CSCE 641 Computer Graphics:
Radiosity
Jinxiang Chai
Rendering: Illumination Computing
Direct (local) illumination

Light directly from light sources

No shadows
Indirect (global) illumination

Transparent, reflective surfaces, and hard
shadows (Ray tracing)

Diffuse interreflections, color bleeding, and soft
shadow (radiosity)
Rendering: Illumination Computing
Direct (local) illumination

Light directly from light sources

No shadows
Indirect (global) illumination

Transparent, reflective surfaces, and hard
shadows (Ray tracing)

Diffuse interreflections, color bleeding, and soft
shadow (radiosity)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light
sources)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light
sources)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light
sources)
- reflection ray (light reflected by an object)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light
sources)
- reflection ray (light reflected by an object)
- transparent ray (light passing through an object)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light
sources)
- reflection ray (light reflected by an object)
- transparent ray (light passing through an object)
Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light
sources)
- reflection ray (light reflected by an object)
- transparent ray (light passing through an object)
Pros and Cons of Ray Tracing
Advantages of ray tracing

All the advantages of the local illumination
model

Also handles shadows, reflection, and
refraction
Disadvantages of ray tracing

Computational expense

No diffuse inter-reflection between surfaces
(i.e., color bleeding)

Not physically accurate
Radiosity handles these shortcomings for diffuse
surfaces!
Radiosity vs. Local Illumination
Radiosity
Physical Image vs. Radiosity Rendering
Radiostiy
Definition: The radiant (luminous) exitance is the
radiant flux/power per unit area leaving a surface.
d o
M ( x) 
dA
 W   lm

 m2   m2  lux 
Radiosity
Model light effects by considering the physical laws
governing the radiant energy transfer;
The radiosity model computes radiant-energy
interactions between all the surfaces in a scene
Radiosity: Key Idea #1
Diffuse Surface
Radiosity: Key Idea #2
Constant Surface Approximation
Radiosity Equation
N
Radiosityi  Radiosityself ,i   a j i Radiosityj
j 1
Radiosity Equation
N
Radiosity1  Radiosityself ,1   a j 1Radiosityj
j 1
N
Radiosity2  Radiosityself , 2   a j 2 Radiosityj
j 1
N
RadiosityN  Radiosityself , N   a j N Radiosityj
j 1
Radiosity Algorithm
Energy Conservation Equation
N
 i   e ,i   i   j F ( j  i )
j 1
Energy Conservation Equation
N
 i   e ,i   i   j F ( j  i )
j 1
The total rate of radiant energy leaving surface i
per unit square
Energy Conservation Equation
N
 i   e ,i   i   j F ( j  i )
j 1
The rate of energy emitted from surface i per unit area
- zero if surface i is not a light source
Energy Conservation Equation
N
 i   e ,i   i   j F ( j  i )
j 1
Reflectivity factor
Percent of incident light that is reflected in all directions
Energy Conservation Equation
N
 i   e ,i   i   j F ( j  i )
j 1
Form factor
Fractional amount of radiant energy from surface j that
reaches surface i
Compute Form Factors
1
F ( j  i) 
Aj

Ai , A j
cos  x  cos  y
r
2
xy
 V ( x, y )  dAi  dA j
The form factor specifies the fraction of the energy leaving one patch and
arriving at the other. In other words, it is an expression of radiant exchange
between two surface patches!
Compute Form Factors
1
F ( j  i) 
Aj

Ai , A j
cos  x  cos  y
r
2
xy
 V ( x, y )  dAi  dA j
Radiant energy reaching Ax from Ay
Radiant energy leaving Ay in all directions
The form factor specifies the fraction of the energy leaving one patch and
arriving at the other. In other words, it is an expression of radiant exchange
between two surface patches!
Form Factor: Reciprocity
1
F ( j  i) 
Aj
1
F (i  j ) 
Ai

Ai , A j

Ai , A j
cos  x  cos  y
r
2
xy
cos  x  cos  y
r
2
xy
V ( x, y )  dAx  dAy
V ( x, y )  dAy  dAx
F (i  j) Ai  F ( j  i) Aj
Radiosity Equation
Radiosity for each polygon
N
i : i   e,i  i   j F ( j  i)
j 1
Linear system:
-  i : radiosity of patch I (unknown)
-  e,i : emission of patch I (known)
- i : reflectivity of patch I (known)
- Fi  j : form-factor (known)
Linear System
1  1 F11  1 F12
 F
1   2 F22
2 21


...
...

