University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell Reflection Models I Today Types of reflection models The BRDF and

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


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


Types of Reflection Functions

Ideal SpecularReflection LawMirror

Ideal DiffuseLambert’s LawMatte

SpecularGlossyDirectional diffuse


Materials

Plastic Metal Matte

From Apodaca and Gritz, Advanced RenderMan


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


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


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


Gonioreflectometer


Properties of BRDF’s

1. Linearity

2. Reciprocity principle

( ) ( )r r i r i rf fω ω ω ω→ = →

From Sillion, Arvo, Westin, Greenberg


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


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


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


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


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


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= ±


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:


Optical Manhole

From Livingston and Lynch

Total internal reflection

4

3wn =


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


Experiment

Reflections from a shiny floor

From Lafortune, Foo, Torrance, Greenberg, SIGGRAPH 97


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


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=


“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


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


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

Documents

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell Reflection Models I Today Types of reflection models The BRDF and