1. PATH INTEGRAL FORMULATION OF LIGHT TRANSPORT Jaroslav Křivánek Charles University in Prague http://cgg.mff.cuni.cz/~jaroslav/

2. Light transport Geometric optics emit travel absorb scatter 2 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

3. Light transport emit travel absorb scatter light transport path 3 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

4. Light transport Camera response all paths hitting the sensor 4 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

5. Path integral formulation camera resp. (j-th pixel value) all paths measurement contribution function 5 [Veach and Guibas 1995] [Veach 1997] Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

6. Measurement contribution function sensor sensitivity (“emitted importance”) path throughput emitted radiance 6

7. Geometry term 7 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

8. Path integral formulation camera resp. (j-th pixel value) all paths measurement contribution function ? 8 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

9. Path integral formulation all path lengths all possible vertex positions 9 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

10. Path integral pixel value all paths contribution function 10 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

11. RENDERING : EVALUATING THE PATH INTEGRAL

12. Path integral pixel value all paths contribution function Monte Carlo integration 12 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

13. Monte Carlo integration General approach to numerical evaluation of integrals x1x1 f(x)f(x) 01 p(x)p(x) x2x2 x3x3 x4x4 x5x5 x6x6 Integral: Monte Carlo estimate of I: Correct „on average“: 13 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

14. MC evaluation of the path integral Sample path from some distribution with PDF Evaluate the probability density Evaluate the integrand ? ? Path integralMC estimator 14 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

15. Algorithms = different path sampling techniques Path sampling 15 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

16. Algorithms = different path sampling techniques  Path tracing Path sampling 16 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

17. Algorithms = different path sampling techniques  Light tracing Path sampling 17 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

18. Algorithms = different path sampling techniques  Bidirectional path tracing Path sampling 18 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

19. Algorithms = different path sampling techniques Same general form of estimator No importance transport, no adjoint equations!!! Path sampling 19 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

20. PATH SAMPLING & PATH PDF

21. 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 Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

22. Example – Path tracing 1. A priori distrib. 2. Direction sampling 3. Connect vertices 1. 2. 1. 3. 2. 22 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

23. Use of local path sampling Path tracingLight tracing Bidirectional path tracing 23 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

24. Probability density function (PDF) path PDF joint PDF of path vertices 24 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

25. Probability density function (PDF) path PDF joint PDF of path vertices 25 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

26. Probability density function (PDF) path PDF joint PDF of path vertices product of (conditional) vertex PDFs Path tracing example: 26 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

27. Probability density function (PDF) path PDF joint PDF of path vertices product of (conditional) vertex PDFs Path tracing example: 27 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

28. Vertex sampling Importance sampling principle 1. Sample from an a priori distrib. 2. Sample direction from an existing vertex 3. Connect sub-paths BRDF lobe sampling emissionsampling high thruput connections Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

29. BRDF lobe sampling Vertex sampling Sample direction from an existing vertex 29 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

30. Measure conversion BRDF lobe sampling Sample direction from an existing vertex 30 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport w.r.t. area w.r.t. proj. solid angle

31. Summary Path integral pixel value all paths contribution function MC estimator path pdf sampled path 31

32. Summary Algorithms  different path sampling techniques  different path PDF 32 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

33. Time for questions… Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Path Integral Formulation of Light Transport