Metropolis sampling

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Metropolis light transport

Digital Image Synthesis Yung-Yu Chuang

12/27/2007

with slides by Matt Pharr

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

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

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

Binary example I

Binary example II 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

Pseudo code

Pseudo code (expected value) 1D example

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1D example (mutation) 1D example

mutation 1 mutation 2

10,000 iterations

1D example

mutation 1 mutation 2

300,000 iterations

1D example

mutation 1 90% mutation 2

+ 10% mutation 1

Periodically using uniform mutations increases ergodicity

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2D example (image copy) 2D example (image copy)

2D example (image copy)

1 sample per pixel

8 samples per pixel

256 samples per pixel

3D example (motion blur)

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Application to integration Application to integration

Motion blur Motion blur

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Results

Distributed ray tracing Metropolis sampling

Parameter tweaking

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.

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 Bidirectional mutation

Caustic perturbation Lens perturbation and pixel stratification

• Make sure every pixel is covered somehow.

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Results

Bidirectional Path tracing 25 samples per pixel

Results

Metropolis light

transport With the same number of ray queries

Results

Bidirectional path tracing (40 samples per pixel)

Results

Metropolis light transport (average 250 mutations per pixel, same computation time as the above)