高光BRDF化简公式

源：http://graphicrants.blogspot.com/2013/08/specular-brdf-reference.html

Specular BRDF Reference

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(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=roughness2

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.

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Blinn-Phong [2]:

DBlinn(m)=1πα2(n⋅m)(2α2−2)

This is not the common form but follows when power=2α2−2.

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Beckmann [3]:

DBeckmann(m)=1πα2(n⋅m)4exp((n⋅m)2−1α2(n⋅m)2)

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GGX (Trowbridge-Reitz) [4]:

DGGX(m)=α2π((n⋅m)2(α2−1)+1)2

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GGX Anisotropic [5]:

DGGXaniso(m)=1παxαy1((x⋅m)2α2x+(y⋅m)2α2y+(n⋅m)2)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]:

GImplicit(l,v,h)=(n⋅l)(n⋅v)

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Neumann [6]:

GNeumann(l,v,h)=(n⋅l)(n⋅v)max(n⋅l,n⋅v)

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Cook-Torrance [11]:

GCook−Torrance(l,v,h)=min(1,2(n⋅h)(n⋅v)v⋅h,2(n⋅h)(n⋅l)v⋅h)

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Kelemen [7]:

GKelemen(l,v,h)=(n⋅l)(n⋅v)(v⋅h)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(l,v,h)=G1(l)G1(v)

I will define G1 below for each model and skip duplicating the above equation.

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Beckmann [4]:

c=n⋅vα1−(n⋅v)2−−−−−−−−−√

GBeckmann(v)={3.535c+2.181c21+2.276c+2.577c21if c<1.6if c≥1.6

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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]:

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.

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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=α2π−−√

GSchlick(v)=n⋅v(n⋅v)(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=α2

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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 F0 which is the reflectance at normal incidence.

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None:

FNone(v,h)=F0

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Schlick [9]:

FSchlick(v,h)=F0+(1−F0)(1−(v⋅h))5

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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)

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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:

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?"