Image Synthesis
PBR Materials

• Physically-Based Rendering ("PBR") is widely used
• PBR material models [Walter et al. 2007] are nowadays used everywhere for real-time and offline rendering (industry standard)[Disney 2012] [Unreal Engine 2013] [Frostbite 2014]
• The BSDF consists of the following components [Disney 2015]:
  • BRDF Dielectrics
  • BRDF Metals
  • BTDF
  • Emission
  • Clear Coat
  • Sheen
  • Subsurface Scattering
• By combining the individual components, a wide range of different materials can be represented

Microfacets

Macroscopic behavior

$\mathbf{v}$

$-\mathbf{l}$

$\mathbf{h}$

$\mathbf{n}$

• Mathematical modeling of the surface by microfacets (much smaller than a pixel). The rougher the surface, the larger is the variation of their orientation [Cook Torrance 1981]
• The BRDF consists of a diffuse and a specular part
  $\mathrm{f}_r(\mathbf{v}, \mathbf{l}) = \mathrm{f}_d(\mathbf{v}, \mathbf{l}) + \mathrm{f}_s(\mathbf{v}, \mathbf{l})$

• Diffuse part is constant (or zero):
  $\mathrm{f}_d(\mathbf{v}, \mathbf{l}) = \frac{\rho_d}{\pi}$
• The BRDF of the specular part is:
  $\mathrm{f}_s(\mathbf{v}, \mathbf{l}) = \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{D}(\mathbf{h})\,\mathrm{G}(\mathbf{l}, \mathbf{v})}{4\,\,\langle\mathbf{n} \cdot \mathbf{l}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle}$
  with
  • $\mathrm{F}$: Fresnel reflectance
  • $\mathrm{D}$: Normal distribution function (NDF)
  • $\mathrm{G}$: Geometry term

• The ratio of reflected and transmitted light depends on the angle of incidence of the light rays and the refractive indices of the materials

• Light is either reflected, transmitted or absorbed with an angle-dependent relative frequency
• The reflected part is scattered on the microfacets. Macroscopically, this creates the specular part of the reflection.
• In opaque materials, the transmitted parted is randomly deflected below the surface, partially absorbed (wavelength dependent) and emitted at different angles. Macroscopically, this creates the diffuse part of the reflection.
• The more light is reflected, the less light is transmitted (and the diffuse part becomes smaller)

Microfacets

Macroscopic behavior

$\mathbf{n}$

Specular part

Diffuse part

• In metals, the transmitted portion is absorbed, i.e. there is no diffuse reflection
• The reflected part depends on the wavelength, so the specular part is colored

Microfacets

Macroscopic behavior

$\mathbf{n}$

Specular part

$\eta_1$

$\eta_2$

$\theta_1$

$\theta_1$

$\theta_2$

$\mathrm{F}$

$1-\mathrm{F}$

• Reflected part:
  $\begin{align} \mathrm{F}_{\tiny\mbox{para}} &= \frac{\eta_2 \cos \theta_1 - \eta_1 \cos \theta_2}{\eta_2 \cos \theta_1 + \eta_1 \cos \theta_2}\\ \mathrm{F}_{\tiny\mbox{perp}} &= \frac{\eta_2 \cos \theta_2 - \eta_1 \cos \theta_1}{\eta_2 \cos \theta_2 + \eta_1 \cos \theta_1}\\ \mathrm{F} &= \frac{1}{2} \left(\mathrm{F}_{\tiny \mbox{para}}^2 + \mathrm{F}_{\tiny\mbox{perp}}^2\right) \end{align}$
• Refracted part: $1 - F$
• In case of a transition from an optically thinner to optically denser medium ($\eta_1 < \eta_2$), the ray is deviated towards the normal. In the other case, the ray is deviated away from the normal.
• The mathematical relationship between the angle of incidence $\theta_1$ and the angle of the refracted ray $\theta_2$ is:
  $\eta_1 \sin(\theta_1) = \eta_2 \sin(\theta_2)$

• $\mathrm{F}_0$ is the reflectance for perpendicular incidence of light $\theta_1 = 0$:
  $\mathrm{F}_0 = \frac{(\eta_2 - \eta_1)^2}{(\eta_2 + \eta_1)^2} $

• In the microfacet model, the angle $\theta_1$ is the angle between the viewing direction $\mathbf{v}$ and the halfway vector $\mathbf{h}$, because only those microfacets, which have the macroscopic halfway vector as normal, have a reflection in the viewing direction
• According to [Cook Torrance 1981] the reflectance for $\eta_1 = 1$ is calculated by:
  $\begin{align}c &= \langle\mathbf{v} \cdot \mathbf{h}\rangle \\ g &= \sqrt{\eta_2^2+ c^2 - 1}\\ F_{\tiny \mbox{Cook-Torrance}}(\mathbf{v}, \mathbf{h}) &= \frac{1}{2} \left( \frac{g - c}{g + c} \right)^2 \left( 1 + \left( \frac{ c\,(g + c) - 1 }{ c\,(g - c)+ 1 } \right)^2 \right) \end{align}$

