1. Metropolis light transport
Digital Image Synthesis
Yung-Yu Chuang
12/27/2007
with slides by Matt Pharr

2. 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

3. 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

4. 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

5. Metropolis sampling
Detailed balance
stationary distribution

6. Binary example I

7. Binary example II

8. 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

9. 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

10. Pseudo code

11. Pseudo code (expected value)

12. 1D example

13. 1D example (mutation)

14. 1D example
mutation 1
mutation 2
10,000 iterations

15. 1D example
mutation 1
mutation 2
300,000 iterations

16. 1D example mutation 1 90% mutation 2 + 10% mutation 1
Periodically using uniform mutations increases ergodicity

17. 2D example (image copy)

18. 2D example (image copy)

19. 2D example (image copy) 1 sample per pixel 8 samples per pixel

20. 3D example (motion blur)

21. Application to integration

22. Application to integration

23. Motion blur

24. Motion blur

25. Distributed ray tracing
Results
Distributed ray tracing
Metropolis sampling

26. Parameter tweaking

27. 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.

28. Metropolis light transport
Each path is generated by mutating previous path.
Advantages:
Path reuse: efficient
Local exploration: explore important contributions, reducing variance

29. Lighting transport

30. Bidirectional mutation

31. Caustic perturbation

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

33. Results
Bidirectional
Path tracing
25 samples per pixel

34. Results
Metropolis light transport
With the same number of ray queries

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

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