Metropolis Light Transit

1
Metropolis Light Transit

• By Dan Halstead

2
For GLOBAL ILLUMINATION to happen

• NEED The ability to sample light transport
  paths.

3
For GLOBAL ILLUMINATION to happen

• NEED The ability to sample light transport
  paths.
• PROBLEM Many of the paths are unimportant.

4
For GLOBAL ILLUMINATION to happen

• Some previous solutions are
• Find information to guide the path finder.
• Bidirectional path tracing

5
Overview of MLT

• The Basic Idea
• Sample complete paths according to their
  contribution to the image by a random walk
  through path space.

6
Overview of MLT

• The Setup
• Let X be a path.

7
Overview of MLT

• The Setup
• Let X be a path.
• Let f(X) be the image contribution function.

8
Overview of MLT

• The Setup
• Let X be a path.
• Let f(X) be the image contribution function.
• Then we want to sample paths by picking them
  randomly so that their distribution p is
  proportional to f.

9
Overview of MLT

• There are two reasons for this
• We want more detail in brighter areas.
• The value of a pixel will then be proportional to
  the number of times it was sampled in a path.

10
Formulating the Problem

• The light transport equation

11
Formulating the Problem

• The light transport equation
• The measurement equation

12
Formulating the Problem

• The light transport equation
• Expanding the measurement equation

13
Formulating the Problem

• The light transport equation
• Expanding the measurement equation

14
Formulating the Problem

• The light transport equation
• Expanding the measurement equation

15
Formulating the Problem

• Expanding the measurement equation
• This can be simplified to

16
Formulating the Problem

• From this simplified form, we can derive a way to
  estimate mj from a sequence of N paths that are
  chosen randomly according to a distribution p

17
Formulating the Problem

• Now, define B to be the normalization constant
• And suppose that p ( 1 / B ) f. Then

18
Overview of MLT

• In order to sample paths, MLT uses a random walk
  through the space of possible paths.

19
Overview of MLT

• In order to sample paths, MLT uses a random walk
  through the space of possible paths.
• MLT will choose a new path Xi by performing a
  random mutation on Xi-1.

20
Overview of MLT

• In order to sample paths, MLT uses a random walk
  through the space of possible paths.
• MLT will choose a new path Xi by performing a
  random mutation on Xi-1.
• New paths can be rejected!

21
Overview of MLT

• Lastly, upon choosing each new path, MLT will
  update the appropriate image pixel(s).

22
Transitions

• How can this mutation thing possibly work??

23
Transitions

• Example
• Suppose you have an n x n grid.

24
Transitions

• Example
• Suppose you have an n x n grid.

25
Transitions

• On each transition step, you can remain still or
  move to an adjacent square.

26
Transitions

• K(YX) 1 / 5.

27
Transitions

28
Transitions

• p0

29
Transitions

• p1

30
Transitions

• p2

31
Transitions

• pi(X) ? 1 / n2 for all x

32
Transitions

• We want to find K such that
• The system will converge to f.
• It will converge as fast as possible.

33
Transitions

• It will help to break K up into two functions
• K(YX) T(YX) a(YX)
• T is the probability of mutating path x into y.
• a is the probability of this mutation being
  accepted.

34
Acceptance

• To derive the formula for a, note that in
  equilibrium the transition density from x to y
  will be the same as that from y to x
• f(X) T(YX) a(YX) f(Y) T(XY) a(XY)

35
Acceptance

• To verify this

36
Acceptance

• Also, we want to maximize a.
• (This will make the system mutate faster and
  therefore converge faster.)

37
Acceptance

• Given an arbitrary T then, we can derive from
  these two conditions that

38
Mutations

• Mutations should aim to do these things
• Dont make changes that are too small

39
Mutations

• Mutations should aim to do these things
• Dont make changes that are too small
• Dont get stuck

?
40
Mutations

• Mutations should aim to do these things
• Dont make changes that are too small
• Dont get stuck
• Support stratification

41
Mutations

• Mutations should aim to do these things
• Dont make changes that are too small
• Dont get stuck
• Support stratification
• Have low cost

42
Mutations

• Three mutations that satisfy all these goals
• Bidirectional mutations
• Perturbations
• Lens subpath mutations

43
Bidirectional Mutations

• Are the foundation of the MLT algorithm.

