1. SI31 Advanced Computer Graphics AGR
Lecture 15
Radiosity

2. Review First a review of the two rendering approaches we have studied:
Phong reflection model
ray tracing

3. Phong Reflection Model
This is the most common approach to rendering
objects represented as polygonal faces
intensity of faces calculated by Phong local illumination model
polygons projected to viewplane
Gouraud shading applied with Z buffer to determine visibility
I() = Ka()Ia() + ( Kd()( L . N ) + Ks( R . V )n ) I*() / dist

4. Phong Reflection Model
Strengths
simple and efficient
models ambient, diffuse and specular reflection
Limitations
only considers light incident from a light source, and not inter-object reflections - ie it is a local illumination method (ambient term is approximation to global illumination
empirical rather than theoretical base
objects typically have plastic appearance

5. Ray Tracing
Ray traced from viewpoint through pixel until first object intersected
Colour calculated as summation of:
local Phong reflection at that point
specularly reflected light from direction of reflection
transmitted light from refraction direction if transparent

6. Ray Tracing This is done recursively
Colour of light incoming along reflection direction found by:
tracing ray back until it hits an object
colour of light emitted by object is itself summation of local component, reflected component and transmitted component
and so on

7. Ray Tracing - Strengths and Weaknesses
Advantages
increased realism through ability to handle inter-object reflection
Disadvantages
much more expensive than local reflection
still empirical
only handles specular inter-object reflection
entire calculation is view-dependent

8. Radiosity Based on the physics of heat transfer between surfaces
Developed in 1980s at Cornell University in US (Cohen, Greenberg)
Determine energy balance of light transfer between all surfaces in an enclosed space
equilibrium reached between emission of light and partial absorption of light
Assume surfaces are opaque, are perfect diffuse reflectors and are represented as sets of patches Ai

9. Radiosity Examples

10. Radiosity - Definition
Radiosity defined as:
energy (Bi ) per unit area leaving a surface patch (Ai ) per unit time
Bi Ai = Ei Ai + Ri  ( Fj-i Bj Aj) i=1,2,..N j
light
leaving
light
emitted
light
reflected
Ei is the light energy emitted by Ai per unit area
Ri is the fraction of incident light reflected in all directions
Fj-i is the fraction of energy leaving Aj that reaches Ai
N is the number of patches

11. Radiosity - Pictorial Definition
Form factor Fj-i is fraction of
energy leaving Aj that reaches
Ai. It is determined by the
relative orientation of the
patches and distance r
between them Aj j r Nj Ni
Form factors are
hard to calculate!
Let’s assume for now
we can do it. i Ai
Bi Ai = Ei Ai + Ri  (Fj-i Bj Aj) j

12. Simplifying the Equation
Bi Ai = Ei Ai + Ri  (Fj-i Bj Aj) ie
Bi Ai = Ei Ai + Ri  (Bj Fj-i Aj) j j
There is a reciprocity relationship:
Fj-i Aj = Fi-jAi
Hence:
Bi Ai = Ei Ai + Ri  (Bj Fi-j Ai) j
So the radiosity of patch Ai is given by:
Bi = Ei + Ri  ( Bj Fi-j ) j

13. Creating a System of Equations
We get one equation for each patch
assume we can calculate form factors
then N equations for N unknowns B1,B2,..BN
First equation is:
(1-R1F1-1)B1 - (R1F1-2)B2 - (R1F1-3)B (R1F1-N)BN = E1
..and we get in all N equations like this.
Generally Fi-i will be zero - why?
Most of the Ei will be zero - why?
B1 = E1 + R1  ( Bj F1-j ) j

14. Solving the Equations
Equations are solved iteratively - ie a first guess chosen, then repeatedly refined until deemed to converge
Suppose we guess solution as:
B1(0), B2(0), .. BN(0)
Rewrite first equation as:
B1 = E1 + (R1F1-2)B (R1F1-N)BN
Then we can ‘improve’ estimate of B1 by:
B1(1) = E1 + (R1F1-2)B2(0) (R1F1-N)BN(0)
.. and so on for other Bi

15. Solving the Equations This gives an improved estimate:
B1(1), B2(1), .. BN(1)
and we can continue until the iteration converges.
This is known as the Jacobi iterative method
An improved method is Gauss-Seidel iteration
this always uses best available values
eg B1(1) (rather than B1(0) ) used to calculate B2(1), etc

16. Rendering What have we calculated?
The Bi are the intensities of light emanating from each patch
the form factors do not depend on wavelength , but the Ri do
thus Bi depend on , so we need to calculate BiRED, BiGREEN, BiBLUE
We get vertex intensities by averaging the intensities of surrounding faces
Then we can pass to Gouraud renderer code for interpolated shading

17. Pause at this stage Number of equations can be very large
Calculation is not view dependent
Only diffuse reflection
We still have not seen how to calculate the form factors!
Their calculation dominates

18. Form Factors
The calculation of the form factors is unfortunately quite hard
We begin by looking at the form factor between two infinitesimal areas on the patches

19. Form Factors - Notation
Form factor Fdi-dj gives the fraction
of energy reaching dAj from dAi. Aj
dAj Nj j Ni r i Ai
r is distance between elements
Ni, Nj are the normals
i, j are angles made with normals
by line joining elements
dAi

20. Form Factor Calculation : 2D Cross Section View
Draw in 2D but imagine this in 3D:
* create unit hemisphere with dAi at centre
* light emits/reflects equally in all
directions from dAi
* form factor Fdi-dj is fraction of energy
reaching dAj from dAi
dAi Aj
dAj Ai

21. Form Factor Calculation : 2D Cross Section View
We begin calculation by
* projecting dAj onto the surface
of the hemisphere (blue)
* this resulting area is (cos j / r2 ) dAj
dAi Aj
dAj Nj j r
This gives us the relative area of light energy reaching dAj from dAi

22. Form Factor Calculation : 2D Cross Section View
But we need a measure per
unit area of Ai so need to adjust
for orientation of Ai
This gives a ‘corrected’ area as:
( cos i cos j / r2 ) dAj
dAi Aj
dAj i

23. Form Factor Calculation : 2D Cross Section View
Total energy comes from integrating
over whole hemisphere - this comes to 
Hence form factor is given by:
Fdi-dj = ( cos i cos j ) / (  r2 )dAj
dAi

24. Form Factor Calculation : 2D Cross Section View
Finally we need to sum up for
ALL dAj. This means integrating
over the whole of the patch:
Fdi-j =  (cos i cos j ) /(  r2 ) dAj
dAi Aj Ai

25. Form Factor Calculation
Strictly speaking, we should now
integrate over all dAi. In practice,
we take Fdi-j as representative of
Fi-j and this assumption:
Fi-j = Fdi-j
works OK in practice.
dAi Aj Ai
However the calculation of the integral  (cos i cos j ) /(  r2 ) dAj
is extremely difficult. Next lecture will discuss an approximation.