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# Definitions and Review

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
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)
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)

(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

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

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

Next: Form Factors Up: Radiosity-based Global Illumination Previous: Radiosity-based Global Illumination

CS488/688: Introduction to Interactive Computer Graphics
University of Waterloo
Computer Graphics Lab

cs488@cgl.uwaterloo.ca