Local Illumination
MIT EECS 6.837, Durand and Cutler
Outline
•
•
•
•
Introduction
Radiometry
Reflectance
Reflectance Models
MIT EECS 6.837, Durand and Cutler
The Big Picture
MIT EECS 6.837, Durand and Cutler
Radiometry
• Energy of a photon
e 
hc
h  6.63 10

34
J  s c  3 10 m / s
8
• Radiant Energy of n photons
n
hc
i 1
i
Q
• Radiation flux (electromagnetic flux, radiant flux)
Units: Watts
dQ

dt
MIT EECS 6.837, Durand and Cutler
Radiometry
• Radiance – radiant flux per unit solid angle per
unit projected area
– Number of photons arriving
per time at a small area
from a particular direction
d 2
L( ) 
cos  dA d
Watt
Units :
meter 2 steradian
MIT EECS 6.837, Durand and Cutler
Radiometry
• Irradiance – differential flux falling onto differential
area
d
E
dA
Watt
Units :
meter 2
• Irradiance can be seen as a density of the incident flux
falling onto a surface.
• It can be also obtained by integrating the radiance over
the solid angle.
MIT EECS 6.837, Durand and Cutler
Light Emission
• Light sources: sun, fire, light bulbs etc.
• Consider a point light source that emits light
uniformly in all directions
 s cos 
s
E
L
2
2
4d
4d
 s  power of the light source
n

d  distance to the light source
Surface
MIT EECS 6.837, Durand and Cutler
Outline
•
•
•
•
Introduction
Radiometry
Reflectance
Reflectance Models
MIT EECS 6.837, Durand and Cutler
Reflection & Reflectance
• Reflection - the process by which electromagnetic
flux incident on a surface leaves the surface
without a change in frequency.
• Reflectance – a fraction of the incident flux that is
reflected
• We do not consider:
– absorption, transmission, fluorescence
– diffraction
MIT EECS 6.837, Durand and Cutler
Reflectance
• Bidirectional scattering-surface distribution
Function (BSSRDF)
Source: Jensen et.al 01
Surface
MIT EECS 6.837, Durand and Cutler
Reflectance
• Bidirectional scattering-surface distribution
Function (BSSRDF)
dLr ( r , r , xr , yr )
S ( i , i ,  r , r , xi , yi , xr , yr ) 
d i ( i , i , xi , yi )
1
Units :
meter 2 steradian
Surface
MIT EECS 6.837, Durand and Cutler
Reflectance
• Bidirectional Reflectance Distribution Function
(BRDF)
dLr ( r , r )
f r ( i , i ,  r , r ) 
dEi ( i , i )
1
Units :
steradian
Lr
r
Li
i
i
r
MIT EECS 6.837, Durand and Cutler
Isotropic BRDFs
• Rotation along surface normal does not change
reflectance
dLr ( r , d )
f r ( i ,  r , r  i )  f r ( i ,  r , d ) 
dEi ( i , d )
Lr
r
i
d
MIT EECS 6.837, Durand and Cutler
Li
Anisotropic BRDFs
• Surfaces with strongly oriented microgeometry
elements
• Examples:
– brushed metals,
– hair, fur, cloth, velvet
Source: Westin et.al 92
MIT EECS 6.837, Durand and Cutler
Properties of BRDFs
• Non-negativity
f r (i , i , r , r )  0
• Energy Conservation
 f ( , , , )d ( , )  1
r
i
i
r
r
r
r
for all (i , i )

• Reciprocity
f r (i , i , r , r )  f r ( r , r ,i , i )
MIT EECS 6.837, Durand and Cutler
How to compute reflected radiance?
• Continuous version
Lr (r )   f r (i , r )dEi (i ) 

  f r (i , r ) dLi (i ) cos(i  n) di
  ( ,  )

• Discrete version – n point light sources
n
Lr (r )   f r (ij , r )E j 
j 1
n
 sj
j 1
4 d j
  f r (ij , r ) cos  j
2
MIT EECS 6.837, Durand and Cutler
Outline
•
•
•
•
Introduction
Radiometry
Reflectance
Reflectance Models
MIT EECS 6.837, Durand and Cutler
How do we obtain BRDFs?
• Measure
BRDF values
directly
• Analytic Reflectance Models
– Physically-based models
• based on laws on physics
– Empirical models
• “ad hoc” formulas that work
MIT EECS 6.837, Durand and Cutler
Source: Greg Ward
Ideal Diffuse Reflectance
• Assume surface reflects equally in all directions.
• An ideal diffuse surface is, at the microscopic level,
a very rough surface.
– Example: chalk, clay, some paints
Surface
MIT EECS 6.837, Durand and Cutler
Ideal Diffuse Reflectance
• BRDF value is constant
Lr (r )   f r (i , r )dEi (i ) 

dB  dA cosi
n
 f r  dEi (i ) 

 f r Ei
i
dA
Surface
MIT EECS 6.837, Durand and Cutler
dB
Ideal Diffuse Reflectance
• Ideal diffuse reflectors reflect light according to Lambert's
cosine law.
MIT EECS 6.837, Durand and Cutler
Ideal Diffuse Reflectance
• Single Point Light Source
– kd: The diffuse reflection coefficient.
– n: Surface normal.
– l: Light direction.
s
L(r )  k d (n  l)
2
4 d
n

