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 xi to propose x’ If it is accepted, xi+1 = x’ Otherwise, xi+1 = xi Acceptance probability guarantees distribution is the stationary distribution f

a(x→x’) is the acceptance probability of accepting the transition 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

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

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

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

2D example (image copy)

2D example (image copy)

2D example (image copy) 1 sample per pixel 8 samples per pixel

3D example (motion blur)

Application to integration

Application to integration

Motion blur

Motion blur

Distributed ray tracing 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

Lighting transport

Bidirectional mutation

Caustic perturbation

Lens perturbation and pixel stratification Make sure every pixel is covered somehow.

Results Bidirectional Path tracing 25 samples per pixel

Results Metropolis light transport With the same number of ray queries

Bidirectional path tracing (40 samples per pixel) Results Bidirectional path tracing (40 samples per pixel)

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