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PATH INTEGRAL FORMULATION OF LIGHT TRANSPORT Jaroslav Křivánek Charles University in Prague

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

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

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

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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

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

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Path integral formulation camera resp. (j-th pixel value) all paths measurement contribution function ? 7 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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Path integral formulation all path lengths all possible vertex positions 8 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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Path integral pixel value all paths contribution function 9 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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RENDERING : EVALUATING THE PATH INTEGRAL

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Path integral pixel value all paths contribution function Monte Carlo integration 11 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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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“: 12 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

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

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

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

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

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PATH SAMPLING & PATH PDF

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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 BRDF lobe sampling

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Use of local path sampling Path tracingLight tracing Bidirectional path tracing 20 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

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

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

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

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

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BIDIRECTIONAL PATH TRACING

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

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

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

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Multiple Importance Sampling (MIS) f(x)f(x) pa(x)pa(x) pb(x)pb(x) [Veach & Guibas, 95] Combined estimator: xaxa Jaroslav Křivánek – Light Transport Simulation with Vertex Connection and Merging

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Bidirectional path tracing Use all of the above sampling techniques Combine using Multiple Importance Sampling 31 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques

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Naive BPT implementation 32 Jaroslav Křivánek – Bidirectional Path Sampling Techniques

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

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

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

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

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NEARLY THERE…

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

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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) 39 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Introduction

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Joint importance sampling Traditional

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THANK YOU! Time for questions… Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

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