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CS348B Lecture 11 Pat Hanrahan, Spring 2004

Reflection Models

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Reflection models

The reflection equation and the BRDF

Ideal reflection, refraction and diffuse

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Phong and microfacet models

Gaussian height field on surface

Self-shadowing

Torrance-Sparrow model

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Anisotropic: Grooves and hair

Translucency: Skin

Phong Model


Reflection Geometry

ˆ ˆcos iθ = •L N

ˆ ˆcos rθ = •E N

ˆ ˆˆ ˆ ˆ ˆcos ( ) ( )sθ = • = •

N NE R L R E L

ˆ ˆcos sθ ′ = •H N

ˆ ˆcos gθ = •E L

L

HN

Eˆ

ˆ ˆ( )N

R E

ˆˆ ˆ( )

NR L L ˆ

ˆ ˆ( )N

R LN

H E

ˆ ˆˆ

ˆ ˆ+

=+

L EH

L E


Phong Model

E

L

NR(L)

( )ˆˆ ˆ( )

s•

NE R L

E

L

N R(E)

( )ˆˆ ˆ( )

s•

NL R E

( ) ( )ˆ ˆ ˆ ˆ( ) ( )s s

• = •E R L L R EReciprocity:

Distributed light source!

CS348B Lecture 11 Pat Hanrahan, Spring 2004s

Phong Model

Mirror Diffuse


Energy normalization

Energy normalize Phong Model

( )

( )2

2

2

2ˆ

ˆ( )

ˆˆ( )

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

→ = •

≤ •

≤ =+

∫

∫

∫

NN

NR

L R E

L R E

Microfacet Model


Bouguer’s “little faces”

P. Bouguer, Treatise on Optics, 1760


Reflection of the Sun from the Sea

Minnaert, Light and Color in theOutdoors, p. 28

α

2α


Reflection Angles

cosh r θ=

PH H′

βα

γδ

α β γ δ+ = +

β α δ− =2γ α⇒ =

δ

Side viewrθ

βα

Assume L and E are at the same height h

L E


Reflection Angles

H H′

αFront view

b

tanb

hα =

tan

tan

tan cos

b

rh

r

ψ

α

α θ

=

=

=

L

cosh r θ=

P


Analysis on the Sphere

N

Lˆˆ( )

NR L

α

2γ α=

γ

tan tan cosψ α θ=


Solid Angle Distributions

iθ ′

rθ ′

hω

rω

( )

( )

sin

sin 2 2

2sin cos 2

4cos sin

4cos

r r r r

i i i

i i i i

i i i i

i h

d d d

d d

d d

d d

d

ω θ θ ϕ

θ θ ϕ

θ θ θ ϕ

θ θ θ ϕ

θ ω

=

′′=

′ ′ ′=

′ ′ ′=

′=

1

4 cosh

r i

d

d

ωω θ

=′

2

r i r

i

θ θ θ

θ

′ ′= +

′=

r iϕ ϕ=

r iθ θ′ ′=


Microfacet Distributions

iθ ′rθ ′

N HL

E

Microfacet

2

( )cosh h h

H

dA d dAω θ ω =∫

Total projected area

( )hdA ωArea distribution

( ) ( )/h hD dA dAω ω≡Microfacet distribution

2

( )cos 1h h h

H

D dω θ ω =∫

Probability distribution


Microfacet Distribution Functions

Isotropic distributions

Characterize by half-angle

Examples:

Blinn

Torrance-Sparrow

Trowbridge-Reitz

11( ) coscD α α= 1

ln 2

ln cosc

β=

22( )

2 ( ) cD e αα −=2

2c

β=

23

3 2 23

( )(1 )cos 1

cD

cα

α=

− −122

3 2

cos 1

cos 2c

ββ

−=

−

( ) ( )hD Dω α⇒1

( )2

D β =β


Gaussian Rough Surface

Gaussian distribution

of heights

Gaussian distribution

of slopes

Beckmann

2

221( )

2

z

p z e σ

πσ

−=

2

2

tan

2 2

1( )

cosmD e

m

α

απ α

−=

2m

στ

=

σ

τ

θz

x

Torrance-Sparrow Model



( ) ( )h hdA D dAω ω=

( )cos ( )

