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CS 775: Advanced Computer Graphics
Lecture 7 : The BRDF
CS 775: Lecture 7 Parag Chaudhuri, 2012
The BRDF
ωoωi
n⃗
p
Differential Irradiance at p is dE ( p ,ωi)=Li( p ,ωi)cosθid ωi
CS 775: Lecture 7 Parag Chaudhuri, 2012
The BRDF
ωoωi
n⃗
p
Differential Irradiance at p is dE ( p ,ωi)=Li( p ,ωi)cosθid ωi
Reflected differential radiance is then given by dLo( p ,ωo)∝dE ( p ,ωi)
CS 775: Lecture 7 Parag Chaudhuri, 2012
The BRDF
ωoωi
n⃗
p
Differential Irradiance at p is dE ( p ,ωi)=Li( p ,ωi)cosθid ωi
Reflected differential radiance is then given by dLo( p ,ωo)∝dE ( p ,ωi)
f r ( p ,ωo ,ωi)=dLo( p ,ωo)
dE ( p ,ωi)
CS 775: Lecture 7 Parag Chaudhuri, 2012
The BRDF
● Reciprocity
● Energy Conservationf r ( p ,ωo ,ωi)=
dLo( p ,ωo)
dE ( p ,ωi)
f r ( p ,ωo ,ωi)= f r ( p.ωi ,ωo)
∫H 2(n⃗)f r ( p ,ωo ,ω ' )cosθ ' d ω '≤1
Lo( p ,ωo)=∫S 2 f ( p ,ωo ,ωi) Li( p ,ωi)∣cosθi∣d ωi
● Consider a general over the sphere of all directions and we get the BSDF (=BRDF+BTDF)
f ()
CS 775: Lecture 7 Parag Chaudhuri, 2012
The BRDF
● Sources
– Measured Data
– Phenomenological Models
– Simulation
– Physical (Wave) Optics
– Geometric optics
● Types of Surfaces
– Diffuse
– Glossy Specular
– Perfect Specular
– Retro-reflectivehttp://www.lamrug.org/resources/indirectips.html
CS 775: Lecture 7 Parag Chaudhuri, 2012
The Ideal Diffuse BRDF
Let f r ( p ,ωo ,ωi)=k d
∫H 2(n⃗)f ( p ,ωo ,ωi)cosθi d ωi=ρ
● Assume BRDF reflects a fraction of the incoming light
● The quantity is known as the albedo of the surface.
k d∫0
2π
∫0
π /2cosθisin θid θi d ϕi=ρ
2π k d∫0
π/2cosθi sinθid θi=ρ
k d=ρπ
ρ
ρ
CS 775: Lecture 7 Parag Chaudhuri, 2012
The Ideal Diffuse BRDF
Let f r ( p ,ωo ,ωi)=k d
∫H 2(n⃗)f ( p ,ωo ,ωi)cosθi d ωi=ρ
● Assume BRDF reflects a fraction of the incoming light
● The quantity is known as the albedo of the surface.
k d∫0
2π
∫0
π /2cosθisin θid θi d ϕi=ρ
2π k d∫0
π/2cosθi sinθid θi=ρ
k d=ρπ
ρ
ρ
N−L
≡
Remember this?
CS 775: Lecture 7 Parag Chaudhuri, 2012
The Ideal Mirror BRDF
If we assume ∫H 2( n⃗)f ( p ,ωo ,ωi)cosθid ωi=F r (ωi)
● BRDF is zero everywhere except where
and we know ∫ f ( x)δ( x−xo)dx= f ( xo)
then it is easy to see that putting f r ( p ,ωo ,ωi)=F r (ωi)δ(ωi−R(ωo , n⃗))
cosθi
θo=θi
ϕo=ϕi+π
gives us Lo( p ,ωo)=∫H 2( n⃗)f r ( p ,ωo ,ωi)Li ( p ,ωi)cosθi dωi
=F r (ωr )Li ( p , R (ωo , n⃗))=F r (ωr)L i( p ,ωr )
CS 775: Lecture 7 Parag Chaudhuri, 2012
Fresnel Reflectance
● For conducting specular surfaces, the amount of reflected light is given by
● Obtained from solution to Maxwell's equation for reflection off smooth surfaces.
