CSCE 641 Computer Graphics:

Reflection Models

Jinxiang Chai

Motivation

How to model surface properties of objects?

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.

Types of Reflection Functions

Ideal Specular Reflection Law

Mirror

Types of Reflection Functions

Ideal Specular Reflection Law

Mirror

Ideal Diffuse Lambert’s Law

Matte

Types of Reflection Functions

Ideal Specular Reflection Law

Mirror

Ideal Diffuse Lambert’s Law

Matte

Specular Glossy

Directional diffuse

Review: Local Illumination

)( NLCkAkI da

Review: Local Illumination

nsda ERkNLkCAkI )()(

5n

Review: Local Illumination

nsda ERkNLkCAkI )()(

50n

Materials

Plastic Metal Matte

From Apodaca and Gritz, Advanced RenderMan

Mathematical Reflectance Models

How can we mathematically model reflectance property of an arbitrary surface?

- what’s the dimensionality?

- how to use it to compute outgoing radiance given incoming radiance?

Introduction

Radiometry and photometry

- Radiant intensity

- Irradiance

- Radiance

- Radiant exitance (radiosity)

Electromagnetic Spectrum

Visible light frequencies range between:

Red: 4.3x1014 hertz (700nm)

Violet: 7.5x1014 hertz (400nm)

Visible Light

The human eye can see “visible” light in the frequency between 400nm-700nm

redviolet

Spectral Energy Distribution

Three different types of lights

Spectral Energy Distribution

Three different types of lights

How can we measure the energy of light?

Photons

The basic quantity in lighting is the photon

The energy (in Joule) of a photon with wavelength λ is: qλ = hc/λ

- c is the speed of light

In vacuum, c = 299.792.458m/s

- h ≈ 6.63*10-34Js is Planck’s constant

(Spectral) Radiant Energy

The spectral radiant energy, Qλ, in nλ photons with wavelength λ is

The radiant energy, Q, is the energy of a collection of photons, and is given as the integral of Qλ over all possible wavelengths:

qnQ

0

dQQ

Example: Fluorescent Bulb

Radiant Intensity

( )d

Id

W lmcd candela

sr sr

Definition: the radiant power radiated from a point on a light source into a unit solid angle in a particular direction.

- measured in watt per steradian

Radiance

Definition: the radiant power per unit projected surface area per solid angle

2

( , )( , )

( , )

dI xL x

dA

d x

d dA

2 2 2

W cd lmnit

sr m m sr m

dA

d

( , )L x

Radiance

Per unit projected surface area

- radiance varies with direction

- incidence radiance and exitant radiance

Radiance

Ray of light arriving at or leaving a point on a surface in a given direction

Radiance (arriving) Radiance (leaving)

Irradiance

Definition: the power per unit area incident on a surface.

( ) id

E xdA

2

( ) ( , )cosi

H

E x L x d

( , )iL x

dA

d

Radiant Exitance

Definition: The radiant (luminous) exitance is the

energy per unit area leaving a surface.

In computer graphics, this quantity is often

referred to as the radiosity (B)

( ) odM x

dA

Directional Power Leaving a Surface

2 ( , ) ( , )coso od x L x dAd

dA d

( , )oL x

2 ( , )od x

Uniform Diffuse Emitter

( , )oL x

dA

d

2

2

cos

cos

o

H

o

H

M L d

L d

Projected Solid Angle

cos d

2

cosH

d

d cos d

Uniform Diffuse Emitter

( , )oL x

dA

d

2

2

cos

cos

o

H

o

H

o

M L d

L d

L

o

ML

Reflection Models

Outline

- Types of reflection models

- The BRDF and reflectance

- The reflection equation

- Ideal reflection

- Ideal diffuse

- Cook-Torrance Model

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.

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.

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.

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.

Types of Reflection Functions

Ideal Specular Reflection Law

Mirror

Types of Reflection Functions

Ideal Specular Reflection Law

Mirror

Ideal Diffuse Lambert’s Law

Matte

Types of Reflection Functions

Ideal Specular Reflection Law

Mirror

Ideal Diffuse Lambert’s Law

Matte

Specular Glossy

Directional diffuse

Mathematical Reflectance Models

How can we mathematically model reflectance property of an arbitrary surface?

- what’s the dimensionality?

- how to use it to compute outgoing radiance given incoming radiance?

The Reflection Equation

( , )r rL x

( , )i iL x

idi

r

r i

N

The Reflection Equation

( , )r rL x

( , )i iL x

idi

r

r i

N

How to model surface reflectance property?

The Reflection Equation

( , )r rL x

( , )i iL x

idi

r

r i

N

i

rrri E

xLxf

),(),(

How to model surface reflectance property?