l
Surface
MIT EECS 6.837, Durand and Cutler
Ideal Diffuse Reflectance – More Details
• If n and l are facing away from each other, n • l
becomes negative.
• Using max( (n • l),0 ) makes sure that the result is
zero.
– From now on, we mean max() when we write •.
• Do not forget to normalize your vectors for the dot
product!
MIT EECS 6.837, Durand and Cutler
Ideal Specular Reflectance
• Reflection is only at mirror angle.
– View dependent
– Microscopic surface elements are usually oriented in the
same direction as the surface itself.
– Examples: mirrors, highly polished metals.
n
r
  l
Surface
MIT EECS 6.837, Durand and Cutler
Ideal Specular Reflectance
• Special case of Snell’s Law
– The incoming ray, the surface normal, and the reflected
ray all lie in a common plane.
n
r
r l l
Surface
nl sin  l  nr sin  r
nl  nr
l   r
MIT EECS 6.837, Durand and Cutler
Non-ideal Reflectors
• Snell’s law applies only to ideal mirror reflectors.
• Real materials tend to deviate significantly from
ideal mirror reflectors.
• They are not ideal diffuse surfaces either …
MIT EECS 6.837, Durand and Cutler
Non-ideal Reflectors
• Simple Empirical Model:
– We expect most of the reflected light to travel in the
direction of the ideal ray.
– However, because of microscopic surface variations we
might expect some of the light to be reflected just slightly
offset from the ideal reflected ray.
– As we move farther and farther, in the angular sense,
from the reflected ray we expect to see less light
reflected.
MIT EECS 6.837, Durand and Cutler
The Phong Model
• How much light is reflected?
– Depends on the angle between the ideal reflection
direction and the viewer direction .
n
r

Camera


l
v
Surface
MIT EECS 6.837, Durand and Cutler
The Phong Model
• Parameters
– ks: specular reflection coefficient
– q : specular reflection exponent
s
q s
L(r )  k s (cos  )
 k s (v  r )
2
4 d
4 d 2
q
n
r

Camera


l
v
Surface
MIT EECS 6.837, Durand and Cutler
The Phong Model
• Effect of the q coefficient
MIT EECS 6.837, Durand and Cutler
The Phong Model
r  l  2 cos  n
r  2(n  l)n  l
s
L(r )  k s (v  r )

2
4 d
q s
 k s (v  (2(n  l)n  l))
4 d 2
q
n
r
r


Surface
MIT EECS 6.837, Durand and Cutler
l
Blinn-Torrance Variation
• Uses the halfway vector h between l and v.
lv
h
|| l  v ||
s
q s
L(r )  k s (cos  )
 k s (n  h)
2
2
4 d
4 d
q
h
n

Camera
l
v
Surface
MIT EECS 6.837, Durand and Cutler
Phong Examples
• The following spheres illustrate specular reflections
as the direction of the light source and the
coefficient of shininess is varied.
Phong
Blinn-Torrance
MIT EECS 6.837, Durand and Cutler
The Phong Model
• Sum of three components:
diffuse reflection +
specular reflection +
“ambient”.
Surface
MIT EECS 6.837, Durand and Cutler
Ambient Illumination
• Represents the reflection of all indirect illumination.
• This is a total hack!
• Avoids the complexity of global illumination.
L(r )  ka
MIT EECS 6.837, Durand and Cutler
Putting it all together
• Phong Illumination Model
(
L(r )  k a  k d (n  l)  k s (v  r )
MIT EECS 6.837, Durand and Cutler
q
)
s
4 d 2
For Assignment 3
• Variation on Phong Illumination Model
(
)
L(r )  kd La  kd (n  l)  k s (v  r ) q Li
MIT EECS 6.837, Durand and Cutler
Adding color
• Diffuse coefficients:
– kd-red, kd-green, kd-blue
• Specular coefficients:
– ks-red, ks-green, ks-blue
• Specular exponent:
q
MIT EECS 6.837, Durand and Cutler
Phong Demo
MIT EECS 6.837, Durand and Cutler
Fresnel Reflection
• Increasing specularity near grazing angles.
Source: Lafortune et al. 97
MIT EECS 6.837, Durand and Cutler
Off-specular & Retro-reflection
• Off-specular reflection
– Peak is not centered at the reflection direction
• Retro-reflection:
– Reflection in the direction of incident illumination
– Examples: Moon, road markings
MIT EECS 6.837, Durand and Cutler
The Phong Model
• Is it non-negative?
• Is it energy-conserving?
• Is it reciprocal?
• Is it isotropic?
MIT EECS 6.837, Durand and Cutler
Shaders (Material class)
• Functions executed when light interacts with a
surface
• Constructor:
– set shader parameters
• Inputs:
– Incident radiance
– Incident & reflected light directions
– surface tangent (anisotropic shaders only)
• Output:
– Reflected radiance
MIT EECS 6.837, Durand and Cutler
Questions?
MIT EECS 6.837, Durand and Cutler