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Lecture 4: Reflection Models
CS 6620, Spring 2009Kavita Bala
Computer ScienceCornell University
© Kavita Bala, Computer Science, Cornell University
Outline• Light sources
– Light source characteristics– Types of sources
• Light reflection– Physics-based models– Empirical models
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Sources of light radiation• Thermal radiation (“blackbody”)
– Sun, tungsten & tungsten-halogen lamps; arc lamps
• Electric discharge– gas discharge lamps (neon, sodium, mercury vapor)– arc lamps, fluorescent lamps
• Other phenomena– fluorescence (fluorescent lamps, fluorescent dyes)– phosphorescence (CRTs); LEDs; lasers
© Kavita Bala, Computer Science, Cornell University400 500 600 700
1
2
3
4
Wavelength
Brig
htne
ss
Mercurylines
Phosphor emission
400 500 600 700
1
2
3
4
Wavelength
Brig
htne
ss
Mercury vapor lampExamples of light emission
10-2
10-1
1
10
102
103
104
105
106
107
10 100 1000 10000
Wavelength
Inte
nsity
20,000K
10,000K
3000K2000
K1000K
500K
VisibleSpectrum
6000K
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Modeling luminaires• Spectral distribution
– Determined by physics of source– Generally tabulated, often RGB used
• Spatial distribution– Modeled as point or simple area light– Also light probes create high dynamic range inputs
• Directional distribution– Often shaped by reflectors– Tabulated when necessary, cosine lobe is common
approximation
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Directional distributions
Lambertian cosine-power arbitrary
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Lighting w/ Environment Maps• High lighting complexity
• Rich: captures real world
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Image-based lighting• Acquiring lighting information of real
scenes– Image-based techniques
• Use light probe
• Varying exposure
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Mirror Ball
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Sphere Maps• Assume viewing is from infinity• Creation uses photographs or ray tracing
or warping
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Environment Mapping
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Sphere Environment Mapping
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Types of Mappings
Sphere mapping
Cube mapping
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Dynamic Range of Sun
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Multiple Exposures
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Light interaction with matter• Volumetric scattering: interaction in 3D
– Atmosphere, water, semi-transparent objects
• Surface scattering: interaction in 2D– Surfaces of mainly opaque materials– The common case in many scenes– Heavily relied upon for graphics
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Surface reflective characteristics• Spectral distribution
– Responsible for surface color– Tabulate in independent wavelength bands, or RGB
• Spatial distribution– Material properties vary with surface position– Texture maps
• Directional distribution– BRDF — more complex than source– Tabulation is impractical because of dimensionality
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Reflection spectrum
SourceSpectrum
ReflectanceSpectrum
Product
? X
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Directional Distribution
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Reflectance—Three Forms
Ideal diffuse (Lambertian)
Directionaldiffuse
Idealspecular
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Ideal Diffuse Reflection• Characteristic of multiple scattering
materials• An idealization but reasonable for matte
surfaces• Basis of most radiosity methods• BRDF is a constant function
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Directional Diffuse Reflection• Characteristic of most rough surfaces• Described by the BRDF
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Ideal Specular Reflection• Calculated from Fresnel’s equations• Exact for polished surfaces• Basis of early ray-tracing methods
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Fresnel Reflection• Considers light as electromagnetic wave
• Polarization: rotation of electric field
• Effect of Fresnel reflection:– Most objects act as mirror reflectors when light
strikes them at grazing angles
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Grazing Angle
Real photographsReal photographs
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Fresnel Equations
2211 sinsin θηθη =
2211
2211
2112
2112
coscoscoscoscoscoscoscos
θηθηθηθηθηθηθηθη
+−
=
+−
=
s
p
F
F
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Fresnel Reflectance
2)|||(| 22
ps FFF
+= for unpolarized light
• Equations apply for metals and nonmetals– for metals, use complex index η = n+ik– for nonmetals, k=0
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Metal vs. Nonmetal
Metals
Nonmetals (k=0)
00
90
1
θ
Fresnel reflectance
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Fresnel Equations
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Mies van der Rohe’s unbuilt Courtyard House
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Directional Reflectance
