Specular BRDF Reference

$$ \newcommand{\nv}{\mathbf{n}} \newcommand{\lv}{\mathbf{l}} \newcommand{\vv}{\mathbf{v}} \newcommand{\hv}{\mathbf{h}} \newcommand{\mv}{\mathbf{m}} \newcommand{\rv}{\mathbf{r}} \newcommand{\ndotl}{\nv\cdot\lv} \newcommand{\ndotv}{\nv\cdot\vv} \newcommand{\ndoth}{\nv\cdot\hv} \newcommand{\ndotm}{\nv\cdot\mv} \newcommand{\vdoth}{\vv\cdot\hv} $$ While I worked on our new shading model for UE4 I tried many different options for our specular BRDF. Specifically, I tried many different terms for to Cook-Torrance microfacet specular BRDF: $$ f(\lv, \vv) = \frac{D(\hv) F(\vv, \hv) G(\lv, \vv, \hv)}{4(\ndotl)(\ndotv)} $$ Directly comparing different terms requires being able to swap them while still using the same input parameters. I thought it might be a useful reference to put these all in one place using the same symbols and same inputs. I will use the same form as Naty [1], so please look there for background and theory. I'd like to keep this as a living reference so if you have useful additions or suggestions let me know.

First let me define alpha that will be used for all following equations using UE4's roughness: $$ \alpha = roughness^2 $$

Normal Distribution Function (NDF)

The NDF, also known as the specular distribution, describes the distribution of microfacets for the surface. It is normalized [12] such that: $$ \int_\Omega D(\mv) (\ndotm) d\omega_i = 1 $$ It is interesting to notice all models have $\frac{1}{\pi \alpha^2}$ for the normalization factor in the isotropic case.

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Blinn-Phong [2]: $$ D_{Blinn}(\mv) = \frac{1}{ \pi \alpha^2 } (\ndotm)^{ \left( \frac{2}{ \alpha^2 } - 2 \right) } $$ This is not the common form but follows when $power = \frac{2}{ \alpha^2 } - 2$.

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Beckmann [3]: $$ D_{Beckmann}(\mv) = \frac{1}{ \pi \alpha^2 (\ndotm)^4 } \exp{ \left( \frac{(\ndotm)^2 - 1}{\alpha^2 (\ndotm)^2} \right) } $$

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GGX (Trowbridge-Reitz) [4]: $$ D_{GGX}(\mv) = \frac{\alpha^2}{\pi((\ndotm)^2 (\alpha^2 - 1) + 1)^2} $$

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GGX Anisotropic [5]: $$ D_{GGXaniso}(\mv) = \frac{1}{\pi \alpha_x \alpha_y} \frac{1}{ \left( \frac{(\mathbf{x} \cdot \mv)^2}{\alpha_x^2} + \frac{(\mathbf{y} \cdot \mv)^2}{\alpha_y^2} + (\ndotm)^2 \right)^2 } $$

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Geometric Shadowing

The geometric shadowing term describes the shadowing from the microfacets. This means ideally it should depend on roughness and the microfacet distribution.

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Implicit [1]: $$ G_{Implicit}(\lv,\vv,\hv) = (\ndotl)(\ndotv) $$

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Neumann [6]: $$ G_{Neumann}(\lv,\vv,\hv) = \frac{ (\ndotl)(\ndotv) }{ \mathrm{max}( \ndotl, \ndotv ) } $$

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Cook-Torrance [11]: $$ G_{Cook-Torrance}(\lv,\vv,\hv) = \mathrm{min}\left( 1, \frac{ 2(\ndoth)(\ndotv) }{\vdoth}, \frac{ 2(\ndoth)(\ndotl) }{\vdoth} \right) $$

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Kelemen [7]: $$ G_{Kelemen}(\lv,\vv,\hv) = \frac{ (\ndotl)(\ndotv) }{ (\vdoth)^2 } $$

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Smith

The following geometric shadowing models use Smith's method[8] for their respective NDF. Smith breaks $G$ into two components: light and view, and uses the same equation for both: $$ G(\lv, \vv, \hv) = G_{1}(\lv) G_{1}(\vv) $$ I will define $G_1$ below for each model and skip duplicating the above equation.

