path integral formulation of light transport
DESCRIPTION
Path Integral Formulation of Light Transport. Jaroslav Křivánek Charles University in Prague http://cgg.mff.cuni.cz/~jaroslav/. Light transport. emit. travel. reflect. scatter. Geometric optics. Light transport. emit. travel. reflect. scatter. light transport path. - PowerPoint PPT PresentationTRANSCRIPT
PATH INTEGRAL FORMULATION OF LIGHT
TRANSPORT
Jaroslav KřivánekCharles University in Prague
http://cgg.mff.cuni.cz/~jaroslav/
Light transport
Geometric optics
emit
travel
reflect
2Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
scatter
Light transport
Geometric optics
emit
travel
reflect
3Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
scatter
light transport path
Camera response all paths hitting
the sensor
Light transport
4Course: Recent Advances in Light Transport Simulation
Jaroslav 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]
Course: Recent Advances in Light Transport SimulationJaroslav 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
)()()(...)()()()( rssr TGTGxT
0x
1x 1kx
kx
)(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
?
7Course: Recent Advances in Light Transport Simulation
Jaroslav 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
8Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Path integral
)(d)( xxfI jj pi
xel v
alue
all p
aths
cont
ribu
tion
func
tion
9Course: Recent Advances in Light Transport Simulation
Jaroslav 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
11Course: Recent Advances in Light Transport Simulation
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)
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 ][
12Course: Recent Advances in Light Transport Simulation
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
??
x )(xp
)(xp
)(xf j
Path integral
)(d)( xxfI jj )(
)(
xp
xfI jj
MC estimator
13Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Path sampling
14Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Path tracing
Path sampling
15Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Light tracing
Path sampling
16Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Algorithms = different path sampling techniques
Same general form of estimator
Path sampling
)(
)(
xp
xfI jj
17Course: Recent Advances in Light Transport Simulation
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
Course: Recent Advances in Light Transport SimulationJaroslav Křivánek - Path Integral Formulation of Light Transport
BRDF lobesampling
Use of local path sampling
Path tracing Light tracingBidirectionalpath tracing
20Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Probability density function (PDF)
path PDF
),...,()( 0 kxxpxp joint PDF of path vertices
0x
1x
2x3x
21Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
Probability density function (PDF)
path PDF
),...,()( 0 kxxpxp joint PDF of path vertices
0x
1x
2x3x
22Course: Recent Advances in Light Transport Simulation
Jaroslav 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:
23Course: Recent Advances in Light Transport Simulation
Jaroslav 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:
24Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
MC evaluation of the path integral
Sample path
Evaluate the probability density
Evaluate the integrand
x
)(xp
)(xf j
Path integral
)(d)( xxfI jj )(
)(
xp
xfI jj
MC estimator
25Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
BIDIRECTIONAL PATH TRACING
Bidirectional path tracing
Path tracing Light tracingBidirectional
path sampling
27Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
All possible bidirectional techniquesvertex on a light sub-path
vertex on en eye sub-path
28
path tracing
light tracing
Course: Recent Advances in Light Transport SimulationJaroslav Křivánek – Bidirectional Path Sampling Techniques
All possible bidirectional techniquesvertex on a light sub-path
vertex on en eye sub-path
29
path tracing
light tracing
VPLs
no single technique importance samples all the terms
Course: Recent Advances in Light Transport SimulationJaroslav Křivánek – Bidirectional Path Sampling Techniques
Multiple Importance Sampling (MIS)
f(x)
pa(x)pb(x)
[Veach & Guibas, 95]
2/)]()([
)(
xpxp
xfI
ba Combined
estimator:
xaJaroslav Křivánek – Light Transport Simulation with Vertex Connection and Merging
Bidirectional path tracing
Use all of the above sampling techniques
Combine using Multiple Importance Sampling
31Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
Naive BPT implementation
32Jaroslav Křivánek – Bidirectional Path Sampling Techniques
MIS weight calculation
Course: Recent Advances in Light Transport SimulationJaroslav Křivánek - Path Integral Formulation of Light Transport 33
BPT Implementation in practice
34Jaroslav Křivánek – Bidirectional Path Sampling Techniques
BPT Implementation in practice
35Jaroslav Křivánek – Bidirectional Path Sampling Techniques
Results
BPT, 25 samples per pixel PT, 56 samples per pixel
Imag
es:
Eri
c V
each
36Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques
NEARLY THERE…
Summary
Algorithms
different path sampling techniques
different path PDF
38Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Path Integral Formulation of Light Transport
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)
39Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek - Introduction
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
42Course: Recent Advances in Light Transport Simulation
Jaroslav Křivánek – Bidirectional Path Sampling Techniques