44
Bidirectional Mutations

• Are the foundation of the MLT algorithm.
• Consist of randomly removing some subpath, and
  recasting it with a random number of vertices.

45
Bidirectional Mutations

• Are the foundation of the MLT algorithm.
• Consist of randomly removing some subpath, and
  recasting it with a random number of vertices.
• Are responsible for making the large changes.

46
Bidirectional Mutations

• Given a path X x0 x1 xk
• Choose a random subpath to delete.
• Add random numbers of new vertices to the new
  interior endpoints of X.
• Try to connect up the two innermost vertices.
• Test for acceptance of the new path.

47
Bidirectional Mutations

• Choose a random subpath to delete.
• Weight the probability so that smaller subpaths
  are chosen more frequently.

48
Bidirectional Mutations

• Choose a random subpath to delete.
• Example

x1
?
x0
x2
49
Bidirectional Mutations

• Add random numbers of new vertices.
• Choose how many to add from a distribution
  centered around the old number.

50
Bidirectional Mutations

• Add random numbers of new vertices.
• Then choose where to put the break
• (i.e. how many of those to add to the end of the
  first segment vs. the beginning of the last
  segment).

51
Bidirectional Mutations

• Add random numbers of new vertices.
• Add each vertex by sampling a direction according
  to the BRDF, and then casting a ray.

52
Bidirectional Mutations

• Add random numbers of new vertices.
• Example
• Chose to add 2 vertices, with the break between
  them.

?
x0
x2
53
Bidirectional Mutations

• Try to connect up the two innermost vertices.
• Test if there is visibility between the two new
  endpoints. (If the path is obstructed, reject
  the mutation.)

54
Bidirectional Mutations

• Try to connect up the two innermost vertices.
• Example

x1
?
x0
x3
x2
55
Bidirectional Mutations

• Test for acceptance of the new path.
• Use
• Note that much of this can be pre-computed.
• Also, much of it can be cancelled out, since X
  and Y will likely share some vertices.

56
Perturbations

• Are needed when bidirectional mutations will
  nearly always be rejected.
• (e.g. When there are small regions of the path
  space in which paths contribute much more than
  average.)

57
Perturbations

• Are needed when bidirectional mutations will
  nearly always be rejected.
• (e.g. When there are small regions of the path
  space in which paths contribute much more than
  average.)
• Smaller mutations keep them within the high
  contribution region

58
Perturbations

1. Move the pixel location (i.e. the paths
   endpoint) by a random distance (chosen to have
   highest probability between two pre-selected
   values) in a random direction.

59
Perturbations

1. Move the pixel location (i.e. the paths
   endpoint) by a random distance (chosen to have
   highest probability between two pre-selected
   values) in a random direction.
2. Recast rays through all the specular bounces so
   that it retains the same length.

60
Perturbations

• Example

x1
?
x0
x2
61
Lens subpath mutations

• Designed to enhance stratification (i.e. to
  prevent bias in one area of the image plane).

62
Lens subpath mutations

• Designed to enhance stratification (i.e. to
  prevent bias in one area of the image plane).
• Occasionally force the algorithm to choose a new
  pixel location, and not one that was chosen too
  many times before.

63
Optimizations

• Run several copies of the algorithm in parallel.
• Helps to remove startup bias.
• Helps to test for convergence.

64
Optimizations

• Ignore the first several path samples.
• Helps to remove startup bias.

65
Optimizations

• Dont bother with MLT for the direct lighting in
  an image.
• Standard techniques for direct lighting usually
  provide better quality at lower cost.
• This way MLT can devote more effort to the
  indirect lighting.

66
Optimizations

• Do variance reduction on a.
• Instead of randomly accepting or rejecting a
  sample, accept exactly a copies of it.

67
Bidirectional path tracing
68
Metropolis Light Transit
69
(No Transcript)
70
Bidirectional path tracing
71
Metropolis Light Transit