Metropolis sampling

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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 x_i to propose x’

– If it is accepted, x_i+1 = x’

Otherwise, x_i+1 = x_i

• Acceptance probability guarantees distribution is the stationary distribution f

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Metropolis sampling

• Mutations propose

x’

given

x

_i

• 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)