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Path Integral Formulation of Light Transport
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  1. Path Integral Formulation of Light Transport Jaroslav Křivánek Charles University in Prague http://cgg.mff.cuni.cz/~jaroslav/

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

  3. Light transport emit travel reflect scatter light transport path Geometric optics Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

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

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

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

  7. Path integral formulation all paths camera resp. (j-th pixel value) measurementcontributionfunction ?  Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  8. Path integral formulation all pathlengths all possible vertex positions Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  9. Path integral all paths pixel value contributionfunction Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  10. Rendering : Evaluating the path integral

  11. Path integral all paths pixel value contributionfunction • Monte Carlo integration Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  12. Monte Carlo integration Integral: f(x) Monte Carlo estimateof I: p(x) 0 1 Correct „on average“: x5 x3 x1 x4 x2 x6 General approach to numerical evaluation of integrals Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  13. MC evaluation of the path integral MC estimator Path integral ? ?  • Sample path from some distribution with PDF • Evaluate the probability density • Evaluate the integrand Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

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

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

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

  17. Path sampling Algorithms = different path sampling techniques Same general form of estimator Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  18. Path sampling&Path PDF

  19. Local path sampling BRDF lobe 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 Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  20. Use of local path sampling Bidirectionalpath tracing Path tracing Light tracing Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

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

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

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

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

  25. MC evaluation of the path integral MC estimator Path integral • Sample path • Evaluate the probability density • Evaluate the integrand Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  26. Bidirectional path TRACING

  27. Bidirectional path tracing Bidirectionalpath sampling Path tracing Light tracing Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques

  28. All possible bidirectional techniques vertex on a light sub-path vertex on en eye sub-path  path tracing      light tracing Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques

  29. All possible bidirectional techniques vertex on a light sub-path vertex on en eye sub-path  path tracing  no single technique importance samples all the terms  VPLs    light tracing Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques

  30. 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

  31. 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

  32. NaiveBPT implementation Jaroslav Křivánek – Bidirectional Path Sampling Techniques

  33. MIS weight calculation Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  34. BPT Implementation in practice Jaroslav Křivánek – Bidirectional Path Sampling Techniques

  35. BPT Implementation in practice Jaroslav Křivánek – Bidirectional Path Sampling Techniques

  36. Results Images: EricVeach BPT, 25 samples per pixel PT, 56 samples per pixel Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques

  37. Nearly there…

  38. Summary • Algorithms • different path sampling techniques • different path PDF Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport

  39. 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 SimulationJaroslav Křivánek - Introduction

  40. Joint importance sampling Traditional

  41. Thank you! Time for questions… Course: Recent Advances in Light Transport Simulation • Jaroslav Křivánek - Path Integral Formulation of Light Transport

  42. Acknowledgements • Czech Science Foundation • grant no. P202-13-26189S • Images • Eric Tabellion • Marcos Fajardo Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques