PATH INTEGRALFORMULATION OF LIGHTTRANSPORT
Jaroslav Křivánek
Charles University in Prague
http://cgg.mff.cuni.cz/~jaroslav/
Light transport
Geometric optics
emit
travel
absorb
scatter
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Light transport
emit
travel
absorb
scatter
lighttransportpathlighttransportpath
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Light transport
Camera response
all paths hittingthe sensor
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Path integral formulation
camera resp.
(j-th pixel value)
all paths
measurementcontributionfunction
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[Veach and Guibas 1995]
[Veach 1997]
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Measurement contribution function
sensor sensitivity(“emitted importance”)
path
throughput
emitted
radiance
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Geometry term
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Path integral formulation
camera resp.
(j-th pixel value)
all paths
measurementcontributionfunction
?
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Path integral formulation
all pathlengths
all possible
vertex positions
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Path integral
pixel value
all paths
contributionfunction
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
RENDERING : EVALUATING THE PATHINTEGRAL
Path integral
pixel value
all paths
contributionfunction
Monte Carlo integration
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Monte Carlo integration
General approach to numerical evaluation of integrals
x1
f(x)
0
1
p(x)
x2
x3
x4
x5
x6
Integral:
Monte Carlo estimate of I:
Correct „on average“:
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Jaroslav 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
?
?
Path integral
MC estimator
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Path sampling
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Path tracing
Path sampling
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Light tracing
Path sampling
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Bidirectionalpath tracing
Path sampling
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Jaroslav 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
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Jaroslav 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
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Example – Path tracing
1.A priori distrib.
2.Direction sampling
3.Connect vertices
1.
2.
1.
3.
2.
2.
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Use of local path sampling
Path tracing
Light tracing
Bidirectionalpath tracing
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Probability density function (PDF)
path PDF
joint PDF of path vertices
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Probability density function (PDF)
path PDF
joint PDF of path vertices
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Probability density function (PDF)
path PDF
joint PDF of path vertices
product
of (conditional)
vertex PDFs
Path tracing example:
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Probability density function (PDF)
path PDF
joint PDF of path vertices
product
of (conditional)
vertex PDFs
Path tracing example:
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Jaroslav 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 lobeBRDF lobe
samplingsampling
emissionemission
samplingsampling
high thruputhigh thruput
connectionsconnections
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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BRDF lobeBRDF lobe
samplingsampling
Vertex sampling
Sample direction from an existing vertex
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Measure conversion
BRDF lobeBRDF lobe
samplingsampling
Sample direction from an existing vertex
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
w.r.t. area
w.r.t. proj.solid angle
Summary
Path integral
pixel value
all paths
contributionfunction
MC estimator
path pdf
sampled
path
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
Summary
Algorithms
different path sampling techniques
different path PDF
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Jaroslav 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