University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell Reflection...

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