BRDF

Q　　radiant energy: 　　　　J　　　　　　　　

Φ　　radiant flux: 　　　　　　W　　　　　　　　dQ/dt

E　　irradiance:　　　　　　　 W/m²　　　　dΦ/dA

I　　radiant intersity: 　　　　W/sr　　　　　　 dΦ/dω

L　　radiance: 　　　　　　　　W/(m²sr)　　　　d²Φ/dA_projdω

BRDF 双向反射分布函数

$f_{ ext{r}}(omega _{ ext{i}},\,omega _{ ext{r}})\,=\,{frac {operatorname {d} L_{ ext{r}}(omega _{ ext{r}})}{operatorname {d} E_{ ext{i}}(omega _{ ext{i}})}}\,=\,{frac {operatorname {d} L_{ ext{r}}(omega _{ ext{r}})}{L_{ ext{i}}(omega _{ ext{i}})cos heta _{ ext{i}}\,operatorname {d} omega _{ ext{i}}}}$

ω_i入射光= fi(θ,φ)

ω_o出射光 = fo(θ,φ)

azimuth angle φ and zenith angle θ

所以BRDF 是一个四维函数。

https://en.wikipedia.org/wiki/Bidirectional_reflectance_distribution_function

渲染方程：

$L_{{{ ext{o}}}}({mathbf x},\,omega _{{{ ext{o}}}},\,lambda ,\,t)\,=\,L_{e}({mathbf x},\,omega _{{{ ext{o}}}},\,lambda ,\,t) +\,int _{Omega }f_{r}({mathbf x},\,omega _{{{ ext{i}}}},\,omega _{{{ ext{o}}}},\,lambda ,\,t)\,L_{{{ ext{i}}}}({mathbf x},\,omega _{{{ ext{i}}}},\,lambda ,\,t)\,(omega _{{{ ext{i}}}}\,cdot \,{mathbf n})\,operatorname domega _{{{ ext{i}}}}$

dω = sinθdθdφ

推导：

　　　　--PBR，From Theory to Impllementation

Microfacet Theory

f = D()G()R()

Microfacet Distrbution Function,D

Shadowing-Masking Function,G

　　Beckmann Distribution with width parameter b:

　　Phong Distribution with exponent parameter ,p:

　　GGX Distribution with width parameter ,g:

Fresnel reflectance Function,R

Schlick

Local Subsurface scattering :Diffuse

https://en.wikipedia.org/wiki/Diffuse_reflection

Normalized BRDF 的推导

Lambert

Phong

Directional hemispherical reflectance

R(l) <=1

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unity的 BRDF1_Unity_PBS

// Main Physically Based BRDF

// Derived from Disney work and based on Torrance-Sparrow micro-facet model

// BRDF = kD / pi + kS * (D * V * F) / 4

// I = BRDF * NdotL

// * NDF (depending on UNITY_BRDF_GGX):

// a) Normalized BlinnPhong

// b) GGX

// * Smith for Visiblity term

// * Schlick approximation for Fresnel

BRDF = kD/pi + kS *(D*V*F)/4

I = BRDF * NdotL

NDF (depending on UNITY_BRDF_GGX):

　　　　a) Normalized BlinnPhong

　　　　b) GGX

V -Visiblity term -Smith
Schlick approximaation for Fresnel --specular比例

color = diffuseColor* (gi.diffuse + light.color * diffuseTerm)

　　　　+ light.color * specularTerm * FresnelTerm

　　　　+ surfaceReduction *gi.specular * FresnelLerp

// Based on Minimalist CookTorrance BRDF

// Implementation is slightly different from original derivation: http://www.thetenthplanet.de/archives/255

// * NDF (depending on UNITY_BRDF_GGX):

// a) BlinnPhong

// b) [Modified] GGX

// * Modified Kelemen and Szirmay-Kalos for Visibility term

// * Fresnel approximated with 1/LdotH

half3 color = (diffColor + specularTerm * specColor) * light.color * nl

+ gi.diffuse * diffColor

+ surfaceReduction * gi.specular * FresnelLerpFast (specColor, grazingTerm, nv);

// Old school, not microfacet based Modified Normalized Blinn-Phong BRDF

// Implementation uses Lookup texture for performance

// * Normalized BlinnPhong in RDF form

// * Implicit Visibility term

// * No Fresnel term

gi.diffuse

gi.specular