Local Reflection Model
Jian Huang, CS 594, Fall 2002
Phong Reflection
Phong specular highlight is a simplification
Phong Model - Limitations
• The Phong model is based more on common sense
than physics
– Perfect specular reflection only occurs on a perfect
mirror surface stroke by a thin light beam
• It fails to handle two aspects of specular reflection
that are observed in real life:
– intensity varies with angle of incidence of light,
increasing particularly when light nearly parallel to
surface
– colour of highlight DOES depend on material, and also
varies with angle of incidence
Physically Based Specular Reflection
•
•
•
After Phong’s work in 1975, Jim Blinn proposed
physically simulated specular component
In 1983, Cook and Torrance extended this model to
account for the spectral composition of highlights,
ie. dependencies on :
• Material type
• Angle of incidence
With physically based local reflection model, can
computer pre-computer BRDF
Modeling the Micro-geometry
• In reality, surfaces are not perfect mirrors
• A physically based approach models the
surface as micro-facets
• Each micro-facet is a perfect reflecting
surface, ie a mirror, but oriented at an angle
to the average surface normal
average
surface
normal
cross-section
through the
microfaceted
surface
Specular Reflection
• The specular reflection from this surface
depends on three factors:
– the number of facets oriented correctly to the
viewer (remember facets are mirrors)
– incident light may be shadowed, or reflected
light may be masked
– Fresnel’s reflectance equations predict colour
change depending on angle of incidence
Orientation of Facets
• Only a certain proportion (D) of facets will
in a particular direction, e.g. viewing
direction
light
eye
H
A Statistical Distribution
• Cook and Torrance give formula for D in terms of:
– Gaussian distribution: D = k exp[-(a/m)2]
• a: angle of viewer (angle between N and H)
• m: standard deviation of the distribution
• Assumptions:
– Small micro-facets is still larger than the
wavelength of light in size
– Diameter of the light beam can intersect a large
number of micro-facets to be statistically
correct
Shadowing and Masking
• Light can be fully
reflected
• Some reflected light
may hit other facets
• Some incident light
may never reach a facet
Cook and Torrance give formula for G, fraction of reflected light,
depending on angle of incidence and angle of view
Degree of Masking and
Shadowing
• Dependent on the ratio l1/l2
• G = 1 - l1/l2
• L: light vector, V: view vector
• H = (L+V)/2
l2
l1
• For masking: Gm = 2(N.H)(N.V)/V.H
• For shadowing: Gs = 2(N.H)(N.L)/V.H
The Glare Term
• Usually, as the angle between N and V
approaches 90, one sees more and more
glare
– You are seeing more micro-facets
• Need a term to account for this effect:
1/N.V
Recap: Snell’s Law
reflected
ray u
surface
normal
N
1 1
2
v
refracted
ray
incident
r ray
1
surface
2
sin 1  1

sin  2  2
Fresnel Term
In general, light is partly
reflected, partly refracted
N
reflected

Reflectance = fraction reflected
f refracted
Refractive Index:  = sin f / sin 
[Note that  varies with the wavelength of light]
The Fresnel term (the reflectance, F), of a
perfectly smooth surface is given in terms of refractive
index  of material and angle of incidence 
F is wavelength dependent!
Fresnel Term
• Don’t know how to calculate F for arbitrary 
directly, so usually started with a known or
measured F0.
• F is a minimum for incident light normal to the
surface, ie  = 0 : F0 = (  - 1 )2 / (  + 1 )2
• So different F0 for different materials
• The refractive index  of a material depends on the
wavelength,  , so have different F0 for different 
– burnished copper has roughly:
F0,blue = 0.1, F0,green = 0.2, F0,red = 0.5
Fresnel Term
• As  increases from 0 ...
F = F0 + ( 1 - cos  )5 ( 1 - F0 )
– so, as  increases, then F increases until F90 = 1
(independent of  )
• This means that when light is tangential to the
surface:
– full reflectance, independent of 
– reflected colour independent of the material
• Thus reflectance does depend on angle of incidence
• Thus colour of specular reflection does depend on
material and incident light angle
Specular Term
• This leads to:
Rs(  ) = F(  ) D G / (N.V)
where:
D = proportion of microfacets aligned to view
G = fraction of light shadowed or masked
F = Fresnel term
N.V glare effect term
In practice, Rs is calculated for red, green, blue
• Note it depends on angle of incidence and angle of
view
Cook and Torrance Reflection
Model
• The specular term is calculated as described and
combined with a uniform diffuse term:
– Reflection (angle of incidence, viewing angle) =
s Rs + d Rd
(where s + d = 1)
– Known as bi-directional reflectance
• For metals: d = 0, s = 1
• For shiny plastics: d = 0.9, s = 0.1
• Its BRDF does not depend on the incoming azimuth
Aluminium
Bronze
Chrome
Stainless Steel