PATH INTEGRAL FORMULATION OF LIGHT

TRANSPORT

Jaroslav KřivánekCharles University in Prague

http://cgg.mff.cuni.cz/~jaroslav/

Light transport

Geometric optics

emit

travel

absorbscatter

2Jaroslav Křivánek - Path Integral Formulation of Light Transport

Light transport

emit

travel

absorbscatter

light transport path

3Jaroslav Křivánek - Path Integral Formulation of Light Transport

Light transport

Camera response all paths hitting

the sensor

4Jaroslav Křivánek - Path Integral Formulation of Light Transport

)(d)( xxfI jj

Path integral formulation

cam

era

resp

.

(j-th

pix

el v

alue

)al

l pat

hsm

easu

rem

ent

cont

ribu

tion

func

tion

5

[Veach and Guibas 1995][Veach 1997]

Jaroslav Křivánek - Path Integral Formulation of Light Transport

Measurement contribution function

)( 10 xxLe )( 1 kkje xxW

kxxxx 10

sensor sensitivity(“emitted importance”)

paththroughput

)()()()( 110 kkjeej xxWxTxxLxf

emittedradiance

6

)()(...)()()( 11110 kkkss xxGxxxxGxT

0x

1x 1kx

kx

Jaroslav Křivánek - Path Integral Formulation of Light Transport

Geometry term

x

yy

x

)(|cos||cos|

)( 2 yxVyx

yxG yx

7Jaroslav Křivánek - Path Integral Formulation of Light Transport

)(d)( xxfI jj

Path integral formulationca

mer

a re

sp.

(j-th

pix

el v

alue

)al

l pat

hsm

easu

rem

ent

cont

ribu

tion

func

tion

?

8Jaroslav Křivánek - Path Integral Formulation of Light Transport

Path integral formulation

100

1

)(d)(d)(

)(d)(

k M

kkj

jj

k

xAxAxxf

xxfI

all pathlengths

all possible vertex positions

9Jaroslav Křivánek - Path Integral Formulation of Light Transport

Path integral

)(d)( xxfI jj pi

xel v

alue

all p

aths

cont

ribu

tion

func

tion

10Jaroslav Křivánek - Path Integral Formulation of Light Transport

RENDERING :

EVALUATING THE PATH INTEGRAL

Path integral

)(d)( xxfI jj pi

xel v

alue

all p

aths

cont

ribu

tion

func

tion

Monte Carlo integration

12Jaroslav Křivánek - Path Integral Formulation of Light Transport

Monte Carlo integration

General approach to numerical evaluation of integrals

x1

f(x)

0 1

p(x)

x2x3 x4x5 x6

xxfI d)(

)(;)(

)(1

1

xpxxp

xf

NI i

N

i i

i

Integral:

Monte Carlo estimate of I:

Correct „on average“:

IIE ][

13Jaroslav Křivánek - Path Integral Formulation of Light Transport

MC evaluation of the path integral

Sample path from some distribution with PDF

Evaluate the probability density

Evaluate the integrand

??

x )(xp

)(xp

)(xf j

Path integral

)(d)( xxfI jj )(

)(

xp

xfI jj

MC estimator

14Jaroslav Křivánek - Path Integral Formulation of Light Transport

Algorithms = different path sampling techniques

Path sampling

15Jaroslav Křivánek - Path Integral Formulation of Light Transport

Algorithms = different path sampling techniques

Path tracing

Path sampling

16Jaroslav Křivánek - Path Integral Formulation of Light Transport

Algorithms = different path sampling techniques

Light tracing

Path sampling

17Jaroslav Křivánek - Path Integral Formulation of Light Transport

Algorithms = different path sampling techniques

Same general form of estimator

No importance transport, no adjoint equations!!!

Path sampling

)(

)(

xp

xfI jj

19Jaroslav Křivánek - Path Integral Formulation of Light Transport

PATH SAMPLING&

PATH PDF

Local path sampling

Sample one path vertex at a time

1. From an a priori distribution lights, camera sensors

2. Sample direction from an existing vertex

3. Connect sub-paths test visibility between vertices

Jaroslav Křivánek - Path Integral Formulation of Light Transport 21

Example – Path tracing

1. A priori distrib.2. Direction sampling3. Connect vertices

1.

2.

1.

3.2.

2.

22Jaroslav Křivánek - Path Integral Formulation of Light Transport

Use of local path sampling

Path tracing Light tracingBidirectionalpath tracing

23Jaroslav Křivánek - Path Integral Formulation of Light Transport

Probability density function (PDF)

path PDF

),...,()( 0 kxxpxp joint PDF of path vertices

0x

1x

2x3x

24Jaroslav Křivánek - Path Integral Formulation of Light Transport

Probability density function (PDF)

path PDF

),...,()( 0 kxxpxp joint PDF of path vertices

0x

1x

2x3x

25Jaroslav Křivánek - Path Integral Formulation of Light Transport

Probability density function (PDF)

path PDF

),...,()( 0 kxxpxp joint PDF of path vertices

)|( 32 xxp)|( 21 xxp

)( 0xp

)( 3xpproduct of (conditional)vertex PDFs

0x

1x

2x3x

Path tracing example:

26Jaroslav Křivánek - Path Integral Formulation of Light Transport

Probability density function (PDF)

path PDF

),...,()( 0 kxxpxp joint PDF of path vertices

)( 2xp)( 1xp)( 0xp

)( 3xpproduct of (conditional)vertex PDFs

0x

1x

2x3x

Path tracing example:

27Jaroslav Křivánek - Path Integral Formulation of Light Transport

Vertex sampling

Importance sampling principle

1. Sample from an a priori distrib.

2. Sample direction from an existing vertex

3. Connect sub-paths

BRDF lobesampling

emissionsampling

high thruputconnections

Jaroslav Křivánek - Path Integral Formulation of Light Transport 28

BRDF lobesampling

Vertex sampling

Sample direction from an existing vertex

29Jaroslav Křivánek - Path Integral Formulation of Light Transport

Measure conversionBRDF lobesampling

Sample direction from an existing vertex

)()()( yxGyxpyp

x

yy

x

30Jaroslav Křivánek - Path Integral Formulation of Light Transport

)()(

)()(

)(

)(

yxGyxp

yxGyx

xp

xfI

s

jj

w.r.

t. ar

ea

w.r.

t. pr

oj.

solid

ang

le

Summary

Path integral

)(d)( xxfI jj

pixe

l val

ueal

l pat

hsco

ntribu

tion

func

tion

)(

)(

xp

xfI jj

MC estimator

path

pdf

sam

pled

path

kxxx ...0

jekkkssej WxxGxxxxGLxf )()(...)()()( 11110

)()()( 0 kxpxpxp

0x

1x 1kx

kx

31Jaroslav Křivánek - Path Integral Formulation of Light Transport

Summary

Algorithms

different path sampling techniques

different path PDF

32Jaroslav Křivánek - Path Integral Formulation of Light Transport

Time for questions…

Tutorial: Path Integral Methods for Light Transport Simulation

Jaroslav Křivánek – Path Integral Formulation of Light Transport