Metropolis light transport

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Metropolis light transport

• Digital Image Synthesis
• Yung-Yu Chuang
• 12/27/2007

with slides by Matt Pharr
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Metropolis sampling

• Another way to generate samples from a
  distribution (similar to inversion, rejection and
  transform)
• Problem given an arbitrary function
• assuming
• generate a set of samples

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Metropolis sampling

• MS only requires the ability to evaluate f
  without requiring integrating f, normalizing f
  nor inversion.
• Steps
• Generate initial sample x0
• mutating current sample xi to propose x
• If it is accepted, xi1 x
• Otherwise, xi1 xi
• Acceptance probability guarantees distribution is
  the stationary distribution f

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Metropolis sampling

• Mutations propose x given xi
• T(x?x) is the tentative transition probability
  density of proposing x from x
• Being able to calculate tentative transition
  probability is the only restriction for the
  choice of mutations
• a(x?x) is the acceptance probability of
  accepting the transition
• By defining a(x?x) carefully, we ensure

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Metropolis sampling

• Detailed balance

stationary distribution
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Binary example I
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Binary example II
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Acceptance probability

• Does not affect unbiasedness just variance
• Want transitions to happen because transitions
  are often heading where f is large
• Maximize the acceptance probability
• Explore state space better
• Reduce correlation

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Mutation strategy

• Very free and flexible, only need to calculate
  transition probability
• Based on applications and experience
• The more mutation, the better
• Relative frequency of them is not so important

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Pseudo code
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Pseudo code (expected value)
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1D example
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1D example (mutation)
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1D example
mutation 1
mutation 2 10,000 iterations
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1D example
mutation 1
mutation 2 300,000 iterations
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1D example
mutation 1
90 mutation 2 10 mutation 1
Periodically using uniform mutations increases
ergodicity
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2D example (image copy)
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2D example (image copy)
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2D example (image copy)
1 sample per pixel
8 samples per pixel
256 samples per pixel
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3D example (motion blur)
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Application to integration
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Application to integration
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Motion blur
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Motion blur
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Results
Distributed ray tracing
Metropolis sampling
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Parameter tweaking
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Metropolis light transport

• Veach and Guibas introduced Metropolis sampling
  to Graphics from computational physics in their
  SIGGRAPH 1997 paper, Metropolis Light Transport.
• Unbiased and robust (can deal with difficult
  cases such as caustics)
• However, difficult to understand and implement
  efficiently.
• Few implementation exists such as Indigo renderer
  and Kerkythea.

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Metropolis light transport

• Each path is generated by mutating previous path.
  
• Advantages
• Path reuse efficient
• Local exploration explore important
  contributions, reducing variance

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Lighting transport
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Bidirectional mutation
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Caustic perturbation
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Lens perturbation and pixel stratification

• Make sure every pixel is covered somehow.

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Results
Bidirectional Path tracing 25 samples per pixel
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Results
Metropolis light transport With the same number
of ray queries
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Results
Bidirectional path tracing (40 samples per pixel)
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Results
Metropolis light transport (average 250 mutations
per pixel, same computation time as the above)