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Path Integral Formulation of Light Transport

Explore the path integral formulation of light transport with Jaroslav Křivánek. Understand the importance of all paths hitting the sensor, measurement contribution functions, and Monte Carlo integration techniques. Dive into path sampling algorithms and probability density functions for improved light transport simulations.

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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

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