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Beckmann [4]: $$ c = \frac{\ndotv}{ \alpha \sqrt{1 - (\ndotv)^2} } $$ $$ G_{Beckmann}(\vv) = \left\{ \begin{array}{l l} \frac{ 3.535 c + 2.181 c^2 }{ 1 + 2.276 c + 2.577 c^2 } & \quad \text{if $c < 1.6$}\\ 1 & \quad \text{if $c \geq 1.6$} \end{array} \right. $$

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Blinn-Phong:
The Smith integral has no closed form solution for Blinn-Phong. Walter [4] suggests using the same equation as Beckmann.

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GGX [4]: $$ G_{GGX}(\vv) = \frac{ 2 (\ndotv) }{ (\ndotv) + \sqrt{ \alpha^2 + (1 - \alpha^2)(\ndotv)^2 } } $$ This is not the common form but is a simple refactor by multiplying by $\frac{\ndotv}{\ndotv}$.

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Schlick-Beckmann:
Schlick [9] approximated the Smith equation for Beckmann. Naty [1] warns that Schlick approximated the wrong version of Smith, so be sure to compare to the Smith version before using. $$ k = \alpha \sqrt{ \frac{2}{\pi} } $$ $$ G_{Schlick}(\vv) = \frac{\ndotv}{(\ndotv)(1 - k) + k } $$

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Schlick-GGX:
For UE4, I used the Schlick approximation and matched it to the GGX Smith formulation by remapping $k$ [10]: $$ k = \frac{\alpha}{2} $$

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Fresnel

The Fresnel function describes the amount of light that reflects from a mirror surface given its index of refraction. Instead of using IOR we instead use the parameter or $F_0$ which is the reflectance at normal incidence.

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None: $$ F_{None}(\mathbf{v}, \mathbf{h}) = F_0 $$

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Schlick [9]: $$ F_{Schlick}(\mathbf{v}, \mathbf{h}) = F_0 + (1 - F_0) ( 1 - (\vdoth) )^5 $$

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Cook-Torrance [11]: $$ \eta = \frac{ 1 + \sqrt{F_0} }{ 1 - \sqrt{F_0} } $$ $$ c = \vdoth $$ $$ g = \sqrt{ \eta^2 + c^2 - 1 } $$ $$ F_{Cook-Torrance}(\mathbf{v}, \mathbf{h}) = \frac{1}{2} \left( \frac{g - c}{g + c} \right)^2 \left( 1 + \left( \frac{ (g + c)c - 1 }{ (g - c)c+ 1 } \right)^2 \right) $$

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Optimize

Be sure to optimize the BRDF shader code as a whole. I choose these forms of the equations to either match the literature or to demonstrate some property. They are not in the optimal form to compute in a pixel shader. For example, grouping Smith GGX with the BRDF denominator we have this: $$ \frac{ G_{GGX}(\lv) G_{GGX}(\vv) }{4(\ndotl)(\ndotv)} $$ In optimized HLSL it looks like this:

float a2 = a*a;
float G_V = NoV + sqrt( (NoV - NoV * a2) * NoV + a2 );
float G_L = NoL + sqrt( (NoL - NoL * a2) * NoL + a2 );
return rcp( G_V * G_L );

If you are using this on an older non-scalar GPU you could vectorize it as well.

References

[1] Hoffman 2013, "Background: Physics and Math of Shading"
[2] Blinn 1977, "Models of light reflection for computer synthesized pictures"
[3] Beckmann 1963, "The scattering of electromagnetic waves from rough surfaces"
[4] Walter et al. 2007, "Microfacet models for refraction through rough surfaces"
[5] Burley 2012, "Physically-Based Shading at Disney"
[6] Neumann et al. 1999, "Compact metallic reflectance models"
[7] Kelemen 2001, "A microfacet based coupled specular-matte brdf model with importance sampling"
[8] Smith 1967, "Geometrical shadowing of a random rough surface"
[9] Schlick 1994, "An Inexpensive BRDF Model for Physically-Based Rendering"
[10] Karis 2013, "Real Shading in Unreal Engine 4"
[11] Cook and Torrance 1982, "A Reflectance Model for Computer Graphics"
[12] Reed 2013, "How Is the NDF Really Defined?"