Metropolis Light Transport

Eric VeachLeonidas J. Guibas

Computer Science Department Stanford University.

Outline

• Prerequisite

• Overview

• Initial Path

• Mutation

• Result

Markov chain

• Markov chain :– A stochastic process with Markov property.

)|(),...,,,|( 12101 nnnn XxXPXXXXxXP

The range of random variable X is called the state space.

)|( 1 nn XXP is called the transition probability. (one step)

Markov chain

How about two or three steps?

11121122 )|()|()|,()|( nnnnnnnnnnn dXXXPXXPdXXXXPXXP

21112233 )|()|()|()|( nnnnnnnnnn dXdXXXPXXPXXPXXP

nnnnn dXXPXXPXP )()|()( 11

Distribution over states at time n+1

Markov chain

• Stationary distribution (steady-state distribution):

dYYYXPX )()|()(

Markov chain

• Ergodic theory:– If we take stationary distribution as initial

distribution...the average of function f over samples of Markov chain is :

1

0

)()()(1

limn

k dXXXfXfn

Markov chain

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

)|( 1 iXjXPP nnij

OverviewRendering equation

Measurement equation

Expand L by rendering equation

Overview

)()( xdxfm jj

)()...()...( 00 kkk xdAxdAxxd

1

)()(k

kk DD kk

1

fj μ

Overview

• f : the energy of all image plane.

• fj : the energy of pixel j.

)()()( xfxwxf jj

We can think wj restricted f to the pixel j.

Overview

)()( xdxfm jj

Now, we want to solve the integration :

N

i i

iijj

Xp

XfXw

NEm

1 )(

)()(1

Overview

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

)()()|()( 1 ydypyxKxp ii

)()()|()( ydypyxKxp

converge

The same idea ..in this paper, we want stationary distributionProportional to f.

K is transition function

Overview

• If we let

)|()|()|( xyaxyTxyK

And we prefer the stationary distribution propose to f

)|()|()()|()|()(

)|()()|()(

)|()()|()(

yxayxTyfxyaxyTxf

yxKycpxyKxcp

yxKypxyKxp

Mutation determine T(y|x). If we determine a mutation, we mustCalculate a(y|x) to satisfy this equation!

Overview

In this paper, based on bidirectional ray tracing

Based on three strategies, decide T(y|x)

Computed by T(y|x)

Initial Path

In this paper, based on bidirectional ray tracing

Based on three strategies, decide T(y|x)

Computed by T(y|x)

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.

0

)(

0

)1(,...,XX

n

Mutation

In this paper, based on bidirectional ray tracing

Based on three strategies, decide T(y|x)

Computed by T(y|x)

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.

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..

Mutation

• In this paper, three Mutation strategies is presented…– Bidirectional mutations

– Perturbations

– Lens subpath perturbations

• Each mutation decide T(Y|X), we must take a(Y|X) to satisfy :

)|()|()()|()|()( yxayxTyfxyaxyTxf

Roughly, when we use a mutation generate a new path, we can compute T(y|x), then we decide a(y|x). According to a(y|x), we rejector accept the new path.

Bidirectional mutation

ko xxx ...

],[ tspd

If we initially have a path :

x0 x1 x2 x3

For k = 3:

]','[ tspa

probability to delete path from s to t

Probability to add new path of length s’, t’ to vertex s, t.

Pd(1,2)

x0 x1 x2 x3

x0 x1 x2 x3

Pa(1,0)

z

Bidirectional mutation

)|(

)()|( where

)|(

)|()|(

xyT

yfxyR

yxR

xyRxya

)|()|()()|()|()( yxayxTyfxyaxyTxf

We want a(y|x) satisfy:

Compute R(y|x),R(x|y)

z to xfrom sampling ofdensity y probabilit :)( 110 zxxps

z sampling ofdensity y probabilit:)()( 110 zxGzxxps

),,(),(),,(),(),,()( 322210110 xxzfxzGxzxfzxGzxxfyf sss

x0 x1 x2 x3z

),(),,(]1,0[),(),,(]0,1[]2,1[)|( 223110 zxGzxxppzxGzxxpppxyT sasad

y:x0 x1 x2 x3

x:

For compute R(y|x):

]0,0[]3,1[/),,(),(),,( 32121210 adss ppxxxfxxGxxxf

For R(x|y):

Perturbations

ko xxx ...If we initially have a path :

x0 x1 x2 x3

For k = 3:

x0 x1 x2 x3

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

Perturbations

• Perturbations has two type:– Lens perturbations.

• Handle (L|D)DS*E

– Caustic perturbations.• Handle (L|D)S*DE

Perturbations

• How about (L|D)DS*DS*DE?

Lens subpath mutations

• Substitute len subpath (xt,…,xk) to another one to achieve the goal of Stratification (L|D)S*E.

– 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.

Result

Result

Result

Result