Microfacets

Macroscopic behavior

$\mathbf{v}$

$-\mathbf{l}$

$\mathbf{h}$

$\mathbf{n}$

• Alternatively, the Schlick approximation [Schlick 1994] of the Fresnel reflectance can be used, which is faster to compute:
  $F_{\tiny \mbox{Schlick}}(\mathbf{v}, \mathbf{h}) = \mathrm{F}_0 + \left(1.0 − \mathrm{F}_0\right) \left(1.0 − \langle\mathbf{v} \cdot \mathbf{h}\rangle \right)^5$
• The figure shows the comparison for glass with $\mathrm{F}_0 = 0.04$

$F_{\tiny \mbox{Cook-Torrance}}$

$F_{\tiny \mbox{Schlick}}$

Angle in degree

Reflectance

• The distribution of the microfacet normals changes depending on the roughness of the surface
• For a rougher surface, the distribution around the surface normal is more scattered
• There are several suggestions for the normal distribution in the literature [Walter et al. 2007]:
  
  $\mathrm{D}_{\tiny \mbox{Blinn}}(\mathbf{h}) = \frac{1}{ \pi \alpha^2 } \langle\mathbf{n} \cdot \mathbf{h}\rangle^{n_s}$   with  $n_s = 2\alpha^{-2} - 2$
  $\mathrm{D}_{\tiny \mbox{Beckmann}}(\mathbf{h}) = \frac{1}{ \pi \alpha^2 \langle\mathbf{n} \cdot \mathbf{h}\rangle^4 } \exp{ \left( \frac{ \langle\mathbf{n} \cdot \mathbf{h}\rangle^2 - 1}{\alpha^2 \langle\mathbf{n} \cdot \mathbf{h}\rangle^2} \right) }$
  $\mathrm{D}_{\tiny \mbox{GGX}}(\mathbf{h}) = \frac{\alpha^2}{\pi \left(\langle\mathbf{n} \cdot \mathbf{h}\rangle^2 (\alpha^2-1)+1\right)^2}$
• where $\alpha = r_p^2$ with the perceived roughness $r_p$ in range [0.0, 1.0]

• Depending on the direction of incidence of the light and the viewing direction of the observer, shadowing and masking effects occur at the microfacets
• [Blinn 1977] and [Cook Torrance 1981] assume the microfacets are V-shaped and derive a geometry factor $0 \le G \le 1$ from this model.
  

  No interference

  Masking effect

  Shadowing effect

  $\mathbf{n}$

  $-\mathbf{l}$

  $\mathbf{v}$

  $\mathbf{h}$

  $\mathbf{n}$

  $-\mathbf{l}$

  $\mathbf{v}$

  $\mathbf{h}$

  $\mathbf{n}$

  $-\mathbf{l}$

  $\mathbf{v}$

  $\mathbf{h}$
  $\mathrm{G}_{\tiny \mbox{Cook-Torrance}}(\mathbf{l}, \mathbf{v}) = \operatorname{min}\left( 1, \frac{ 2 \langle\mathbf{n} \cdot \mathbf{h}\rangle \langle\mathbf{n} \cdot \mathbf{v}\rangle }{ \langle\mathbf{v} \cdot \mathbf{h}\rangle}, \frac{ 2 \langle\mathbf{n} \cdot \mathbf{h}\rangle \langle\mathbf{n} \cdot \mathbf{l}\rangle }{ \langle\mathbf{v} \cdot \mathbf{h}\rangle} \right)$

