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PATH INTEGRAL FORMULATION OF LIGHT TRANSPORT Jaroslav Kivnek Charles University in Prague http://cgg.mff.cuni.cz/~jaroslav/ Slide 2 Light transport Geometric optics emit travel reflect 2 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport scatter Slide 3 Light transport Geometric optics emit travel reflect 3 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport scatter light transport path Slide 4 Camera response all paths hitting the sensor Light transport 4 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 5 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 Kivnek - Path Integral Formulation of Light Transport Slide 6 Measurement contribution function sensor sensitivity (emitted importance) path throughput emitted radiance 6 Slide 7 Path integral formulation camera resp. (j-th pixel value) all paths measurement contribution function ? 7 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 8 Path integral formulation all path lengths all possible vertex positions 8 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 9 Path integral pixel value all paths contribution function 9 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 10 RENDERING : EVALUATING THE PATH INTEGRAL Slide 11 Path integral pixel value all paths contribution function Monte Carlo integration 11 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 12 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 Kivnek - Path Integral Formulation of Light Transport Slide 13 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 Kivnek - Path Integral Formulation of Light Transport Slide 14 Algorithms = different path sampling techniques Path sampling 14 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 15 Algorithms = different path sampling techniques Path tracing Path sampling 15 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 16 Algorithms = different path sampling techniques Light tracing Path sampling 16 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 17 Algorithms = different path sampling techniques Same general form of estimator Path sampling 17 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 18 PATH SAMPLING & PATH PDF Slide 19 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 Kivnek - Path Integral Formulation of Light Transport BRDF lobe sampling Slide 20 Use of local path sampling Path tracingLight tracing Bidirectional path tracing 20 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 21 Probability density function (PDF) path PDF joint PDF of path vertices 21 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 22 Probability density function (PDF) path PDF joint PDF of path vertices 22 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 23 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 Kivnek - Path Integral Formulation of Light Transport Slide 24 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 Kivnek - Path Integral Formulation of Light Transport Slide 25 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 Kivnek - Path Integral Formulation of Light Transport Slide 26 BIDIRECTIONAL PATH TRACING Slide 27 Bidirectional path tracing Path tracingLight tracing Bidirectional path sampling 27 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek Bidirectional Path Sampling Techniques Slide 28 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 Kivnek Bidirectional Path Sampling Techniques Slide 29 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 Kivnek Bidirectional Path Sampling Techniques Slide 30 Multiple Importance Sampling (MIS) f(x)f(x) pa(x)pa(x) pb(x)pb(x) [Veach & Guibas, 95] Combined estimator: xaxa Jaroslav Kivnek Light Transport Simulation with Vertex Connection and Merging Slide 31 Bidirectional path tracing Use all of the above sampling techniques Combine using Multiple Importance Sampling 31 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek Bidirectional Path Sampling Techniques Slide 32 Naive BPT implementation 32 Jaroslav Kivnek Bidirectional Path Sampling Techniques Slide 33 MIS weight calculation Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport 33 Slide 34 BPT Implementation in practice 34 Jaroslav Kivnek Bidirectional Path Sampling Techniques Slide 35 BPT Implementation in practice 35 Jaroslav Kivnek Bidirectional Path Sampling Techniques Slide 36 Results BPT, 25 samples per pixelPT, 56 samples per pixel Images: Eric Veach 36 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek Bidirectional Path Sampling Techniques Slide 37 NEARLY THERE Slide 38 Summary Algorithms different path sampling techniques different path PDF 38 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 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) 39 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Introduction Slide 40 Joint importance sampling Traditional Slide 41 THANK YOU! Time for questions Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek - Path Integral Formulation of Light Transport Slide 42 Acknowledgements Czech Science Foundation grant no. P202-13-26189S Images Eric Tabellion Marcos Fajardo 42 Course: Recent Advances in Light Transport Simulation Jaroslav Kivnek Bidirectional Path Sampling Techniques