Transcript Recent Advances in Light Transport Simulation

PATH INTEGRAL
FORMULATION OF LIGHT
TRANSPORT
Jaroslav Křivánek
Charles University in Prague
http://cgg.mff.cuni.cz/~jaroslav/
Light transport

Geometric optics
emit
travel
scatter
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
reflect
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Light transport

Geometric optics
emit
travel
scatter
reflect
light
transport
path
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Light transport

Camera response
 all paths hitting
the sensor
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Path integral formulation
[Veach and Guibas 1995]
[Veach 1997]
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Measurement contribution function
emitted
radiance
sensor sensitivity
path
throughput (“emitted importance”)
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Path integral formulation
? 
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Path integral formulation
all path all possible
lengths vertex positions
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Path integral
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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RENDERING :
EVALUATING THE PATH
INTEGRAL
Path integral

Monte Carlo integration
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Monte Carlo integration

General approach to numerical evaluation of integrals
Integral:
f(x)
Monte Carlo estimate of I:
p(x)
0
x5 x3 x1 x4 x2
x6
1 Correct „on average“:
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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MC evaluation of the path integral
Path integral
MC estimator

Sample path
from some distribution with PDF

Evaluate the probability density

Evaluate the integrand
?
?

Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Path sampling

Algorithms = different path sampling techniques
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Path sampling

Algorithms = different path sampling techniques

Path tracing
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Path sampling

Algorithms = different path sampling techniques

Light tracing
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Path sampling

Algorithms = different path sampling techniques

Same general form of estimator
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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PATH SAMPLING
&
PATH PDF
Local path sampling

1.
Sample one path vertex at a time
From an a priori distribution

lights, camera sensors
2.
Sample direction from an existing vertex
3.
Connect sub-paths

test visibility between vertices
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
BRDF lobe
sampling
Use of local path sampling
Path tracing
Light tracing
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Bidirectional
path tracing
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Probability density function (PDF)
path PDF
joint PDF of path vertices
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Probability density function (PDF)
path PDF
joint PDF of path vertices
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Probability density function (PDF)
path PDF
joint PDF of path vertices
product
of (conditional)
vertex PDFs
Path tracing example:
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Probability density function (PDF)
path PDF
joint PDF of path vertices
product
of (conditional)
vertex PDFs
Path tracing example:
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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MC evaluation of the path integral
Path integral

Sample path

Evaluate the probability density

Evaluate the integrand
MC estimator
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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BIDIRECTIONAL PATH
TRACING
Bidirectional path tracing
Path tracing
Light tracing
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
Bidirectional
path sampling
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All possible bidirectional techniques
vertex on a light sub-path
vertex on en eye sub-path

path tracing





Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
light tracing
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All possible bidirectional techniques
vertex on a light sub-path
vertex on en eye sub-path

path tracing

no single technique importance
 terms VPLs
samples all the



Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
light tracing
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Multiple Importance Sampling (MIS)
[Veach & Guibas, 95]
Combined
estimator:
f(x)
pa(x)
pb(x)
xa
Jaroslav Křivánek – Light Transport Simulation with Vertex Connection and Merging
Bidirectional path tracing

Use all of the above sampling techniques

Combine using Multiple Importance Sampling
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
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Naive BPT implementation
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MIS weight calculation
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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BPT Implementation in practice
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
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BPT Implementation in practice
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
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Images: Eric Veach
Results
BPT, 25 samples per pixel
PT, 56 samples per pixel
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
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NEARLY THERE…
Summary

Algorithms

different path sampling techniques

different path PDF
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Jaroslav Křivánek - Path Integral Formulation of Light Transport
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Why is the path integral view so
useful?

Identify source of problems


High contribution paths sampled with low probability
Develop solutions

Advanced, global path sampling techniques

Combined path sampling techniques (MIS)
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Introduction
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Joint
importance
sampling
Traditional
THANK YOU!
Time for questions…
Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Acknowledgements

Czech Science Foundation


grant no. P202-13-26189S
Images


Eric Tabellion
Marcos Fajardo
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Jaroslav Křivánek – Bidirectional Path Sampling Techniques
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