CS 551/651:
Advanced Computer Graphics
Advanced Ray Tracing
Radiosity
David Luebke
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7/27/2016
Administrivia
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My penance: Ray tracing homeworks very
slow to grade
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Many people didn’t include READMEs
Many (most) people didn’t include
workspace/project files or Makefiles
Some people’s don’t work
Nobody’s works perfectly
Quiz 1: Tuesday, Feb 20
David Luebke
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7/27/2016
Recap: Stochastic Sampling
Sampling theory tells us that with a regular
sampling grid, frequencies higher than the
Nyquist limit will alias
 Q: What about irregular sampling?
 A: High frequencies appear as noise, not
aliases
 This turns out to bother our visual system less!
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David Luebke
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Recap: Stochastic Sampling
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Poisson distribution:
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Completely random
Add points at random until area is full.
Uniform distribution: some neighboring samples
close together, some distant
David Luebke
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7/27/2016
Recap: Stochastic Sampling
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Poisson disc distribution:
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Poisson distribution, with minimum-distance
constraint between samples
Add points at random, removing again if they are
too close to any previous points
Jittered distribution
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Start with regular grid of samples
Perturb each sample slightly in a random direction
More “clumpy” or granular in appearance
David Luebke
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7/27/2016
Recap: Stochastic Sampling
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Spectral characteristics of these distributions:
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Poisson: completely uniform (white noise). High
and low frequencies equally present
Poisson disc: Pulse at origin (DC component of
image), surrounded by empty ring (no low
frequencies), surrounded by white noise
Jitter: Approximates Poisson disc spectrum, but
with a smaller empty disc.
David Luebke
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Recap: Nonuniform
Supersampling
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To be correct, need to modify filtering step:
David Luebke
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Recap: Nonuniform
Supersampling
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Approximate answer: weighted average filter
  I(i, j) h(x-i, y-j)
I’(x,y) =   h(x-i, y-j)
Correct answer: multistage filtering
 Real-world answer: ignore the problem
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David Luebke
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7/27/2016
Recap: Antialiasing and
Texture Mapping
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Texture mapping is uniquely harder
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Potential for infinite frequencies
Texture mapping is uniquely easier
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Textures can be prefiltered
David Luebke
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7/27/2016
Recap: Antialiasing and Texture
Mapping
Issue in prefiltering texture is how much
texture a pixel filter covers
 Simplest prefiltering scheme: MIP-mapping
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Idea: approximate filter size, ignore filter shape
Create a pyramid of texture maps
Each level doubles filter size
David Luebke
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7/27/2016
Recap: Mip-Mapping
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Tri-linear mip-mapping
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Pixel size  depth d
Linearly interpolate colors within 2 levels
closest to d
Linearly interpolate color between levels
according to d
David Luebke
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7/27/2016
Distributed Ray Tracing
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Distributed ray tracing is an elegant technique
that tackles many problems at once
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Stochastic ray tracing: distribute rays
stochastically across pixel
Distributed ray tracing: distribute rays
stochastically across everything
David Luebke
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7/27/2016
Distributed Ray Tracing
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Distribute rays stochastically across:
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Pixel for antialiasing
Light source for soft shadows
Reflection function for soft (glossy) reflections
Time for motion blur
Lens for depth of field
Cook: 16 rays suffice for all of these
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I done told you wrong: 4x4, not 8x8
David Luebke
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7/27/2016
Distributed Ray Tracing
Distributed ray tracing is basically a Monte
Carlo estimation technique
 Practical details:
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Use lookup tables (LUTs) for distributing rays
across functions
See W&W Figure 10.14 p 263
David Luebke
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7/27/2016
Backwards Ray Tracing
Traditional ray tracing traces rays from the
eye, through the pixel, off of objects, to the
light source
 Backwards ray tracing traces rays from the
light source, into the scene, into the eye
 Why might this be better?
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David Luebke
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7/27/2016
Backwards Ray Tracing
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Backwards ray tracing can capture:
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Indirect illumination
Color bleeding
Caustics
 Remember
what a caustic is?
 Give some examples
 Remember where caustics get the name?
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Backwards Ray Tracing
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Usually implies two passes:
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Rays are cast from light into scene
Rays are cast from the eye into scene, picking up
illumination showered on the scene from the first
pass
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Backwards Ray Tracing
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Q: How might these two passes “meet in the
middle?”
David Luebke
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7/27/2016
Backwards Ray Tracing
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Arvo: illumination maps tile surfaces with
regular grids, like texture maps
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Shoot rays outward from lights
Every ray hit deposits some of its energy into
surface’s illumination map
 Ignore
first generation hits that directly illuminate
surface (Why?)
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Eye rays look up indirect illumination using
bilinear interpolation
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7/27/2016
Backwards Ray Tracing
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Watt & Watt, Plate 34
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Illustrates Arvo’s method of backwards ray tracing
using illumination maps
Illustrates a scene with caustics
Related idea: photon maps
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7/27/2016
Advanced Ray Tracing Wrapup
Backwards ray tracing accounts for indirect
illumination by considering more general paths
from light to eye
 Distributed ray tracing uses a Monte Carlo
sampling approach to solve many ray-tracing
aliasing problems
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David Luebke
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7/27/2016
Next Up: Radiosity
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Ray tracing:
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Models specular reflection easily
Diffuse lighting is more difficult
Radiosity methods explicitly model light as an
energy-transfer problem
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Models diffuse interreflection easily
Shiny, specular surfaces more difficult
David Luebke
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7/27/2016
Radiosity Introduction
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First lighting model: Phong
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Still used in interactive graphics
Major shortcoming: local illumination!
After Phong, two major approaches:
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Ray tracing
Radiosity
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7/27/2016
Radiosity Introduction
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Ray tracing: ad hoc approach to simulating
optics
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Deals well with specular reflection
Trouble with diffuse illumination
Radiosity: theoretically rigorous simulation of
light transfer
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Very realistic images
But makes simplifying assumption: only diffuse
interaction!
David Luebke
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7/27/2016
Radiosity Introduction
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Ray-tracing:
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Computes a view-dependent solution
End result: a picture
Radiosity:
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Models only diffuse interaction, so can compute a
view-independent solution
End result: a 3-D model
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7/27/2016
Radiosity
Basic idea: represent surfaces in environment
as many discrete patches
 A patch, or element, is a polygon over which
light intensity is constant
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David Luebke
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7/27/2016
Radiosity
Model light transfer between patches as a
system of linear equations
 Solving this system gives the intensity at each
patch
 Solve for R, G, B intensities and get color at
each patch
 Render patches as colored polygons in
OpenGL. Voila!
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David Luebke
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7/27/2016
Fundamentals
Theoretical foundation: heat transfer
 Need system of equations that describes
surface interreflections
 Simplifying assumptions:
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Environment is closed
All surfaces are Lambertian reflectors
David Luebke
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Radiosity
The radiosity of a surface is the rate at which
energy leaves the surface
 Radiosity = rate at which the surface emits
energy + rate at which the surface reflects
energy
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Notice: previous methods distinguish light sources
from surfaces
In radiosity all surfaces can emit light
Thus: all emitters inherently have area
David Luebke
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7/27/2016
Radiosity
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Break environment up into a finite number n of
discrete patches
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Patches are opaque Lambertian surfaces of finite
size
Patches emit and reflect light uniformly over their
entire surface
Q: What’s wrong with this model?
 Q: What can we do about it?
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David Luebke
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7/27/2016
Radiosity
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Then for each surface i:
Bi = Ei + i  Bj Fji (Aj / Ai)
where
Bi, Bj= radiosity of patch i, j
Ai, Aj= area of patch i, j
Ei = energy/area/time emitted by i
i = reflectivity of patch i
Fji = Form factor from j to i
David Luebke
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7/27/2016
Form Factors
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Form factor: fraction of energy leaving the
entirety of patch i that arrives at patch j,
accounting for:
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The shape of both patches
The relative orientation of both patches
Occlusion by other patches
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Form Factors
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Some examples…
Form factor:
nearly 100%
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Form Factors
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Some examples…
Form factor:
roughly 50%
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Form Factors
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Some examples…
Form factor:
roughly 10%
David Luebke
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7/27/2016
Form Factors
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Some examples…
Form factor:
roughly 5%
David Luebke
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7/27/2016
Form Factors
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Some examples…
Form factor:
roughly 30%
David Luebke
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7/27/2016
Form Factors
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Some examples…
Form factor:
roughly 2%
David Luebke
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Form Factors
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In diffuse environments, form factors
obey a simple reciprocity relationship:
Ai Fij = Ai Fji
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Which simplifies our equation:
Bi = Ei + i
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Rearranging to:
Bi - i
David Luebke
 Bj Fij
 Bj Fij = Ei
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7/27/2016
Form Factors
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So…light exchange between all patches
becomes a matrix:
1  1F11  1F12
 F
1


