csce 641 computer graphics: reflection models jinxiang chai
TRANSCRIPT
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CSCE 641 Computer Graphics:
Reflection Models
Jinxiang Chai
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Motivation
How to model surface properties of objects?
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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.
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Types of Reflection Functions
Ideal Specular Reflection Law
Mirror
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Types of Reflection Functions
Ideal Specular Reflection Law
Mirror
Ideal Diffuse Lambert’s Law
Matte
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Types of Reflection Functions
Ideal Specular Reflection Law
Mirror
Ideal Diffuse Lambert’s Law
Matte
Specular Glossy
Directional diffuse
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Review: Local Illumination
)( NLCkAkI da
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Review: Local Illumination
nsda ERkNLkCAkI )()(
5n
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Review: Local Illumination
nsda ERkNLkCAkI )()(
50n
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Materials
Plastic Metal Matte
From Apodaca and Gritz, Advanced RenderMan
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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?
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Introduction
Radiometry and photometry
- Radiant intensity
- Irradiance
- Radiance
- Radiant exitance (radiosity)
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Electromagnetic Spectrum
Visible light frequencies range between:
Red: 4.3x1014 hertz (700nm)
Violet: 7.5x1014 hertz (400nm)
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Visible Light
The human eye can see “visible” light in the frequency between 400nm-700nm
redviolet
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Spectral Energy Distribution
Three different types of lights
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Spectral Energy Distribution
Three different types of lights
How can we measure the energy of light?
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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
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(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
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Example: Fluorescent Bulb
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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
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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
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Radiance
Per unit projected surface area
- radiance varies with direction
- incidence radiance and exitant radiance
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Radiance
Ray of light arriving at or leaving a point on a surface in a given direction
Radiance (arriving) Radiance (leaving)
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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
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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
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Directional Power Leaving a Surface
2 ( , ) ( , )coso od x L x dAd
dA d
( , )oL x
2 ( , )od x
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Uniform Diffuse Emitter
( , )oL x
dA
d
2
2
cos
cos
o
H
o
H
M L d
L d
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Projected Solid Angle
cos d
2
cosH
d
d cos d
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Uniform Diffuse Emitter
( , )oL x
dA
d
2
2
cos
cos
o
H
o
H
o
M L d
L d
L
o
ML
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Reflection Models
Outline
- Types of reflection models
- The BRDF and reflectance
- The reflection equation
- Ideal reflection
- Ideal diffuse
- Cook-Torrance Model
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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.
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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.
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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.
![Page 34: CSCE 641 Computer Graphics: Reflection Models Jinxiang Chai](https://reader035.vdocuments.site/reader035/viewer/2022062809/5697bf881a28abf838c896af/html5/thumbnails/34.jpg)
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.
![Page 35: CSCE 641 Computer Graphics: Reflection Models Jinxiang Chai](https://reader035.vdocuments.site/reader035/viewer/2022062809/5697bf881a28abf838c896af/html5/thumbnails/35.jpg)
Types of Reflection Functions
Ideal Specular Reflection Law
Mirror
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Types of Reflection Functions
Ideal Specular Reflection Law
Mirror
Ideal Diffuse Lambert’s Law
Matte
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Types of Reflection Functions
Ideal Specular Reflection Law
Mirror
Ideal Diffuse Lambert’s Law
Matte
Specular Glossy
Directional diffuse
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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?
![Page 39: CSCE 641 Computer Graphics: Reflection Models Jinxiang Chai](https://reader035.vdocuments.site/reader035/viewer/2022062809/5697bf881a28abf838c896af/html5/thumbnails/39.jpg)
The Reflection Equation
( , )r rL x
( , )i iL x
idi
r
r i
N
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The Reflection Equation
( , )r rL x
( , )i iL x
idi
r
r i
N
How to model surface reflectance property?
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The Reflection Equation
( , )r rL x
( , )i iL x
idi
r
r i
N
i
rrri E
xLxf
),(),(
How to model surface reflectance property?
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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
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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
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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
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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
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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
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Gonioreflectometer
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Scattering Models
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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
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Properties of BRDF’s
From Sillion, Arvo, Westin, Greenberg
1. Linearity
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Properties of BRDF’s
( ) ( )r r i r i rf f
From Sillion, Arvo, Westin, Greenberg
1. Linearity
2. Reciprocity principle
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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
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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
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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
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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
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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]
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BRDF
What’s the BRDF for a mirror surface?
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Law of Reflection
r i
rii r
mod2r i
N RI
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Ideal Reflection (Mirror)
, ( , ) ( , )r m r r i r rL L ri
( , )i i iL ( , )r r rL
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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
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, ,( , ) ( , ; , ) ( , )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
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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
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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(
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Phong Model
E
L
NR(L) E
L
N R(E)
Reciprocity:
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s
Phong Model
Mirror Diffuse
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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]
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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]
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Cook-Torrance Model
More sophisticated for specular reflection
Surface is made of reflecting microfacets
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Cook-Torrance Model
H: angular bisector of vector L and V
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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.
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Varying m
m = 0.2 m = 0.6
Highly directional vs. spread-out
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Cook-Torrance Model
H: angular bisector of vector L and V
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Mask and Shadow Term
No interference
shadow mask
The geometrical attenuation factor G accounts for the shadowing andmasking of one facet by another.
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Cook-Torrance Model
H: angular bisector of vector L and V
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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
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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