Computing Form-Factors
How to compute form factors Fij ® for radiative exchange 
between 2 surfaces with areas Ai, Aj:
- First consider 2 infinitesimal surfaces dAi, dAj.

            

Solid angle subtended by dAj (as seen from dAi) is

	dw = Cosfj   dAj
		    r2

From eq. 1 and 3a	1 ® i = dp / (Cosf dw)
			3a ® P = ip

Radiant energy leaving Ai and incident on dAj  is 

dPi dAi = ii Cosfi dw dAi = Pi Cosfi Cosfj dAi dAj
				                pr2

Note: total energy leaving dAi into hemisphere is Pi dAi.

Now form factor is fraction of energy incident on dAj so

	FdAi - dAj = Pi Cosfi Cosfj dAi dAj / pr2 = Cosfi Cosfj dAj 
                                                 Pi dAi                                    pr2

So fraction leaving dAi and incident on Aj is

	FdAi - Aj  ò Aj Cosfi Cosfj dAj
 		               pr2

then form factor between surfaces Ai, Aj is area average of above 

	FAi - Aj = Fij = (1/Ai ò Ai ò Aj Cosfi Cosfj dAi dAj
						pr 2
A more efficient method of computing Fij  is to transform area integrals into contour integrals using Stokes theorem and get

Fij = [1/(2pAi)] ò Cj ò Ci [ln(r) dXi dXj + ln(r) dYi dYj = ln(r) dZi dZj]

From derivation of F's some possible shortcuts:

1. Ai Fij = Aj Fij	so if know Ai, Aj, Fij can determine Fji

2. From conservation of energy ®  all energy leaving a surface must be accounted for so (N surfaces)
		Sj=1N Fij = 1
3. For a plane or convex surface (doesn't see itself) ® Fii = 0

So general algorithm:
   
   -  Read in scene description (polygon mesh)

   -  Subdivide polygons into elements (sub-polygon) - the smaller the element the more  accurate the results, since compute 1 radiosity value/patch.

   -  Compute form factors between all elements.

   -  Now once Fij's are computed, put into matrix equations for radiosities(bj) and solve        for bj's.

   -  Smooth shading across elements (similar to Gouraud)

   -  Display  image.

To be really precise would solve matrix for each l (since r [assume continuous] is f(l), but usually just for RGB values).
Above derivation ignores possible occlusion of surfaces, so good only for very simple scenes - no shadows or penumbras. Cohen and Greenberg extended for more complex scenes.

A geometric analog of form factor integral was developed by Nusselt:
	Consists of projecting element area (Aj) onto hemisphere covering Ai, then projecting down to a circle, and determine fraction of unit circle covered.

Now could decompose hemisphere into small d solid angles representing d fij's.
Then F for a patch would be S dF's covered when patch projected onto hemisphere.

Problem: difficult to create equal sized elements on sphere and set of linear coordinates to describe surface location on sphere.
So instead of using a sphere, center a cube on patch ® hemi-cube.

We can do this since any 2 patches whose projections occupy some area and location on hemisphere give same form factor - also true for any other surrounding surface.

                            

                   

Faces of hemi-cube are divided into source "pixels" and patches are projected onto "pixels" (might have 50 × 50, 100 × 100, time ¬ accuracy trade-off).

Handles visible surface and shadow problem:
If >= 2 patches project onto same pixel do depth determination ® so have type of depth buffer ® "item buffer" ® which patch is visible at each "pixel".

Now can compute a delta form factor for each pixel in hemi-cube.







                                               



So to find Fij (hemi-cube on i):
- for each "pixel" occupied by patch j sum delta form factors.

	Fij = Sq=1R DFq

D Fq = delta form factor associated with pixel q on hemi-cube.
R = # of hemi-cube pixels covered by projection of patch onto hemi-cube.

To solve matrix equation:
Now make guess at initial set of radiosities (emission light source only, so only non-zero).
Solve for b's, in terms of other b's, iterate process until self consistent - usually 
6-8 iteration.

Now have computed radiosity (intensity) for each patch.
To render, want smooth shading so do Gouraud type bi-linear interpolation across patch.

So to render, find ray from eye through pixel and intersection with visible patch.
This X, Y, Z values ® U, V within patch and do interpolation.

Note: Advantage for animation ® if lights not changed (only view) the F's only computed once.
Example: F's computation - 90 % of total, but 0 for other views.