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CS348b Lecture 10 Pat Hanrahan, Spring 2016
Reflection Models I: Ideal Materials
Today
■ Types of reflection models
■ The BRDF and reflectance
■ The reflection equation
■ Ideal reflection and refraction
■ Fresnel effect
■ Ideal diffuse
Next lecture (Thursday)
■ Glossy and specular reflection models
■ Rough surfaces and microfacets
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Reflection Models
Definition: Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident side without change in frequency.
Properties
■ Spectra and Color
■ Polarization
■ Directional distribution
Gonio-reflectometer
CS348b Lecture 10 Pat Hanrahan, Spring 2016
CS348b Lecture 10 Pat Hanrahan, Spring 2016
The BRDF
Bidirectional Reflectance-Distribution Function
( ) 1( ) r i rr i r
i
dLfdE srω ω
ω ω→ # $→ ≡ & '( )
( , )r rdL x ω
( , )i iL x ω
idωiθ
rθ
rφiφ
N
The Reflection Equation
CS348b Lecture 10 Pat Hanrahan, Spring 2016
2
( , ) ( , ) ( , )cosr r r i r i i i iH
L x f x L x dω ω ω ω θ ω= →∫
( , )r rL x ω
( , )i iL x ω
idωiθ
rθ
rφ iφ
N
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Properties of BRDF’s
1. Linearity
From Sillion, Arvo, Westin, Greenberg
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Properties of BRDF’s
2. Reciprocity principle
( ) ( )r r i r i rf fω ω ω ω→ = →
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Properties of BRDF’s
3. Isotropic vs. anisotropic
( , ; , ) ( , , )r i i r r r i r r if fθ ϕ θ ϕ θ θ ϕ ϕ= −
( , , ) ( , , ) ( , , )r i r r i r r i i r r i r r if f fθ θ ϕ ϕ θ θ ϕ ϕ θ θ ϕ ϕ− = − = −
Reciprocity and isotropy
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Properties of BRDF’s
4. Energy Conservation iΩ
rΩ
d�r
d�i
=
R⌦r
Lr
(!r
) cos ✓r
d!rR
⌦iLr
(!o
) cos ✓i
d!i
1
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Types of Reflection Functions
Ideal Specular
■ Reflection Law
■Mirror
Ideal Diffuse
■ Lambert’s Law
■Matte
Specular
■ Glossy
■ Directional diffuse
Law of Reflection
CS348b Lecture 10 Pat Hanrahan, Spring 2016
r iθ θ=
rθiθiϕ rϕ
( )mod2r iϕ ϕ π π= +
N RI
ˆ ˆ ˆ ˆ ˆ ˆ( ) 2 cos 2( )θ+ − = = − •R I N I N Nˆ ˆ ˆ ˆ ˆ2( )= − •R I I N N
Ideal Reflection (Mirror)
CS348b Lecture 10 Pat Hanrahan, Spring 2016
,(cos cos )( , ; , ) ( )
cosi r
r m i i r r i ri
f δ θ θθ ϕ θ ϕ δ ϕ ϕ π
θ−
= − ±
, ( , ) ( , )r m r r i r rL Lθ ϕ θ ϕ π= ±
, ,( , ) ( , ; , ) ( , )cos cos
(cos cos ) ( ) ( , )cos coscos
( , )
r m r r r m i i r r i i i i i i
i ri r i i i i i i
i
i r r
L f L d d
L d d
L
θ ϕ θ ϕ θ ϕ θ ϕ θ θ ϕ
δ θ θδ ϕ ϕ π θ ϕ θ θ ϕ
θ
θ ϕ π
=
−= − ±
= ±
∫
∫
rθiθ
( , )i i iL θ ϕ ( , )r r rL θ ϕ
Recall: Snell’s Law
CS348b Lecture 10 Pat Hanrahan, Spring 2016
sin sini i t tn nθ θ=
ˆ ˆ ˆ ˆi tn n× = ×N I N T
tθ
iθ
N
T
I
iϕ tϕ
t iϕ ϕ π= ±
Recall: Law of Refraction
CS348b Lecture 10 Pat Hanrahan, Spring 2016
tθ
iθ
N
T
I
/i tn nµ =
ˆ ˆ ˆµ γ= +T I N
2 2 2ˆ ˆ ˆ1 2µ γ µγ= = + + •T I N
( )( ){ }{ }
12
12
22
2 2
ˆ ˆ ˆ ˆ1 1
cos 1 sin
cos coscos cos
i i
i t
i t
γ µ µ
µ θ µ θ
µ θ θ
µ θ θ
= − • ± − − •
= ± −
= ±
= −
I N I N
1γ µ← = −
( )22 ˆ ˆ1 (1 ) 0µ− − • <I N
Total internal reflection:
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Experiment
Reflections from a shiny floor
Reflection is greater at glancing angles
From Lafortune, Foo, Torrance, Greenberg, SIGGRAPH 97
Fresnel Reflectance
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Dielectric (Glass n=1.5)
5( ) (0) (1 (0))(1 cos )F F Fθ θ= + − −Schlick Approximation
F(0)=0.04
Fresnel Reflectance
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Metal (Aluminum)
Gold F(0)=0.82 Silver F(0)=0.95
Reflection from Metals
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Measured Reflectance
Reflectance of Copper as a function of wavelength and angle of incidence
2π
λ
ρ
Light spectra
Copper spectraθ
Cook-Torrance Reflection Model
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Measured Reflectance
Reflectance of Copper as a function of wavelength and angle of incidence
2π
λ
ρ
Light spectra
Copper spectraθ
Approximated Reflectance
( ) (0)(0) ( / 2)( / 2) (0)F FR R RF F
θπ
π
# $−= + & '−( )
Cook-Torrance approximation
CS348b Lecture 10 Pat Hanrahan, Spring 2016
Ideal Diffuse Reflection
Assume light is equally likely to be reflected in any output direction
, ,
,
,
( ) ( )cos
( )cos
r d r r d i i i i
r d i i i i
r d
L f L d
f L d
f E
ω ω θ ω
ω θ ω
=
=
=
∫∫
( )cos cosr r r r r r r rM L d L d Lω θ ω θ ω π= = =∫ ∫,
, ,r d dr
d r d r d
f EM L f fE E E
π ρπρ π
π= = = = ⇒ =
cosd d s sM E Eρ ρ θ= =Lambert’s Cosine Law
,r df c=
CS348b Lecture 10 Pat Hanrahan, Spring 2016
“Diffuse” Reflection
Theoretical
■ Bouguer - Special micro-facet distribution
■ Seeliger - Subsurface reflection
■Multiple surface or subsurface reflections
Experimental
■ Pressed magnesium oxide powder
■ Almost never valid at high angles of incidence
Paint manufactures attempt to create ideal diffuse