Radiosity

1
Radiosity

• What can we do with scan line conversion and ray
  tracing
• What we cant do
• Radiosity

2
Types of Illumination

• Scan-line Techniques
• Diffuse reflection of light sources
• Specular reflection of light sources
• Ray tracing and move advanced scan line
  techniques
• Specular reflection from other objects
• Transmission
• Seem like anything is missing, here?

T T
3
The Big Lie!!!

• Ambient Light!
• Does not really exist
• Its faking light reflecting from other surfaces

4
What really happens

• Light reflects off of surfaces and illuminates
  other surfaces
• Every surface affects every other surface
• One hacky solution
• Assume all of the light from all surfaces is
  averaged into a single illumination lighting that
  goes everywhere
• Thats what weve been calling ambient light.

5
Ambient Light is an approximation

• Many elements of natural lighting are not modeled
  well by ambient lighting at all
• Corners
• Interiors
• Moving items

6
Characteristics of Diffuse Reflection
Light
How will OpenGL draw this box? What about ray
tracing? What about nature?
Box
7
Our box on edge
8
Complex Lighting Environments

• This is really a complicated lighting
  environment.
• There is specular and diffuse reflection
  everywhere in the box.
• Ray tracing can approximate the specular
  reflection fairly well, but not the diffuse
  reflection.
• Lets just look at diffuse reflection for now.
• Note In the real world diffuse and specular
  reflection interact in a complex manner. Well
  get to combining the two later.

9
Radiosity

• Fundamental principle
• Conservation of energy
• Light is either absorbed or reflected!

10
Radiosity

• Every surface is treated as a diffuse reflector
  and emitter and all surfaces are assumed to
  conserve light energy.
• The rate at which energy leaves the surface, the
  radiosity, is the sum of the rates at which the
  surface reflects or transmits energy from other
  surfaces and the rate at which the surface emits
  energy.
• Note we no longer have disjoint light sources.
• Every object in the space is treated the same.
• A light source is simply a surface that emits
  energy.

11
How do we solve this?

• Simple example two surfaces

12
Solving this

• Every surfaces radiosity is dependent upon the
  light falling on that surface from other
  surfaces,
• which is a function of their radiosity,
• which will be dependent upon every other
  surfaces radiosity
• which will be dependent upon every other
  surfaces radiosity
• which will be dependent upon every other
  surfaces radiosity
• Is it any surprise that we will be either solving
  or approximating solutions to simultaneous
  equations?

13
Pure Radiosity vs. Real World

• Pure Radiosity systems dont have light sources,
  surfaces are light sources
• Lighted panels are easy to do, as are light
  bulbs, etc.
• Neon is just too cool
• But, many systems compute reflection from light
  sources, then augment with the radiosity solution.

14
Problems with our box

• What do we model?
• Surfaces?
• Points?

15
Patches

• Radiosity systems break surfaces into smaller
  areas called patches
• Surfaces are too big for a single radiosity
• Points are just too small

16
The Radiosity Equation

• Bi - The radiosity of patch i.
• the rate at which light is emitted. This is
  units of energy per unit area per time.
• This is what we are computing. Note that this
  may be a spectra (color). Do all this stuff for
  R, G, and B.
• Ei - The emissivity of the patch.
• This is the light generated by the patch and is
  of the same units as Bi, will be zero for
  anything that is not a light source.

17
The Radiosity Equation

• ri - The reflectivity of the surface.
• This is equivalent to kdr in the Hall model.
• Bj - The radiosity of some other patch j.
• Fj-i - The form factor.
• What fraction of energy leaving patch j will
  arrive at patch i.
• This is dependent upon shape, relative angles,
  intervening patches, etc.
• Ai and Aj are the areas of the patches i and j.

18
Whats that Aj/Ai ratio?

• For patch j of area Aj, the amount of energy
  leaving per unit area is Bj.
• Since the area can be of any size, we are
  interested in the total amount of energy!
• This is BjAj.
• But, Bi is also energy per unit area.
• So, we divide the total energy getting to Bi from
  Bj by Ai.
• Hence, the ratio Aj/Ai.

19
A Simplification

• The unit area energy leaving patch j is scaled by
  Fj-iAj.
• Likewise, unit area energy leaving patch i is
  scaled by Fi-jAi.
• These are symmetrical paths
• So, Fi-jAi Fj-iAj. So, Fj-iAj/Ai Fi-j
• We can plug that into the above equation and
  ELIMINATE the areas

20
How do we solve this?

• We have one massive simultaneous equation
• Note that the problem is O(N2)
• We can solve using Gauss-Seidel iteration, a
  numerical method for solving simultaneous
  equations.

21
Solving
22
TNTArray2D A(patchcnt,
patchcnt) TNTArray1D
b(patchcnt) // Create matrix A and vector
b for(i0 ifor(int j0 j Aij -m_patchesi.r
m_patchesi.ffj if(i j)
Aij 1. bi
m_patchesi.e // Solve for x
JAMAQR qr(A) TNTArray1D x qr.solve(b) // Result is in x
for(i0 im_patchesi.b xi
23
But, these are radiosities for patches

• Wont the image look patchy?
• No...
• We average the colors for patches to make vertex
  colors and interpolate

24
View Independence

• Note that all of the diffuse lighting is view
  independent
• We can compute it once and display it afterwards
  with varying views
• Or, we can add this component to other computed
  light components
• Use ray tracing to compute the specular surface
  colors, radiosity to compute the diffuse

25
Examples
OpenGL Rendering
Radiosity Rendering
26
Form Factors
Ai Area of patch i Hij 1 if path from
dAi to dAj exists.
nj
fj
ni
fi
r
Patch j
Patch i
27
Form Factor Computations

• Direct analytical computation of form factors is
  not practical
• Life is too short for numerical integration
• Instead, well use an approximation technique

28
Cohen and Greenberg Form Factor Estimation
Algorithm

• We are interested in determining Fij, the form
  factor for light traveling from patch i to patch
  j.
• Well approximate this using the following
  technique

29
Hemicube

• Create a half cube - called a hemicube -
  consisting of rectangular square areas called
  cells

30
Projecting on Hemicube

• Project all the surfaces onto the surface of the
  hemicube. This will take 5 different projections
  to accomplish (why?)

31
Think of this as viewing

• When I view from the center of a patch, the
  following affects the form factor
• How big it appears
• The angles to the surface
• How far to the projection
• Note the far and large is the same as near and
  smaller
• If we use a z-buffer like algorithm, we can tell
  whats closer and not occluded

32
Item Buffers

• Suppose we used a z-buffer, but instead of
  putting colors in each pixel, we put some
  identification for an object
• Point to data structure
• Integer id, etc
• This is called an item buffer
• We use item buffers to do the projection onto the
  hemicube

33
Projecting on a Hemicube

• We have 5 item buffers, each one for the cells on
  one face of the hemicube

34
Cells equal parts of the form factor

• Suppose cell p has item j in it.
• We add DFp to Fij
• For top faces
• For side faces

(xp,yp,zp) is center of cell. DA is the area of
the cell.
35
Substructuring

• How much do we subdivide?
• Too much is costly, not enough looks bad

Substructuring Compute at one level of
subdivision and further divide if we get large
differences in intensities What will we have to
recompute?
36
Substructuring

• Very effective at finding shadow lines, and
  varying gradients