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