• Another well-known model from [Smith 1967] combines the geometry factor from a component for light direction and one for the view direction:
  $\mathrm{G}_{\tiny \mbox{Smith}}(\mathbf{l}, \mathbf{v}) = \mathrm{G}_1(\mathbf{l}) \mathrm{G}_1(\mathbf{v}) $
• Different $\mathrm{G}_1$ according to [Walter et al. 2007] are:
  $\mathrm{G}_{1\ \tiny \mbox{Beckmann}}(\mathbf{v}) =\left\{ \begin{array}{l l} \frac{ 3.535 \,c \,+ \,2.181 \,c^2 }{ 1 \,+ \,2.276 \,c \,+ \,2.577 \,c^2 } & : \,\, c < 1.6\\ 1 & : \,\, c \ge 1.6\\ \end{array} \right.$    mit   $c = \frac{\langle\mathbf{n} \cdot \mathbf{v}\rangle}{ \alpha \sqrt{1 - \langle\mathbf{n} \cdot \mathbf{v}\rangle^2} }$
  $\mathrm{G}_{1\ \tiny \mbox{GGX}}(\mathbf{v}) = \frac{ 2 \, \langle\mathbf{n} \cdot \mathbf{v}\rangle }{ \langle\mathbf{n} \cdot \mathbf{v}\rangle + \sqrt{ \alpha^2 + (1 - \alpha^2)\langle\mathbf{n} \cdot \mathbf{v}\rangle^2 } }$
• Fast approximation [Schlick 1994] for the GGX variant:
  $\mathrm{G}_{1\ \tiny \mbox{Schlick-GGX}}(\mathbf{v}) =\frac{\langle\mathbf{n} \cdot \mathbf{v}\rangle}{\langle\mathbf{n} \cdot \mathbf{v}\rangle(1 - k) + k }$    mit   $k = \frac{\alpha}{2}$

• The specular part of the GGX microfacet BRDF is:
  $\begin{align}L_{o,s}(\mathbf{v}) &= \int\limits_\Omega \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{D}(\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v})}{4\,\,\langle\mathbf{n} \cdot \mathbf{l}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l}) \cos(\theta) \, d\omega \end{align}$
• This is an integral over all incoming light directions $\mathbf{l}$
• The normal distribution function $\mathrm{D}(\mathbf{h})$ of the microfacets is a function of the halfway vector $\mathbf{h}$
  $\mathrm{D}_{\tiny \mbox{GGX}}(\mathbf{h}) = \frac{\alpha^2}{\pi \left(\langle\mathbf{n} \cdot \mathbf{h}\rangle^2 (\alpha^2-1)+1\right)^2}$
• Since this distribution is given as an analytic function, the integral should be solved by importance sampling over the halfway vector

$\mathbf{v}$

$-\mathbf{l}$

$\mathbf{h}$

$\mathbf{n}$

• Relation between $\mathbf{l}$ and $\mathbf{h}$
• As can be seen from in the figure, $\mathbf{l}$ can be computed by reflecting $\mathbf{v}$ at $\mathbf{h}$
• It follows:
  $\mathbf{l} = 2 \,(\mathbf{v}^\top \mathbf{h}) \, \mathbf{h} - \mathbf{v}$

• Integration by substitution (change of variables):
  $\begin{align}L_{o,s}(\mathbf{v}) &= \int\limits_{\Omega_h} \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{D}(\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v})}{4\,\,\langle\mathbf{n} \cdot \mathbf{l}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l}) \cos(\theta) \, d\omega\\ &= \int\limits_{\Omega_h} \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{D}(\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v})}{4\,\,\langle\mathbf{n} \cdot \mathbf{l}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l}) \cos(\theta) \, \frac{d\omega}{d\omega_h} \,d\omega_h\\ &= \int\limits_{\Omega_h} \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{D}(\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v})}{4\,\,\langle\mathbf{n} \cdot \mathbf{l}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l}) \cos(\theta) \, \frac{\sin(\theta)\, d\theta \,d\phi}{\sin(\theta_h) \,d\theta_h d\phi_h} \,d\omega_h \end{align}$
  

  $d\omega = \sin(\theta)\, d\phi\, \cdot d\theta $

with $\theta^\ast = 2 \,\theta_h^\ast$ and $\phi^\ast = \phi_h^\ast$

• Importance sampling:
  $\begin{align}L_{o,s}(\mathbf{v}) &= \int\limits_{0}^{2\pi}\,\int\limits_{0}^{\frac{\pi}{2}}\underbrace{ \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{D}(\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v})}{\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l}) \,\langle\mathbf{v} \cdot \mathbf{h}\rangle \,\sin(\theta_h)}_{\mathrm{f}(\theta_h, \phi_h)} \, d\theta_h \, d\phi_h\\ &\approx \frac{1}{N} \sum_{n=1}^{N} \frac{\mathrm{f}(\theta_{h}, \phi_{h})}{\mathrm{p}(\theta_{h}, \phi_{h})}\\[1.5ex] &=\frac{1}{N} \sum_{n=1}^{N} \frac{\frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{D}(\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v})}{\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l}) \,\langle\mathbf{v} \cdot \mathbf{h}\rangle \,\sin(\theta_h)}{\mathrm{D}(\mathbf{h}) \cos(\theta_h) \sin(\theta_h)}\\[1.5ex] &= \frac{1}{N} \sum_{n=1}^{N} \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v}) \,\langle\mathbf{v} \cdot \mathbf{h}\rangle}{\langle\mathbf{n} \cdot \mathbf{h}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l})\\ \end{align}$

