Definitions and Review

Radiosity:

Diffuse interaction of light

• Ambient term is an approximation to diffuse interaction of light
• Would like a better model of this ambient light
• Idea: ``Discretize'' environment and determine interaction between each pair of pieces.
• Models ambient light under following assumptions:
  • Conservation of energy in closed environment
  • Only diffuse reflection of light
• All energy emitted or reflected accounted for by its reflection or absorption elsewhere
• All light interactions computed once, in a view independent way
  • Multiple views can then be rendered just using hidden surface removal
  • No mirror/specular reflections
  • Ideal for architectural walkthroughs

Radiance:

Electromagnetic energy flux, the amount of energy traveling

• at some point x
• in a specified direction ,
• per unit time
• per unit area perpendicular to the direction
• per unit solid angle
• for a specified wavelength
• denoted by

Spectral Properties:

Total energy flux comes from flux at each wavelength

• 

Picture:

For the indicated situation is

• energy radiated through differential solid angle
• through/from differential area dx
• not perpendicular to direction (projected area is )
• during differential unit time dt

Power:

Energy per unit time (as in the picture)

• 

Radiosity:

Total power leaving a surface point per unit area

(integral is over the hemisphere above the surface point)

Bidirectional Reflectance Distribution Function:

• is a surface property at a point
• relates energy in to energy out
• depends on incoming and outgoing directions
• varies from wavelength to wavelength
• Definition: Ratio
  • of radiance in the outgoing direction
  • to radiant flux density for the incoming direction

Energy Balance Equation:

• is the radiance
  • at wavelength
  • leaving point x
  • in direction ,
• is the radiance emitted by the surface from the point
• is the incident radiance impinging on the point
• is the BRDF at the point
  • describes the surface's interaction with light at the point
• the integration is over the hemisphere above the point

Radiosity Approach to Global Illumination:

• Assume that all wavelengths act independently
  • 
  • 
• Assume that all surfaces are purely Lambertian
  • 
• As a result of the Lambertian assumption
  • 
  • 

Simple Energy Balance (Hemisphere Based):

Multiplying by and letting :

But, in general

So we let

for the appropriate point y.

Picture (Scene Based Energy Balance):

Visibility:

We use a term to pick out the special y

Area:

We convert to surface area

Simple Energy Balance (Scene Based):

Piecewise Constant Approximation:

• Approximate the integral by breaking it into a summation over patches
• Assume a constant (average) radiosity on each patch

  

  

• Solve only for a per-patch average density
  • 
  • 
  • 

Piecewise Constant Radiosity Approximation:

Form Factor:

Note, by symmetry, that we have

Linear Equations:

(by Summetry)

Radiosity:

(Summary)

• Idea is to discretize scene into n patches (polygons) each of which emits and reflects light uniformly under its entire area.
• Then radiosity emitted by patch i is given by

  

  where

  • : radiosity in energy/unit-time/unit-area
  • : light emitted from patch i
  • : patch i's reflectivity
  • : Form factor specifying fraction of energy leaving j that reaches i (accounts for shape, orientation, occulsion)
  • : Area of patches

Radiosity:

Full Matrix Solution

• The equations are

  

  or

  

• In matrix form

  

• 's are only unknowns
• If (true for polygonal patches) then diagonal is 1
• Solve 3 times to get RBG values of

Advanced Radiosity:

Galerkin Method

• Instead of assuming that B(x) is piecewise constant, express the radiosity in terms of a linear combination of basis functions

  

• The energy equations are satisfied only to within a residual

  

Minimize Residual Function

• Define an inner product (to quantify residual size)

  

• r(x) is made small when for all k.
  
• This leads to linear equations in the coefficients .
  
• These equations are the ones already seen when the basis functions are the characteristic functions over patches in the scene