Pbr

Physically-based Rendering

• notations:
  • $\mathbf{p}$ is the shading point.
  • ${\omega }_o$ is the outgoing view direction (shading point to eye)
  • ${\omega }_i$ is the outgoing lighting direction (shading point to light)
  • $\mathbf{n}$ is the normal vector
  • $\mathbf{t}=2(\mathbf{n}\cdot {\omega }_o)\mathbf{n}-{\omega }_o$ is the reflective direction
  • $\mathbf{h}=\frac{w_i+w_o}{||w_i+w_o||}$ is the halfway vector (different from $\mathbf{n}$ !)
  • $m$ is the metalness
  • $\rho$ is the roughness
  • $\mathbf{a}$ is the ambient color (base color)
  • $s({\omega }_i)$ is the occlusion probability from the shading point to light.
• rendering Equation:

$$\begin{array}{c}\mathbf{c}({\omega }_o)={\int }_{\mathrm{\Omega }}L({\omega }_i)f({\omega }_i,{\omega }_o)({\omega }_i\cdot \mathbf{n})d{\omega }_i\end{array}$$

• micro-facet BRDF (Cook-Torrance 1982): $$ f(\omega_i, \omega_o) = (1 - m) \frac {\mathbf a} {\pi} + \frac {DFG} {4(\omega_i \cdot \mathbf n)(\omega_o \cdot \mathbf n)} $$ The BRDF can be divided into diffuse + specular terms, where the specular term includes:

  • F = Fresnel term
  $$\begin{gathered} F_0=m\ast \mathbf{a}+(1-m)\ast 0.04 \\ F=F_0+(1-F_0)(1-(\mathbf{h}\cdot {\omega }_o){)}^5 \end{gathered}$$
  • G = Geometry term (Schlick-GGX) $$ k = \rho^4 / 2 \ g(\mathbf v) = \frac {\mathbf n \cdot \mathbf v} {k + (1 - k)\mathbf n \cdot \mathbf v}\ G = g(\omega_o) g(\omega_i) $$
  • D = Normal Distribution (Trowbridge-Reitz GGX) $$ \alpha = \rho^2 \ D = \frac {\alpha^2} {\pi((\mathbf n \cdot \mathbf h)(\alpha^2 - 1) + 1)^2} $$

• split-sum approximation (Karis and Games 2013) $$ \int_\Omega L(\omega_i)\frac {DFG} {4(\omega_i \cdot \mathbf n)(\omega_o \cdot \mathbf n)} (\omega_i \cdot \mathbf n)d \omega_i \ = \int_\Omega L(\omega_i)Dd\omega_i \int_\Omega \frac {DFG} {4(\omega_o \cdot \mathbf n)} d \omega_i \ $$

• lighting representation $$ L(\omega_i) = (1 - s(\omega_i)) g_\text{direct}(\omega_i) + s(\omega_i) g_\text{indirect}(\omega_i, \mathbf p) \ s(\omega_i) = g_\text{occ}(\omega_i, \mathbf p) $$

  • direct term (from light): only dependent on the outgoing light direction.
  • indirect term (from other reflective surfaces): also dependent on the current shading point.

Metallic/Roughness v.s. Specular/Glossiness

Except for the common metallic-roughness model, there is also specular-glossiness model for PBR (e.g., Unity).

Both models can be converted between: https://kcoley.github.io/glTF/extensions/2.0/Khronos/KHR_materials_pbrSpecularGlossiness/examples/convert-between-workflows/

$$\begin{array}{c}glossiness=1-roughness\\ diffuse=(1-metallic)\ast basecolor\\ specular=metallic\ast basecolor\end{array}$$

A detailed documentation can be found: https://kcoley.github.io/glTF/extensions/2.0/Khronos/KHR_materials_pbrSpecularGlossiness/