Path Integral Formulation of Light Transport
Description: Path Integral Formulation of Light Transport Jaroslav Křivánek Charles University in Prague http:cgg.mff.cuni.czjaroslav Light transport Geometric optics emit travel absorb scatter 2 Jaroslav Křivánek - Path Integral Formulation of
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slide1. Path Integral Formulation of Light Transport Jaroslav Křivánek
Charles University in Prague
http://cgg.mff.cuni.cz/~jaroslav/<br>
slide2. Light transport Geometric optics emit travel absorb
scatter 2 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide3. Light transport emit travel absorb
scatter light transport path 3 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide4. Light transport Camera response
all paths hitting the sensor 4 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide5. Path integral formulation camera resp.
(j-th pixel value) all paths measurement contribution function 5 [Veach and Guibas 1995]
[Veach 1997] Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide6. Measurement contribution function 6 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide7. Geometry term 7 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide8. Path integral formulation camera resp.
(j-th pixel value) all paths measurement contribution function  ? 8 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide9. Path integral formulation all path lengths all possible
vertex positions 9 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide10. Path integral 10 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide11. Rendering : Evaluating the path integral<br>
slide12. Path integral Monte Carlo integration 12 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide13. Monte Carlo integration General approach to numerical evaluation of integrals f(x) 0 1 Integral: Monte Carlo estimate of I: Correct „on average“: 13 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide14. MC evaluation of the path integral Sample path from some distribution with PDF
Evaluate the probability density
Evaluate the integrand  ? ? Path integral 14 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide15. Algorithms = different path sampling techniques Path sampling 15 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide16. Algorithms = different path sampling techniques
Path tracing Path sampling 16 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide17. Algorithms = different path sampling techniques
Light tracing Path sampling 17 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide18. Algorithms = different path sampling techniques
Bidirectional path tracing Path sampling 18 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide19. Algorithms = different path sampling techniques
Same general form of estimator
No importance transport, no adjoint equations!!! Path sampling 19 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide20. Path sampling & Path PDF<br>
slide21. Local path sampling Sample one path vertex at a time
From an a priori distribution
lights, camera sensors
Sample direction from an existing vertex
Connect sub-paths
test visibility between vertices Jaroslav Křivánek - Path Integral Formulation of Light Transport 21<br>
slide22. Example – Path tracing A priori distrib.
Direction sampling
Connect vertices 1. 2. 1. 3. 2. 2. 22 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide23. Use of local path sampling Path tracing Light tracing Bidirectional path tracing 23 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide24. Probability density function (PDF) 24 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide25. Probability density function (PDF) 25 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide26. Probability density function (PDF) Path tracing example: 26 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide27. Probability density function (PDF) Path tracing example: 27 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide28. Vertex sampling Importance sampling principle
Sample from an a priori distrib.
Sample direction from an existing vertex
Connect sub-paths BRDF lobe
sampling emission
sampling high thruput
connections Jaroslav Křivánek - Path Integral Formulation of Light Transport 28<br>
slide29. Vertex sampling Sample direction from an existing vertex 29 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide30. Measure conversion Sample direction from an existing vertex 30 Jaroslav Křivánek - Path Integral Formulation of Light Transport w.r.t. area w.r.t. proj. solid angle<br>
slide31. Summary Path integral pixel value all paths contribution function 31 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide32. Summary Algorithms
different path sampling techniques
different path PDF 32 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide33. Time for questions… Tutorial: Path Integral Methods for Light Transport Simulation
Jaroslav Křivánek – Path Integral Formulation of Light Transport<br>
Charles University in Prague
http://cgg.mff.cuni.cz/~jaroslav/<br>
slide2. Light transport Geometric optics emit travel absorb
scatter 2 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide3. Light transport emit travel absorb
scatter light transport path 3 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide4. Light transport Camera response
all paths hitting the sensor 4 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide5. Path integral formulation camera resp.
(j-th pixel value) all paths measurement contribution function 5 [Veach and Guibas 1995]
[Veach 1997] Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide6. Measurement contribution function 6 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide7. Geometry term 7 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide8. Path integral formulation camera resp.
(j-th pixel value) all paths measurement contribution function  ? 8 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide9. Path integral formulation all path lengths all possible
vertex positions 9 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide10. Path integral 10 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide11. Rendering : Evaluating the path integral<br>
slide12. Path integral Monte Carlo integration 12 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide13. Monte Carlo integration General approach to numerical evaluation of integrals f(x) 0 1 Integral: Monte Carlo estimate of I: Correct „on average“: 13 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide14. MC evaluation of the path integral Sample path from some distribution with PDF
Evaluate the probability density
Evaluate the integrand  ? ? Path integral 14 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide15. Algorithms = different path sampling techniques Path sampling 15 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide16. Algorithms = different path sampling techniques
Path tracing Path sampling 16 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide17. Algorithms = different path sampling techniques
Light tracing Path sampling 17 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide18. Algorithms = different path sampling techniques
Bidirectional path tracing Path sampling 18 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide19. Algorithms = different path sampling techniques
Same general form of estimator
No importance transport, no adjoint equations!!! Path sampling 19 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide20. Path sampling & Path PDF<br>
slide21. Local path sampling Sample one path vertex at a time
From an a priori distribution
lights, camera sensors
Sample direction from an existing vertex
Connect sub-paths
test visibility between vertices Jaroslav Křivánek - Path Integral Formulation of Light Transport 21<br>
slide22. Example – Path tracing A priori distrib.
Direction sampling
Connect vertices 1. 2. 1. 3. 2. 2. 22 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide23. Use of local path sampling Path tracing Light tracing Bidirectional path tracing 23 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide24. Probability density function (PDF) 24 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide25. Probability density function (PDF) 25 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide26. Probability density function (PDF) Path tracing example: 26 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide27. Probability density function (PDF) Path tracing example: 27 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide28. Vertex sampling Importance sampling principle
Sample from an a priori distrib.
Sample direction from an existing vertex
Connect sub-paths BRDF lobe
sampling emission
sampling high thruput
connections Jaroslav Křivánek - Path Integral Formulation of Light Transport 28<br>
slide29. Vertex sampling Sample direction from an existing vertex 29 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide30. Measure conversion Sample direction from an existing vertex 30 Jaroslav Křivánek - Path Integral Formulation of Light Transport w.r.t. area w.r.t. proj. solid angle<br>
slide31. Summary Path integral pixel value all paths contribution function 31 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide32. Summary Algorithms
different path sampling techniques
different path PDF 32 Jaroslav Křivánek - Path Integral Formulation of Light Transport<br>
slide33. Time for questions… Tutorial: Path Integral Methods for Light Transport Simulation
Jaroslav Křivánek – Path Integral Formulation of Light Transport<br>