The Reflection Equation

( , )r rL x

( , )i iL x

idi

r

r i

N

i

rrri E

xLxf

),(),(

How to model surface reflectance property?

for a given incoming direction, the amount of light that is reflected in a certain outgoing direction

The Reflection Equation

2

( , ) ( , ) ( , )cosr r r i r i i i i

H

L x f x L x d

( , )r rL x

( , )i iL x

idi

r

r i

N

The Reflection Equation

2

( , ) ( , ) ( , )cosr r r i r i i i i

H

L x f x L x d

( , )r rL x

( , )i iL x

idi

r

r i

N

Directional irradiance

The BRDF

Bidirectional Reflectance Distribution Function

( ) 1( ) r i r

r i ri

dLf

dE sr

( , )r rdL x

( , )i iL x

idi

r

ri

N

Dimensionality of the BRDF

This is 6-D function

- x: 2D position

- (θi,φi): incoming direction

- (θr,φr): outgoing direction

Usually represented as 4-D

- ignoring x

- homogenous material property

Gonioreflectometer

Scattering Models

The BSSRDF

Bidirectional Surface Scattering Reflectance-Distribution Function

( , , )( , , ) r i i r r

i i r ri

dL x xS x x

d

( , )r rdL x

( , )i iL x

idi

r

ri

N

rx ix

Translucency

Properties of BRDF’s

From Sillion, Arvo, Westin, Greenberg

1. Linearity

Properties of BRDF’s

( ) ( )r r i r i rf f

From Sillion, Arvo, Westin, Greenberg

1. Linearity

2. Reciprocity principle

Properties of BRDF’s

3. Isotropic vs. anisotropic

( , ; , ) ( , , )r i i r r r i r r if f

( , , ) ( , , ) ( , , )r i r r i r r i i r r i r r if f f

Reciprocity and isotropy

Properties of BRDF’s

3. Isotropic vs. anisotropic

4. Energy conservation

( , ; , ) ( , , )r i i r r r i r r if f

( , , ) ( , , ) ( , , )r i r r i r r i i r r i r r if f f

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

ir

Definition: Reflectance is ratio of reflected to incident power

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

ir

Definition: Reflectance is ratio of reflected to incident power

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

ir

Definition: Reflectance is ratio of reflected to incident power

Conservation of energy: 0 < < 1

Units: [dimensionless], fr [1/steradians]

BRDF

What’s the BRDF for a mirror surface?

Law of Reflection

r i

rii r

mod2r i

N RI

Ideal Reflection (Mirror)

, ( , ) ( , )r m r r i r rL L ri

( , )i i iL ( , )r r rL

Ideal Reflection (Mirror)

,

(cos cos )( , ; , ) ( )

cosi r

r m i i r r i ri

f

, ( , ) ( , )r m r r i r rL L ri

( , )i i iL ( , )r r rL

, ,( , ) ( , ; , ) ( , )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

Ideal Reflection (Mirror)

,

(cos cos )( , ; , ) ( )

cosi r

r m i i r r i ri

f

, ( , ) ( , )r m r r i r rL L ri

( , )i i iL ( , )r r rL

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

( )cos cosr r r r r r r rM L d L d L ,

, ,r d dr

d r d r d

f EM Lf f

E E E

cosd d s sM E E Lambert’s Cosine Law

,r df c

Diffuse Light

C = intensity of point light source

kd = diffuse reflection coefficient

= angle between normal and direction to light

)()cos( NLkCkCI dd

LN

Surface

NL)cos(

Phong Model

E

L

NR(L) E

L

N R(E)

Reciprocity:

s

Phong Model

Mirror Diffuse

BRDF Models

Phenomenological

- Phong [75]

- Blinn-Phong [77]

- Ward [92]

- Larfortune et al [97]

- Ashikhmin et al [00]

Physical - Cook-torrence [81]

- He et al [91]

Data-driven

[Matusik et al 2003]

[Cook-Torrence 81]

[Larfortune et al 97]

BRDF Models

Phenomenological

- Phong [75]

- Blinn-Phong [77]

- Ward [92]

- Larfortune et al [97]

- Ashikhmin et al [00]

Physical - Cook-torrence [81]

- He et al [91]

Data-driven

[Matusik et al 2003]

[Cook-Torrence 81]

[Larfortune et al 97]

Cook-Torrance Model

More sophisticated for specular reflection

Surface is made of reflecting microfacets

Cook-Torrance Model

H: angular bisector of vector L and V

Microfacet Slope Distribution (D)

D: density of microfacets along H

m: root mean square of slopes of microfacets (i.e. roughness)

Beckman function:

The facet slope distribution function D represents the fraction of the facets thatare oriented in the direction H.

Varying m

m = 0.2 m = 0.6

Highly directional vs. spread-out

Cook-Torrance Model

H: angular bisector of vector L and V

Mask and Shadow Term

No interference

shadow mask

The geometrical attenuation factor G accounts for the shadowing andmasking of one facet by another.

Cook-Torrance Model

H: angular bisector of vector L and V

Fresnel Term

n: refractive index

The Fresnel term F describes how light is reflected from each smooth microfacet. It is a function of incidence angle and wavelength

Cook-Torrance Model for Metals

Measured Reflectance

Reflectance of Copper as afunction of wavelength and

angle of incidence

2

Light spectra

Copper spectra

Approximated Reflectance

( ) (0)(0) ( / 2)

( / 2) (0)

F FR R R

F F

Cook-Torrance approximation