Radiosity

Basic Theory

Radiosity is one of the three major methods to create graphics images, with ray tracing and "conventional" scan-line rendering being the other two. Scan-line rendering handles only directly reflected light (from a light source to the object to the eye) and has no real method to handle object-object reflections, except by tricks such as environment maps. Ray tracing, e.g. the Whitted global illumination model, deals with object-to-object specular reflections but ignores object-to-object diffuse reflections, which is approximated as the ambient term. 

Ray tracing is view dependent and changing the view means recomputing the entire scene, i.e. no effort is saved between scenes. The major portion of the work performed in a radiosity computation is view independent. This makes it difficult to include specular interactions in radiosity. But this view independence is very good for an application such as an architectural walkthrough. A single radiosity based image may take ten times the amount of compuytation of a ray tracered image but for an animation sequence, the total time may be much shorter.

Radiosity computations

Radiosity 

Actually there are four mechanisms of light transport between surfaces (Wall87):
   1. diffuse to diffuse
   2. specular to diffuse
   3. diffuse to specular
   4. specular to specular ® ray tracing

The diffuse reflections may account for major part of light energy in many complex scenes.
Note:	Can always add on a specular component since small part of total energy.

Radiosity is a method to determine intensity of all diffuse reflected light (and magnitude of ambient light) based on energy principles. It makes the assumption that all surfaces are ideal diffuse reflectors or emitters. The analysis is similar to the calculation of radiative heat exchange in enclosures from thermal engineering.

Theory and Mathematical Formulation
Radiant energy (in form of visible light) is assumed to emanate in all directions from area dA.
               
Radiant intensity in particular viewing direction V
®	I = dp / (Cosf dw)						(1)

I = radiant energy leaving/unit time per unit projected area per unit solid angle 
	(watts/(meter2 steradians).
dp = radiant energy leaving surface in direction f within a solid angle dw per unit time, per unit surface area (unprojected).

Note: Human eye senses intensity - receives energy within a solid angle dw defined by pupil size - so energy is appropriate quantity to be concerned about.

For ideal diffuse reflection, distribution of reflected light energy 
® 	dp/dw = K Cos f (K constant)				(2)

Now intensity is f(projected area) and projected area a Cos f.
Therefore intensity of reflected light is
      I = 	dp/dw = K Cos f = K ®  constant in all direction
	Cos f	     Cos f	

Total energy leaving a surface is integral(1) over hemisphere above surface
 (solid angle 2p)

	P = ò 2p(dp) = ò 2p(i Cosf dw)

where P = total energy leaving surface per unit time and area (watts/meter2).

For ideal diffuse surface:
	P = I ò 2p(Cosf dw = Ip)				(3a)

Now look at diffuse light reflection within an environment.
- The light energy at any surface includes radiation arriving from all directions in space.
- So construct hypothetical "enclosure" where an "enclosure" is a set of surfaces that completely define illuminating environment (light sources and reflecting walls).
Example: enclosure consisting of N surfaces.
                            
For surface j, incident radiation (light) is Hj = S from all other surfaces in scene.
The reflected radiation is Bj.

Assumptions about surfaces in enclosure:
	- ideal diffuse reflectors or
	- ideal diffuse transmitter or combination of the two
	- each surface has uniform illumination, reflection, and emission intensities
              (can always subdivide surfaces until true).

For diffuse light sources - treated as surfaces of enclosure with certain illuminating intensities.
For directional light sources - identify surfaces illuminated by light source and treat as diffuse light sources (from reflection) or could model as diffuse light panel.
So result is all light treated as diffuse.
 
Now define:
Form factor º fraction of radiant light energy leaving one particular surface which strikes a second surface.
For ideal diffuse emission or reflection:
	form factor is f(shape, size, position, and orientation of surfaces).

The radiosity is the hemispherical integral of energy leaving a surface.
Radiosity ® Bj = Ej + rj Hj

	Ej = rate of direct energy emission from surface;
	rj = reflectivity of j (fraction of incident light that is reflected)
	Hj = incident radiant energy at j.

Note: We are treating emitted and reflected light together so we only deal with one radiosity for each surface Bj.

Now consider Hj  (incident flux on surface j) = S fluxes from all other surfaces
 "visible" to j in enclosure.
The fraction of flux leaving surface i (Bi) reaching surface j is the form factor (Fij).
So	
	Hj = Si=1N Bj Fij	Note: j might see itself (e.g. concave surface)

So Bj = Ej + rj Si=1N Bj Fij	for j = 1, N ,- so need to compute radiosities.

So need to compute Ej, rj, Fij for each surface and solve system of N linear equations for Bj's.  Ej's are emission sources (lights) - if Ej's all = 0 then no light so Ej's come from diffuse light panel or first reflection of directional light source from a surface.

Note: Ej, rj may be f(l) over visible spectrum.

Now actually want radiant intensity (eye sense intensity) rather than energy so divide by p 
(eqn.3a) to give bj = ej + rj Si=1N bj Fij