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Path Integral Formulation of Light Transport

Path Integral Formulation of Light Transport . Jaroslav Křivánek Charles University in Prague http://cgg.mff.cuni.cz/~jaroslav/. Light transport. emit. travel. absorb scatter. Geometric optics. Light transport. emit. travel. absorb scatter. light transport path. Light transport.

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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 absorb scatter Geometric optics Jaroslav Křivánek - Path Integral Formulation of Light Transport

  3. Light transport emit travel absorb scatter light transport path Jaroslav Křivánek - Path Integral Formulation of Light Transport

  4. Light transport • Camera response • all paths hitting the sensor Jaroslav 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] Jaroslav Křivánek - Path Integral Formulation of Light Transport

  6. Measurement contribution function sensor sensitivity(“emitted importance”) emitted radiance path throughput Jaroslav Křivánek - Path Integral Formulation of Light Transport

  7. Geometry term Jaroslav Křivánek - Path Integral Formulation of Light Transport

  8. Path integral formulation all paths camera resp. (j-th pixel value) measurementcontributionfunction ?  Jaroslav Křivánek - Path Integral Formulation of Light Transport

  9. Path integral formulation all pathlengths all possible vertex positions Jaroslav Křivánek - Path Integral Formulation of Light Transport

  10. Path integral all paths pixel value contributionfunction Jaroslav Křivánek - Path Integral Formulation of Light Transport

  11. Rendering : Evaluating the path integral

  12. Path integral all paths pixel value contributionfunction • Monte Carlo integration Jaroslav Křivánek - Path Integral Formulation of Light Transport

  13. 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 Jaroslav Křivánek - Path Integral Formulation of Light Transport

  14. MC evaluation of the path integral MC estimator Path integral ? ?  • Sample path from some distribution with PDF • Evaluate the probability density • Evaluate the integrand Jaroslav Křivánek - Path Integral Formulation of Light Transport

  15. Path sampling Algorithms = different path sampling techniques Jaroslav Křivánek - Path Integral Formulation of Light Transport

  16. Path sampling • Algorithms = different path sampling techniques • Path tracing Jaroslav Křivánek - Path Integral Formulation of Light Transport

  17. Path sampling • Algorithms = different path sampling techniques • Light tracing Jaroslav Křivánek - Path Integral Formulation of Light Transport

  18. Path sampling Algorithms = different path sampling techniques Same general form of estimator No importance transport, no adjoint equations!!! Jaroslav Křivánek - Path Integral Formulation of Light Transport

  19. Path sampling&Path PDF

  20. 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. Example – Path tracing 2. 1. 1. 3. 2. 2. A priori distrib. Direction sampling Connect vertices Jaroslav Křivánek - Path Integral Formulation of Light Transport

  22. Use of local path sampling Bidirectionalpath tracing Path tracing Light tracing Jaroslav Křivánek - Path Integral Formulation of Light Transport

  23. Probability density function (PDF) path PDF joint PDF of path vertices Jaroslav Křivánek - Path Integral Formulation of Light Transport

  24. Probability density function (PDF) path PDF joint PDF of path vertices Jaroslav Křivánek - Path Integral Formulation of Light Transport

  25. Probability density function (PDF) path PDF product of (conditional) vertex PDFs joint PDF of path vertices Path tracing example: Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

  27. Vertex sampling emission sampling BRDF lobe sampling high thruput connections • Importance sampling principle • Sample from an a priori distrib. • Sample direction from an existing vertex • Connect sub-paths Jaroslav Křivánek - Path Integral Formulation of Light Transport

  28. Vertex sampling BRDF lobe sampling • Sample direction from an existing vertex Jaroslav Křivánek - Path Integral Formulation of Light Transport

  29. Measure conversion BRDF lobe sampling w.r.t. proj.solid angle w.r.t. area • Sample direction from an existing vertex Jaroslav Křivánek - Path Integral Formulation of Light Transport

  30. Summary Path integral MC estimator all paths pixel value contributionfunction sampled path path pdf Jaroslav Křivánek - Path Integral Formulation of Light Transport

  31. Summary • Algorithms • different path sampling techniques • different path PDF Jaroslav Křivánek - Path Integral Formulation of Light Transport

  32. Time for questions… • Tutorial: Path Integral Methods for Light Transport Simulation Jaroslav Křivánek – Path Integral Formulation of Light Transport

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