• The overall result is:
  $\begin{align}L_{o,s}(\mathbf{v}) &\approx \frac{1}{N} \sum_{n=1}^{N} \frac{\mathrm{F}(\mathbf{v},\mathbf{h})\,\mathrm{G}(\mathbf{l},\mathbf{v}) \,\langle\mathbf{v} \cdot \mathbf{h}\rangle}{\langle\mathbf{n} \cdot \mathbf{h}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l})\\ \end{align}$
  with $\mathbf{h} = \begin{pmatrix}\sin(\theta_h) cos(\phi_h) \\ \sin(\theta_h) \sin(\phi_h)\\ \cos(\theta_h)\end{pmatrix}$  and   $\mathbf{l} = \operatorname{reflect}(-\mathbf{v}, \mathbf{h})$
  where $\phi_h = 2 \pi \,h_2(n)$ and $\theta_h = \arccos\left(\sqrt{\frac{1 - h_3(n)}{h_3(n) (\alpha^2-1) + 1} }\right)$
  and $\operatorname{reflect}$ refers to the GLSL function reflect:
  $\operatorname{reflect}(\mathbf{i}, \mathbf{n}) = \mathbf{i} - 2 \,(\mathbf{i}^\top \mathbf{n}) \, \mathbf{n}$

• The behavior of the transmission should correspond to that of the specular part of the reflection
• Therefore, the same sampling approach as for the reflection can be used, except that $\mathbf{l}$ is now obtained by the refraction of the view vector $\mathbf{v}$ at the halfway-vector $\mathbf{h}$:
  $\begin{align}L_{o,t}(\mathbf{v}) &\approx \frac{1}{N} \sum_{n=1}^{N} \frac{(1 - \mathrm{F}(\mathbf{v},\mathbf{h}))\,\mathrm{G}(\mathbf{l},\mathbf{v},\mathbf{h}) \,\langle\mathbf{v} \cdot \mathbf{h}\rangle}{\langle\mathbf{n} \cdot \mathbf{l}\rangle\,\,\langle\mathbf{n} \cdot \mathbf{v}\rangle} \,L_i(\mathbf{l})\\ \end{align}$
  with $\mathbf{h} = \begin{pmatrix}\sin(\theta_h) cos(\phi_h) \\ \sin(\theta_h) \sin(\phi_h)\\ \cos(\theta_h)\end{pmatrix}$  and   $\mathbf{l} = \operatorname{refract}(-\mathbf{v}, \mathbf{h}, \eta)$
  where $\phi_h = 2 \pi \,h_2(n)$ und $\theta_h = \arccos\left(\sqrt{\frac{1 - h_3(n)}{h_3(n) (\alpha^2-1) + 1} }\right)$
  and $\operatorname{refract}$ refers to the GLSL function refract

• [Phong 1975] Bui-T. Phong: Illumination for computer generated pictures. Communications of the ACM, 18(6):311-317, June 1975
• [Blinn 1977] James F. Blinn: Models of light reflection for computer synthesized pictures. SIGGRAPH 1977, pp. 192-198, July 1977
• [Cook Torrance 1981] R. L. Cook and K. E. Torrance: A reflectance model for computer graphics. TOG 1(1):7-24, Jan. 1981.
• [Disney 2012] Brent Burley: Physically Based Shading at Disney SIGGRAPH 2012 Course: Practical Physically Based Shading in Film and Game Production
• [Disney 2015] Brent Burley: Extending the Disney BRDF to a BSDF with Integrated Subsurface Scattering, SIGGRAPH 2015 Course: Practical Physically Based Shading in Film and Game Production
• [Unreal Engine 2013] Brian Karis: Real Shading in Unreal Engine 4, SIGGRAPH 2013 Course: Physically Based Shading in Theory and Practice
• [Frostbite 2014] S. Lagarde, C. de Rousiers: Moving Frostbite to PBR, SIGGRAPH 2014 Course: Physically Based Shading in Theory and Practice

• [Schlick 1994] Christophe Schlick: An Inexpensive BRDF Model for Physically-Based Rendering. Computer Graphics Forum, 13 (3), 233–246, 1994.
• [Walter 2005] B. Walter: Notes on the Ward BRDF. Technical report PCG-05-06, Cornell University
• [Walter et al. 2007] B. Walter, S. R. Marschner, H. Li, K. E. Torrance: Microfacet Models for Refraction through Rough Surfaces. Eurographics Symposium on Rendering, 2007.
• [Smith 1967] B. Smith: Geometrical shadowing of a random rough surface. IEEE Transactions on Antennas and Propagation 15(5):668-671, Sep. 1967