F
2
21
2
22

 


  n Fn1  n Fn 2
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David Luebke
 1F1n   B1   E1 


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

 2 F2 n   B2   E2 

     


   
 1  n Fnn   Bn   En 
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What do the various terms mean?
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Form Factors
1 - 1F11
- 2F21
.
.
.
- pnFn1
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David Luebke
- 1F12
1 - 2F22
.
.
.
- nFn2
… - 1F1n
… - 2F2n
…
.
…
.
…
.
… 1 - nFnn
B1
B2
.
.
.
Bn
E1
E2
.
.
.
En
Note: Ei values zero except at emitters
Note: Fii is zero for convex or planar patches
Note: sum of form factors in any row = 1 (Why?)
Note: n equations, n unknowns!
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7/27/2016
Radiosity
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Now “just” need to solve the matrix!
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W&W: matrix is “diagonally dominant”
Thus Guass-Siedel must converge
End result: radiosities for all patches
 Solve RGB radiosities separately, color each
patch, and render!
 Caveat: actually, color vertices, not patches
(see F&vD p 795)
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David Luebke
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7/27/2016
Radiosity
Q: How many form factors must be computed?
 A: O(n2)
 Q: What primarily limits the accuracy of the
solution?
 A: The number of patches
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David Luebke
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7/27/2016
Radiosity
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Where we go from here:
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Evaluating form factors
Progressive radiosity: viewing an approximate
solution early
Hierarchical radiosity: increasing patch resolution
on an as-needed basis
David Luebke
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The End
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