university of texas at austin cs395t - advanced image synthesis spring 2007 don fussell reflection...
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University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Reflection Models I
TodayTypes of reflection models
The BRDF and reflectance
The reflection equation
Ideal reflection and refraction
Fresnel effect
Ideal diffuse
Next lectureGlossy and specular reflection models
Rough surfaces and microfacets
Self-shadowing
Anisotropic reflection models
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
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 [Moon Spectra]PolarizationDirectional distribution
TheoriesPhenomenologicalPhysical
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Types of Reflection Functions
Ideal SpecularReflection LawMirror
Ideal DiffuseLambert’s LawMatte
SpecularGlossyDirectional diffuse
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Materials
Plastic Metal Matte
From Apodaca and Gritz, Advanced RenderMan
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
The Reflection Equation
2
( , ) ( , ) ( , )cosr r r i r i i i i
H
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
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
The BRDF
Bidirectional Reflectance-Distribution Function
( ) 1( ) r i r
r i ri
dLf
dE sr
ω ωω ω → ⎡ ⎤→ ≡ ⎢ ⎥⎣ ⎦
( , )r rdL x ω
( , )i iL x ω
idωiθ
rθ
rφiφ
N
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
The BSSRDF
Bidirectional Surface Scattering Reflectance-Distribution Function
( , , )( , , ) r i i r r
i i r ri
dL x xS x x
d
ω ωω ω →→ ≡
Φ
( , )r rdL x ω
( , )i iL x ω
idωiθ
rθ
rφiφ
N
rx ix
Translucency
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Gonioreflectometer
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Properties of BRDF’s
1. Linearity
2. Reciprocity principle
( ) ( )r r i r i rf fω ω ω ω→ = →
From Sillion, Arvo, Westin, Greenberg
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Properties of BRDF’s
3. Isotropic vs. anisotropic
4. Energy conservation
( , ; , ) ( , , )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
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Energy Conservation
( )cos
( )cos
( ) ( )cos cos
( )cos
1
r
i
r i
i
r r r r
r
i i i i i
r i r i i i i r r
i i i i
L dd
d L d
f L d d
L d
ω θ ω
ω θ ω
ω ω ω θ ω θ ω
ω θ ω
Ω
Ω
Ω Ω
Ω
Φ=
Φ
→
=
≤
∫
∫
∫∫
∫
iΩrΩ
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
The Reflectance
Definition: Reflectance is ratio of reflected to incident power
Conservation of energy: 0 < < 13 by 3 set of possibilities:
Units: [dimensionless], fr [1/steradians]
( )cos cos
( )cos
( ) ( )cos cos
( )cos
r i
i
r i
i
r i r i i r r
i r
i i
r i r i i i i r r
i i i i
f d d
d
f L d d
L d
ω ω θ ω θ ω
θ ω
ω ω ω θ ω θ ω
ω θ ω
Ω Ω
Ω
Ω Ω
Ω
→
Ω → Ω ≡
→
≤
∫∫
∫
∫∫
∫
iΩrΩ
{ } { }2 2, , , ,i i i r r rd H d Hω ωΩ × Ω
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Law of Reflection
r iθ θ=
rθiθ iϕ rϕ
( )mod 2r iϕ ϕ π π= +
NRI
ˆ ˆ ˆ ˆ ˆ ˆ( ) 2 cos 2( )θ+ − = =− •R I N I N N
ˆ ˆ ˆ ˆ ˆ2( )= − •R I I N N
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Ideal Reflection (Mirror)
,
(cos cos )( , ; , ) ( )
cosi r
r m i i r r i ri
fδ 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 cos
cos
( , )
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 θ δ ϕ ϕ π θ ϕ θ θ ϕθ
θ ϕ π
=
−= − ±
= ±
∫
∫
rθiθ
( , )i i iL θ ϕ ( , )r r rL θ ϕ
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Snell’s Law
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= ±
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Law of Refraction
tθ
iθ
N
T
I
/i tn nμ =
ˆ ˆ ˆ ˆμ× = ×N T N I
ˆ ˆ ˆ( ) 0μ× − =N T I
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 cos
cos 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:
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Optical Manhole
From Livingston and Lynch
Total internal reflection
4
3wn =
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Fresnel ReflectanceMetal (Aluminum) Dielectric (N=1.5)
5( ) (0) (1 (0))(1 cos )F F Fθ θ= + − −Schlick Approximation
Glass n=1.5 F(0)=0.04Diamond n=2.4 F(0)=0.15
Gold F(0)=0.82Silver F(0)=0.95
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Experiment
Reflections from a shiny floor
From Lafortune, Foo, Torrance, Greenberg, SIGGRAPH 97
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Cook-Torrance Model for Metals
Measured Reflectance
Reflectance of Copper as afunction of wavelength and
angle of incidence
2
π λ
Light spectra
Copper spectraθ
Approximated Reflectance
( ) (0)(0) ( / 2)
( / 2) (0)
F FR R R
F F
θππ
⎡ ⎤−= + ⎢ ⎥−⎣ ⎦
Cook-Torrance approximation
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Ideal Diffuse Reflection
Assume light is equally likely to be reflected in any output direction (independent of input 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 Lf f
E E E
π π ππ
= = = = ⇒ =
cosd d s sM E E θ= =Lambert’s Cosine Law
,r df c=
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
“Diffuse” Reflection
TheoreticalBouguer - Special micro-facet distribution
Seeliger - Subsurface reflection
Multiple surface or subsurface reflections
ExperimentalPressed magnesium oxide powder
Almost never valid at high angles of incidence
Paint manufactures attempt to create ideal diffuse
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Phong Model
E
L
NR(L)
( )ˆ ˆ( )s
•E R L
E
L
N R(E)
( )ˆ ˆ( )s
•L R E
( ) ( )ˆ ˆ ˆ ˆ( ) ( )s s
• = •E R L L R EReciprocity:
Distributed light source!
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussells
Phong Model
Mirror Diffuse
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell
Properties of the Phong Model
Energy normalize Phong Model
( )
( )2
2
2
2
ˆ( )
ˆ( )
ˆ ˆ( ) ( ) cos
ˆ ˆ( )
2cos
1
s
r i i
H
s
ir
H
s
H
H d
d
ds
ρ ω θ ω
ω
πθ ω
→ = •
≤ •
≤ =+
∫
∫
∫
N
R
L R E
L R E