VIEWS: 3 PAGES: 17

• pg 1
```									Global Illumination (2)
Next
• We have the form factors
• How do we find the radiosity
solution for the scene?
– The "Full Matrix" Radiosity Algorithm
– Gathering & Shooting
• Meshing

Mesh Surfaces into Elements

Compute Form Factors
Between Elements

Solve Linear System

Reconstruct and
Display Solution

Bi  Ei  i  B j Fij
Form Factor of surface j
Radiosity of surface i                             relative to surface i

Emissivity of surface i         Radiosity of surface j

Reflectivity of surface i
accounts for the
physical
relationship
between the two
Surface j    surfaces
will absorb a certain
percentage of light
energy which strikes
the surface
Surface i

n
Bi Ai  Ei Ai   i  F ji B j A j           Bi
j 1
Ai Fij  A j F ji
n
Bi  Ei   i  Fij B j
j 1
n
Bi   i  Fij B j  Ei
j 1

1  1 F11  1 F12         1 F1n   B1   E1 
 F       1   2 F22      2 F2 n   B2   E2 
     2 21                                  
                                        
                                          
   n Fn1   n Fn 2     1   n Fnn   Bn   En 
• The "full matrix" radiosity solution calculates the form
factors between each pair of surfaces in the
environment, then forms a series of simultaneous linear
equations.

1  1 F11  1 F12         1 F1n   B1   E1 
 F       1   2 F22      2 F2 n   B2   E2 
     2 21                                  
                                        
                                          
   n Fn1   n Fn 2     1   n Fnn   Bn   En 

• This matrix equation is solved for the "B" values, which
can be used as the final intensity (or color) value of
each surface.
• This method produces a complete solution, at the
substantial cost of
– first calculating form factors between each pair of surfaces
– and then the solution of the matrix equation.

• Each of these steps can be quite expensive if the
number of surfaces is large: complex environments
typically have upwards of ten thousand surfaces, and
environments with one million surfaces are not
uncommon.

• This leads to substantial costs not only in computation
time but in storage.
Next
• We have the form factors
• How do we find the radiosity
solution for the scene?
– The "Full Matrix" Radiosity Algorithm
– Gathering & Shooting
• Meshing
Solve [F][B] = [E]
• Direct methods: O(n3)

– Gaussian elimination
• Goral, Torrance, Greenberg, Battaile, 1984

• Iterative methods: O(n2)

Energy conservation
¨diagonally dominant ¨  iteration converges

– Gauss-Seidel, Jacobi: Gathering
• Nishita, Nakamae, 1985
• Cohen, Greenberg, 1985

– Southwell: Shooting
• Cohen, Chen, Wallace, Greenberg, 1988
Gathering
• In a sense, the light
leaving patch i is
determined by
gathering in the light
from the rest of the
environment
n
Bi  Ei   i  B j Fij
Bi  Ei    i Fij B j
n
j 1
j 1
Bi due to B j  i B j Fij
Gathering

• Gathering light through
a hemi-cube allows
be updated.

Bi  Ei    i Fij B j
n

j 1
Gathering
Successive Approximation
Shooting
• Shooting light through a
single hemi-cube allows
the whole environment's
updated simultaneously.

For all j       B j  B j  Bi  j E ji 

Fij Ai
where   F ji 
Aj
Shooting

Mesh Surfaces into Elements

Compute Form Factors
Between Elements

Solve Linear System