MonteCarlo Methods

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Monte-Carlo Methods
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Topics

• Kajiyas paper
• Showed that existing rendering methods are
  approximations of rendering equation.
• Introduced path tracing.
• More recent work Lafortune, Veach.

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Kajiyas Rendering Equation

• Expressed as point-to-point transfer
• Integral is over S,
• union of all surfaces

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Rendering Equation Terms

• is unoccluded two-point transport intensity
• Energy per unit time per unit area of source per
  unit area of target

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Rendering Equation Terms

• is geometry term
• 0 if x not visible from x, 1/r2 if visible.

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Rendering Equation Terms

• is unoccluded emittance term

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Rendering Equation Terms

• is unoccluded three-point transport reflectance
  (scattering term)
• Intensity scattered to x by x originating from
  x

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Why express as point-to-point?

• Wants to set up for a path tracing solution
• Point-to-point transfer of energy

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Relationship to OtherRendering Methods

• Compared rendering eqn. to conventional polygon
  rendering, ray tracing and distributed ray
  tracing.
• Let scattering,

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Relationship to Utah Approx.

• Geometry term, g, only computed to eye.
• g? is ambient term.
• Scattering operator, M, only operates on point
  sources (?0), ignores visibility to light (no
  shadows).
• M is now only sum over lights, not integration
• Later extensions for shadows and area lights

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Relationship to Ray Tracing

• M0 now one reflection, one refraction, and cosine
  for diffuse term
• Computes visibility, g, of point lights
  (generates shadow)
• M still small sum

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Relationship to Distributed Ray Tracing

• Similar equation to Whitteds
• Now need to evaluate M as integral
• M now distribution around reflection, refraction,
  and shadow ray
• Ambient term still elusive

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Relationship to Radiosity

• Solves energy balance, but only for diffuse.
• Derives equations for radiosity from the one
  presented.
• Points out that solving visibility is expensive
  and that you may not need radiosity for all
  surfaces (on other hand, you might).

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Kajiyas Path Tracing
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Method

• At each hit,
• One ray cast based on specular, diffuse, and
  transmission coefficients
• One random ray per light
• Constant number of rays per pixel (40)

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Markov Chains for Solution
p 0.11
p 0.08
p 1
p 0.02
Absorbing state
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Algorithm

• Choose pt. x visible from eye
• Add in radiated intensity
• For length of Markov path
• Select pt. x and compute g(x, x)
• Calculate reflectance ?(x, x, x), multiply by
  ?(x, x)
• Add contribution to pixel

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Sampling

• Most important factor in Monte Carlo
• Need to avoid bias
• But also need to make most out of few rays
• Otherwise, noisy images
• Kajiya discusses several ways.
• Better to look in more modern reference
• Glassner
• Recent dissertations

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Path Tracing

• Postulates that even for ray tracing, following
  one path (probabilistically) is better.
• Why? Most contribution from first ray.
• Need to be careful about proportion of
  reflection, refraction, and shadow rays.

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Results
256 x 256 image
Ray Traced (no ambient)
Path Traced
Light scattered by sphere
401 minutes
533 minutes
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Results
Objects are gray, except for spheres and
base. Color bleeding Caustics
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Current Methods

• Bi-directional path tracing (Lafortune and Veach)
• Metropolis (Veach)

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Pure Path Tracing
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Pure Path Tracing
Best for big luminaires. If lights small, few
hits and large variance.
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With Shadow Ray to Lights
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With Shadow Ray to Lights
Small lights OK. Best for specular surfaces.
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Light Tracing
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Light Tracing
Small lights OK. Best for caustics.
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Bi-Directional Path Tracing
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Bi-Directional Path Tracing
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Generating Samples

• Generated as groups
• Prefix from light joined with suffix from (to)
  eye (if edge is not obstructed)
• Russian roulette to cut off path
• Contribution must be multiplied by probability of
  generating path

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Results (from Veach)
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Metropolis

• Method for importance sampling
• A path is a sequence of points from a light
  to the eye.
• Let be the image contribution function,
  a measure of contribution over path
• is flux
  contributed by paths D
• Strategy generate sequence of paths
• with probability proportional to

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Mutations

• New path Xi1 mutated from Xi
• Probability of rejecting each mutation keeps
  paths distributed according to contribution
• Discard start-up to reduce bias
• Can be run from a set of seed paths
• Run some bi-directional paths to find good seeds

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Implementation

• Path selection important
• minimize rejected paths, but not too correlated
• finds sub-paths and replace
• or perturb vertices of path (for caustics, etc)
• Mutation of lens path, (LD)DSE, to cover
  image pixels
• Adds direct lighting
• rejects path if found by Metropolis

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Results
Light for this example comes only through crack
in doorway
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Results
There are specific mutations to capture caustics.
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Advantages

• Works well for difficult lighting
• because it stays in important area
• Like bi-directional path tracing theres just a
  little work to get a new path

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References

• Kajiya, Jim, The Rendering Equation, SIGGRAPH
  86.
• Lafortune papers and thesis
• Veach papers and thesis
• Jensen papers
• Shirley draft of MC book on his web page
• Kalos Whitlock, Monte Carlo Methods, 1986.