Post on 03-Jan-2016
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PATH INTEGRAL FORMULATION OF LIGHT
TRANSPORT
Jaroslav KřivánekCharles University in Prague
http://cgg.mff.cuni.cz/~jaroslav/
Light transport
Geometric optics
emit
travel
absorbscatter
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Light transport
emit
travel
absorbscatter
light transport path
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Light transport
Camera response all paths hitting
the sensor
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)(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
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)(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
?
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Path integral formulation
100
1
)(d)(d)(
)(d)(
k M
kkj
jj
k
xAxAxxf
xxfI
all pathlengths
all possible vertex positions
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Path integral
)(d)( xxfI jj pi
xel v
alue
all p
aths
cont
ribu
tion
func
tion
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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
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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 ][
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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
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Algorithms = different path sampling techniques
Path sampling
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Algorithms = different path sampling techniques
Path tracing
Path sampling
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Algorithms = different path sampling techniques
Light tracing
Path sampling
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Algorithms = different path sampling techniques
Same general form of estimator
No importance transport, no adjoint equations!!!
Path sampling
)(
)(
xp
xfI jj
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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.
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Use of local path sampling
Path tracing Light tracingBidirectionalpath tracing
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Probability density function (PDF)
path PDF
),...,()( 0 kxxpxp joint PDF of path vertices
0x
1x
2x3x
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Probability density function (PDF)
path PDF
),...,()( 0 kxxpxp joint PDF of path vertices
0x
1x
2x3x
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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:
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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:
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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
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Measure conversionBRDF lobesampling
Sample direction from an existing vertex
)()()( yxGyxpyp
x
yy
x
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)()(
)()(
)(
)(
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
sam
pled
path
kxxx ...0
jekkkssej WxxGxxxxGLxf )()(...)()()( 11110
)()()( 0 kxpxpxp
0x
1x 1kx
kx
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Summary
Algorithms
different path sampling techniques
different path PDF
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Time for questions…
Tutorial: Path Integral Methods for Light Transport Simulation
Jaroslav Křivánek – Path Integral Formulation of Light Transport