r∥=(η2+k 2)cosθi
2−2ηcosθi+1
(η2+k 2)cosθi2+2ηcosθi+1
for parallel polarized light
r ⊥=(η2+k 2)cosθi
2−2ηcosθi+cosθi2
(η2+k 2)cosθi2+2ηcosθi+cosθi
2for perpendicular polarized light
F r=12(r∥
2+r ⊥
2) for unpolarized light
CS 775: Lecture 7 Parag Chaudhuri, 2012
Fresnel Reflectance
● Fresnel Reflectance for pure aluminum at 550 nanometers
http://www.graphics.cornell.edu/~westin/misc/fresnel.html
CS 775: Lecture 7 Parag Chaudhuri, 2012
Specular Transmittance
● BTDF is zero everywhere except where
f t ( p ,ωo ,ωi)=ηo
2
ηi2 (1−F r (ωi))
δ(ωi−T (ωo , n⃗))
∣cosθi∣
cosθod θocosθi d θi
=ηiηo
CS 775: Lecture 7 Parag Chaudhuri, 2012
Fresnel Reflectance
● For dielectric specular surfaces, the amount of reflected light is given by
r∥=ηt cosθi−ηi cosθtηt cosθi+ηi cosθt
for parallel polarized light
r ⊥=ηi cosθi−ηt cosθtηi cosθi+ηt cosθt
for perpendicular polarized light
F r=12(r∥
2+r ⊥
2) for unpolarized light
CS 775: Lecture 7 Parag Chaudhuri, 2012
Fresnel Reflectance
● For a typical dielectric
http://www.graphics.cornell.edu/~westin/misc/fresnel.html
η=1.5
CS 775: Lecture 7 Parag Chaudhuri, 2012
Glossy BRDF
f Phong=k s (ωo . R (ωi , n̂))g
ωi
f Blinn−Phong=k s ' (ωhalf . n̂)h
ωr
ωo
n⃗
ωi
ωr
ωhalf
n⃗
Phenomenological
Also, phenomenological but gives less error
Experimental Validation of Analytical BRDF Models, Addy Ngan Fredo Durand, Wojciech Matusik , Technical Sketch, SIGGRAPH2004
ωo
CS 775: Lecture 7 Parag Chaudhuri, 2012
Glossy BRDFs
Experimental Validation of Analytical BRDF Models, Addy Ngan Fredo Durand, Wojciech Matusik , Technical Sketch, SIGGRAPH2004
PhongBlinnPhong
CS 775: Lecture 7 Parag Chaudhuri, 2012
Microfacet BRDFs
n⃗ f
Described by a function giving the distribution of microfacet normals, with respect to the surface normal . Greater variation indicates a rougher surface.
n⃗
n⃗ fn⃗
Also, necessary to describe the BRDF for the individual facets.
First described by TorranceSparrow (1967).
CS 775: Lecture 7 Parag Chaudhuri, 2012
Microfacet BRDFs
ωi
In BlinnPhong, the distribution of microfacet normals is approximated by an exponential falloff [Blinn 1977].
n⃗
The most likely orientation in this model is in the direction of the surface normal direction, falling off to no microfacets oriented perpendicular to the normal. For smooth surfaces the fall of is fast compared to a slow fall of for rough surfaces.
D(ωh)∝(ωh⋅n̂)h
ωo
ωh
D(ωh)=h+22π
(ωh⋅n̂)h
CS 775: Lecture 7 Parag Chaudhuri, 2012
Microfacet BRDFs
ωi
The complete TorrenceSparrow BRDF isn⃗
The most likely orientation in this model is in the direction of the surface normal direction, falling off to no microfacets oriented perpendicular to the normal. For smooth surfaces the fall of is fast compared to a slow fall of for rough surfaces.
The model also assumes facets are along infinitely long Vshaped grooves.
f ( p ,ωo ,ωi)=D(ωh)G(ωo ,ωi)F r (ωi)
4 cosθocosθi
ωo
ωh
D(ωh) gives microfacet distribution
G (ωo ,ωi) resolves visibility between a given pair of directions
F r (ωi) is the Fresnel Term
CS 775: Lecture 7 Parag Chaudhuri, 2012
Microfacet BRDFs
Experimental Validation of Analytical BRDF Models, Addy Ngan Fredo Durand, Wojciech Matusik , Technical Sketch, SIGGRAPH2004
[2000]
[1981,1982]
[1977]
[1992][1997]
CS 775: Lecture 7 Parag Chaudhuri, 2012
Microfacet BRDFs
Anisotropic BRDFs
http://www.evermotion.org/tutorials/show/7875/anisotropicshadertutorialusingvray15finalsp1
CS 775: Lecture 7 Parag Chaudhuri, 2012
Microfacet BRDFs
● Oren-Nayar Diffuse BRDF [1994]
– Symmetric V-shaped grooves
– Gaussian Distribution of Microfacets
– Each facet is perfectly Lambertian
http://en.wikipedia.org/wiki/Oren–Nayar_reflectance_model
http://www.larrygritz.com/arman/materials.html
OrenNayar Ideal Diffuse
CS 775: Lecture 7 Parag Chaudhuri, 2012
The Rendering Equation
● Now that we know the 4D-5D BRDFs, we can write the rendering equation as:
● Next question: How to solve the rendering equation for all points in the environment.
Lo( p ,ωo)=Le( p ,ωo)+∫Ω
f r ( p ,ωo ,ωi)L i ( p ,ωi)cosθi d ωi