( )cos ( )h i i i i h h

i i i i h h

d L d dA d

L d D dAd

ω θ ω ω ωω θ ω ω ω

′ ′Φ =′ ′=

( )cos

( )cos ( )r i r r r

i i i i h h

dL d dA

L d D d dA

ω ω θ ωω θ ω ω ω

∴ →′ ′=

r hd dΦ = Φ

( )cosr r i r r rd dL d dAω ω θ ωΦ = →ˆ ˆcos iθ = •L Nˆ ˆcos iθ ′ = •L H

iθ ′N H

L



( ) ( )

( )( )

( )

( )cos ( )

cos ( )cos

( )cos

cos cos

( )

4cos cos

r i rr i r

i i

i i i i h h

r r i i i i

h hi

i r r

h

i r

dLf

dE

L d D d dA

d dA L d

D d

d

D

ω ωω ωω

ω θ ω ω ωθ ω ω θ ω

ω ωθθ θ ωωθ θ

→→ ≡

′ ′=

′=

=

N HL

E

i id dω ω′ =

( )cos

( )cos ( )r i r r r

i i i i h h

dL d dA

L d D d dA

ω ω θ ωω θ ω ω ω→

′ ′=

Self-Shadowing


Shadows on Rough Surfaces

Without self-shadowing With self-shadowing


Self-Shadowing Function

Probability of shadowing

( )2 2/ 222 ( ) erfc / 2mm

e mµµ µπ µ

− Λ = −

( )11 erfc / 2

2( )

1 ( )

mS

µθ

µ

− =+ Λ

σ

τ

θz

x


Self-Shadowing Function

From Smith, 1967

W=m


Self-Consistency Condition

The sum of the areas of the illuminated surface projected onto the plane normal to the direction of incidence is independent of the roughness of the surface, and equal to the projected area of the underlying mean plane.

( ) ( )cos cosS D d αθ α θ ω θ′ =∫


Torrance-Sparrow Theory

( )

( ) ( ) ( ) ( )

4 cos cos

r i r

i i r

i r

f

F S S D

ω ωθ θ θ α

θ θ

→′

=


Torrance-Sparrow Comparison

Magnesium Oxide


Torrance-Sparrow Comparison

Aluminum

Self-Shadowing

V-Groove Model


Self-Shadowing: V-Groove Model

Assumptions (Torrance-Sparrow)

1. Symmetric, longitudinal, isotropically-distributed

2. Upper edges lie in plane

1aG =

ˆ ˆ ˆ ˆ2( )( )ˆ ˆ( )bG

• •=

•N H N E

H E

ˆ ˆ ˆ ˆ2( )( )ˆ ˆ( )cG

• •=

•N H N L

H L

min( , , )a b cG G G G=


Self-Shadowing: V-Groove Model

1

sin1

sinˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2( )( )

ˆ ˆ

ˆ ˆ ˆ ˆ2( )( )ˆ ˆ

mG

lm

l

= −

= −

• − • + • •=

•• •

=•

H E H E N H N EH E

N H N EH E

α

m l

sin

sin

m m

l l=

sin cos

cos sini

i

l

l

θθ′=′=

( )( )

2

2

sin sin 2

sin cos 2 cos sin 2

cos cos 2 sin sin 2

cos (1 2 sin ) sin 2 cos sin

cos (1 2 cos ) sin 2 cos sin

cos 2 cos cos cos sin sin

cos 2 cos cos

cos 2 cos cos

ˆ

i i

i i

i i

i i i

i i

i r

m l

l l

ψψ ψ

θ ψ θ ψ

θ ψ θ ψ ψ

θ α θ α α

θ α α θ α θ

θ α α θ

θ α θ

= += +

′ ′= +

′ ′= − +

′ ′= − +

′ ′ ′′= − −

′ ′= − +

′= −

= •H ( ) ( )ˆ ˆ ˆ ˆ ˆ2− • •E N H N E

iθ ′

ψ ψsin cos

cos sin

ψ αψ α==

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