f(l,v)=D(h)F(v,h)G(l,v,h)4(n⋅l)(n⋅v)
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:
α=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:
∫ΩD(m)(n⋅m)dωi=1
It is interesting to notice all models have
1πα2 for the normalization factor in the isotropic case.
Blinn-Phong [2]:
DBlinn(m)=1πα2(n⋅m)(2α2−2)
This is not the common form but follows when
power=2α2−2 .
Beckmann [3]:
DBeckmann (m)=1πα2(n⋅m)4exp((n⋅m)2−1α2(n⋅m)2)
GGX (Trowbridge-Reitz) [4]:
DGGX(m)=α2π((n⋅m)2(α2−1)+1)2
GGX Anisotropic [5]:
DGGXaniso (m)=1παxαy1((x⋅m)2α 2x+(y⋅m)2α 2y+(n⋅m)2)2
Geometric Shadowing
The geometric shadowing term describes the shadowing from the microfacets. This means ideally it should depend on roughness and the microfacet distribution.Implicit [1]:
GImplicit (l,v,h)=(n⋅l)(n⋅v)
Neumann [6]:
GNeumann (l,v,h)=(n⋅l)(n⋅v)max(n⋅l,n⋅v)
Cook-Torrance [11]:
GCook−Torrance (l,v,h)=min(1,2(n⋅h)(n⋅v)v⋅h,2(n⋅h)(n⋅l)v⋅h)
Kelemen [7]:
GKelemen (l,v,h)=(n⋅l)(n⋅v)(v⋅h)2
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(l,v,h)=G1(l)G1(v)
I will define
G1 below for each model and skip duplicating the above equation.
Beckmann [4]:
c=n⋅vα1−(n⋅v)2−−−−−−−−−√
GBeckmann (v)={3.535c+2.181c21+2.276c+2.577c21if c<1.6if c≥1.6
Blinn-Phong:
The Smith integral has no closed form solution for Blinn-Phong. Walter [4] suggests using the same equation as Beckmann.
GGX [4]:
GGGX(v)=2(n⋅v)(n⋅v)+α2+(1−α2)(n⋅v)2−−−−−−−−−−−−−−−−√
This is not the common form but is a simple refactor by multiplying by
n⋅vn⋅v ．
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=α2π−−√
GSchlick (v)=n⋅v(n⋅v)(1−k)+k
Schlick-GGX:
For UE4, I used the Schlick approximation and matched it to the GGX Smith formulation by remapping k [10]:
k=α2
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 F0 which is the reflectance at normal incidence.None:
FNone (v,h)=F0
Schlick [9]:
FSchlick (v,h)=F0+(1−F0)(1−(v⋅h))5
Cook-Torrance [11]:
η=1+F0−−√1−F0−−√
c=v⋅h
g=η2+c2−1−−−−−−−−−√
FCook−Torrance(v,h)=12(g−cg+c)2(1+((g+c)c−1(g−c)c+1)2)
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:
GGGX(l)GGGX(v)4(n⋅l)(n⋅v)
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?"