  N FN 1   N FN 2
A
 1 F1 N   1    e,1 
 



...   2 F2 N    2   e, 2 

  ...   ... 
...
...

  
... 1   N FN  N   N   e,3 
...
X
=
B
Radiosity Algorithm
Form Factors for Infinitesimal Surfaces
• Visibility
- if not visible, receive zero
power
F ( j  i) 
cos x  cos y
r
2
xy
dAiV ( x, y)
Form Factors for Subdivided Patches
• Visibility
1
F ( j  i) 
- if not visible, receive zero
Aj
power

Ai , A j
cos  x  cos  y
r
2
xy
 V ( x, y )  dAi  dA j
Form Factor: How to compute?
Closed Form
- analytical
Hemicube
Form Factor: Analytical
1
F ( j  i) 
Aj

Ai , A j
cos  x  cos  y
r
2
xy
V ( x, y )  dAx  dAy
Form Factor: How to compute?
Closed Form
- analytical
Hemicube
Form Factor: Nusselt Analog
Nusselt developed a geometric analog which allows the simple and
accurate calculation of the form factor between a surface and a point
on a second surface.
3D diagram
Form Factor: Nusselt Analog
The form factor is, then, the area projected on the base of the
hemisphere divided by the area of the base of the hemisphere, or
(A/B)
A
B
2D diagram
Form Factor: Nusselt Analog
Form Factor: Nusselt Analog
So how can we use Nusselt Analog to compute the form factor?
Form Factor: Nusselt Analog
So how can we use Nusselt Analog to compute the form factor?
- answer: precomputing
Form Factor: HemiCube
Form Factor: HemiCube
• Project path on hemicube
• Add hemicube cells to compute form factors
A
B
2D diagram
Precomputing Form Factor
How to calculate the form factor for each cell?
Delta Form Factor: Top Face
Top of hemicube
Delta Form Factors: Side Faces
Side of hemicube
The Hemicube in Action
Form Factors: HemiCube
Form Factors
Radiosity Algorithm
How to Solve Linear System?
Matrix conversion
Iterative approaches
- Jacobian (gathering)
- Gauss-Seidel (gathering)
- progressive refinement (shooting)
Matrix Conversion
 1  1  1 F11  1 F12
     F
1   2 F22
2 21
 2
 ...  
...
...
  
 N    N FN 1   N FN 2
...  1 F1 N 
...   2 F2 N 

...
...

... 1   N FN  N 
- Computational cost: O(N3)
- Very slow for a large set of polygons
1
  e,1 
 
 e, 2 
 ... 



 e,3 
Iterative Approaches
N
 i   e ,i   i   j F ( j  i )
j 1
Jacobian Iterations
For all patches i, i=1,…,N, i0  e,i
While not converged:
for all patches i=1,…,N

N
iter
i
1
  e,i  i   iter
F ( j  i)
j
j 1
Jacobian Iterations
For all patches i, i=1,…,N, i0  e,i
While not converged:
for all patches i=1,…,N

N
iter
i
1
  e,i  i   iter
F ( j  i)
j
j 1
Update of one patch requires evaluation of N Form Factors
What’s the computational cost?
Successive Approximation
Le
K Le
Le
Le  K Le
K K Le
Le 
K 2 Le
K K K Le
Le 
K 3 Le
Rendering
- The final Φi's can be used in place of intensities in a
standard renderer (Gouraud, per-veretx shading)
- Radiosities are constant over the extent of a patch
- A standard renderer requires vertex intensities (or
radiosities)
- If the radiosities of surrounding patches are know,
vertex radiosities can be estimated using bilinear
interpolation
Vertex Intensity: Bilinear Interpolation
Consolation Room
Theatre
Steel Mills
Radiosity: Benefit
 Global illumination method: modeling diffuse interreflection
 Color bleeding: a red wall next to a white one casts a
reddish glow on the white wall near the corner
 Soft shadows – an “area” light source casts a soft
shadow from a polygon
 No ambient term hack, so when you want to look at
your object in low light, you don’t have to adjust
parameters of the objects – just the intensities of the
lights!
 View independent: it assigns a brightness to every
surface
Radiosity: Limitation
 Radiation is uniform in all directions
 Radiosity is piecewise constant
– usual renderings make this assumption, but then interpolate cheaply to
fake a nice-looking answer
– this introduces quantifiable errors
 No surface is transparent or translucent
 Reflectivity is independent of directions to source and
destination