughness^2}{\pi (\cos^2\theta_h(Roughness^2 - 1) + 1)^2})F) N \cdot L$