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:

• 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
  $$F_0 = m * \mathbf a + (1 - m) * 0.04 \\ F = F_0 + (1 - F_0)(1-(\mathbf h \cdot \omega_o))^5$$

  • 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/

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