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##### Path Integral Formulation of Light Transport

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**Path Integral Formulation of Light Transport**Jaroslav Křivánek Charles University in Prague http://cgg.mff.cuni.cz/~jaroslav/**Light transport**emit travel reflect scatter Geometric optics Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**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**Light transport**• Camera response • all paths hitting the sensor Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**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**Measurement contribution function**sensor sensitivity(“emitted importance”) emitted radiance path throughput**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**Path integral formulation**all pathlengths all possible vertex positions Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**Path integral**all paths pixel value contributionfunction Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**Path integral**all paths pixel value contributionfunction • Monte Carlo integration Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**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**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**Path sampling**Algorithms = different path sampling techniques Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**Path sampling**• Algorithms = different path sampling techniques • Path tracing Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**Path sampling**• Algorithms = different path sampling techniques • Light tracing Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**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**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**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**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**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**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**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**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**Bidirectional path tracing**Bidirectionalpath sampling Path tracing Light tracing Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques**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**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**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**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**NaiveBPT implementation**Jaroslav Křivánek – Bidirectional Path Sampling Techniques**MIS weight calculation**Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**BPT Implementation in practice**Jaroslav Křivánek – Bidirectional Path Sampling Techniques**BPT Implementation in practice**Jaroslav Křivánek – Bidirectional Path Sampling Techniques**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**Summary**• Algorithms • different path sampling techniques • different path PDF Course: Recent Advances in Light Transport SimulationJaroslav Křivánek- Path Integral Formulation of Light Transport**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**Joint importance sampling**Traditional**Thank you!**Time for questions… Course: Recent Advances in Light Transport Simulation • Jaroslav Křivánek - Path Integral Formulation of Light Transport**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