Metropolis Light Transport

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Metropolis Light Transport

• Eric Veach
• Leonidas J. Guibas
• Computer Science Department Stanford University.

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Outline

• Prerequisite
• Overview
• Initial Path
• Mutation
• Result

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Markov chain

• Markov chain
• A stochastic process with Markov property.

The range of random variable X is called the
state space.
is called the transition probability. (one step)
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Markov chain

• How about two or three steps?

Distribution over states at time n1
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Markov chain

• Stationary distribution (steady-state
  distribution)

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Markov chain

• Ergodic theory
• If we take stationary distribution as initial
  distribution...the average of function f over
  samples of Markov chain is

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Markov chain

• If the state space is finite...transition
  probability can be represented as a transition
  matrix P.

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Overview
Rendering equation
Measurement equation
Expand L by rendering equation
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Overview
µ
fj
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Overview

• f the energy of all image plane.
• fj the energy of pixel j.

We can think wj restricted f to the pixel j.
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Overview
Now, we want to solve the integration
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Overview

• As the Monte-Carlo Method, we want the
  probability distribution p proportional to f.

K is transition function
converge
The same idea ..in this paper, we want stationary
distribution Proportional to f.
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Overview

• If we let

And we prefer the stationary distribution propose
to f
Mutation determine T(yx). If we determine a
mutation, we must Calculate a(yx) to satisfy
this equation!
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Overview
In this paper, based on bidirectional ray tracing
Based on three strategies, decide T(yx)
Computed by T(yx)
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Initial Path
In this paper, based on bidirectional ray tracing
Based on three strategies, decide T(yx)
Computed by T(yx)
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Initial Path

• How to choose Initial Path? (Have a good start!)
• Run n copies of the algorithm and accumulate all
  samples into one imege.
• Sample n paths
• Resample form the n paths to obtain relative
  small number n paths.
• If n 1, then just take the mean value.

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Mutation
In this paper, based on bidirectional ray tracing
Based on three strategies, decide T(yx)
Computed by T(yx)
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Properties of good mutations

• High acceptance probability.
• Prevent the path sequence x, x, x, x, x, x,
• Large changes to the path.
• Prevent path sequence with high correration
• Ergodicity.
• Ensure random walk converge to an ergodic state.

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Properties of good mutations

• Changes to the image location.
• Minimize correlation between image plane.
• Stratification.
• Uniform distribute on image plane.
• Low cost.
• As the word says..

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Mutation

• In this paper, three Mutation strategies is
  presented
• Bidirectional mutations
• Perturbations
• Lens subpath perturbations
• Each mutation decide T(YX), we must take a(YX)
  to satisfy

Roughly, when we use a mutation generate a new
path, we can compute T(yx), then we decide
a(yx). According to a(yx), we reject or accept
the new path.
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Bidirectional mutation
If we initially have a path
For k 3
probability to delete path from s to t
x1
x2
x3
x0
Probability to add new path of length s, t to
vertex s, t.
Pd(1,2)
x1
x2
x3
x0
Pa(1,0)
z
x0
x1
x2
x3
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Bidirectional mutation
We want a(yx) satisfy
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Compute R(yx),R(xy)
y
x
z
x0
x1
x2
x3
x1
x2
x3
x0
For compute R(yx)
For R(xy)
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Perturbations
If we initially have a path
For k 3
x1
x2
x3
x0
x1
x2
x3
x0
Choose a subpath and move the vertices
slightly. In the case above, the subpath is x1-x3.
Main interest perturbations is subpath consist
xk-1 - xk
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Perturbations

• Perturbations has two type
• Lens perturbations.
• Handle (LD)DSE
• Caustic perturbations.
• Handle (LD)SDE

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Perturbations

• How about (LD)DSDSDE?

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Lens subpath mutations

• Substitute len subpath (xt,,xk) to another one
  to achieve the goal of Stratification (LD)SE.
• Initialize by n initial path seeds . Then store
  the current lens subpath xe. At most reuse xe a
  fix number ne times.
• We sequentially mutate n initial path seeds.
• Generate new len subpath by case a ray through a
  point on image plane , follows zero or more
  specular bounce until a non-specular vertex.

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Result
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Result
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Result
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Result