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Classes of Models for the BRDF• Plausible simple functions
– Phong 1975;
• Physics-based models– Cook/Torrance, 1981; He et al. 1992;
• Empirically-based models– Ward 1992, Lafortune model
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Diffuse = )( LNkd ⋅ Specular = ns VRk )( ⋅
Phong Reflection ModelMirror
Reflection Vector
Diffuse Specular
LVR
d
n
sr kNRkf +
ΨΘ
=Ψ↔Θ).(
).()(
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The Blinn-Phong ModelHalf-Vector
LV
HSpecular
d
n
sr kN
HNkf +Ψ
=Ψ↔Θ).().()(
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The Modified Blinn-Phong Model
dn
sr kHNkf +=Ψ↔Θ ).()(
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The Phong Model• Computationally simple• Visually pleasing
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Phong: Reality Check
Real photographsReal photographs
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Phong: Reality CheckReal photographs
Phong model
Therefore, physically-based models
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Physics-based modelPhong model
Phong: Reality Check
• Computationally simple, visually pleasing • Doesn’t represent physical reality
– Energy not conserved– Not reciprocal (can be fixed with modification)– Maximum always in specular direction
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Cook-Torrance BRDF Model• A microfacet model
– Surface modeled as random collection of planar facets
– Incoming ray hits exactly one facet, at random• Input: probability distribution of facet angle
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Result of Cook-Torrance• Plastic has substrate that is white with
embedded pigment particles– Colored diffuse component– White specular component
• Metal – Specular component depends on metal– Negligible diffuse component
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Rob Cook’s vases
Source: Cook, Torrance 1981
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Cook-Torrance BRDF Model• A microfacet model
– Surface modeled as random collection of planar facets
– Incoming ray hits exactly one facet, at random• Input: probability distribution of facet angle
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N
VL
Hα
θ θ
Facet Reflection• H vector used to define facets that
contribute
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))(( VNLNDGFRs ⋅⋅
=π
Fresnel Reflectance
Cook-Torrance BRDF Model
• “Specular” term (really directional diffuse)
• Fresnel reflectance for smooth facet
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Cook-Torrance BRDF Model
))(( VNLNDGFRs ⋅⋅
=π
Facet distribution
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Facet Distribution• D function describes distribution of H• Formula due to Beckmann
– derivation based on Gaussian height distribution
2tan
42 cos1 ⎥⎦
⎤⎢⎣⎡−
= mem
Dα
α
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Cook-Torrance BRDF Model
))(( VNLNDGFRs ⋅⋅
=π
Masking/shadowing
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Unoccluded
LV
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Masking
VN.
V
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Self-Shadowing
LN.
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Masking and Shadowing
⎥⎦⎤
⎢⎣⎡
⋅⋅⋅
⋅⋅⋅
=)(
))((2,)(
))((2,1minHV
LNHNHV
VNHNG
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Rob Cook’s vases
Source: Cook, Torrance 1981
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© Kavita Bala, Computer Science, Cornell University
Classes of Models for the BRDF• Plausible simple functions
– Phong 1975;
• Physics-based models– Cook/Torrance, 1981; He et al. 1992;
• Empirically-based models– Ward 1992, Lafortune model
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Measured BRDFs
White paint
Commercial aluminum
Blue paint
Blue plastic
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Empirical BRDF Representation• Generalized Phong model (Lafortune
1997)• Used to represent:
– Measured data– Wave optics reflectance model
• Features:– Efficient and compact– Easily added to rendering algorithms
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Ward Model• Physically valid
– Energy conserving– Satisfies reciprocity:
• Based on empirical data• Isotropic and anisotropic materials
)()( irrrir ff Θ→Θ=Θ→Θ
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Ward Model: Isotropic
• where, – α is surface roughness
))((
)tanexp(
41 2
2
2LNVN
fh
ss rrrr⋅⋅
−= α
θ
παρ
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Ward Model: Anisotropic
• where, – αx, αy are surface roughness in– are mutually perpendicular to the normal
)1
ˆˆ
2exp(4
1
22
NH
yHxH
VNLNf yx
yxss rr
rr
rr
rr
⋅+
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⋅
−⋅⋅
=αα
απαρ
yx ˆ,ˆyx ˆ,ˆ
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Examples
(0.1, 0.1) (0.1, 0.2)
Images: Simon Premoze
(0.1, 0.5)
(0.1, 1.0)(0.2, 0.2)
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Teapot(0.15, 0.5) (0.5, 0.15)
(0.3, 0.3)
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Conclusions• Light modeling and BRDF modeling
• Shading models:– Physically-based model: Cook-Torrance– Empirically-based model: Ward
– Recent workanisotropic Cook-Torrance[SIG’08]