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

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

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

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

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

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

Geometry term 7 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Path integral formulation camera resp. (j-th pixel value) all paths measurement contribution function ? 8 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Path integral formulation all path lengths all possible vertex positions 9 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Path integral pixel value all paths contribution function 10 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

RENDERING : EVALUATING THE PATH INTEGRAL

Path integral pixel value all paths contribution function Monte Carlo integration 12 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

MC evaluation of the path integral Sample path from some distribution with PDF Evaluate the probability density Evaluate the integrand ? ? Path integralMC estimator 14 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

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

Algorithms = different path sampling techniques  Light tracing Path sampling 17 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Algorithms = different path sampling techniques  Bidirectional path tracing Path sampling 18 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Algorithms = different path sampling techniques Same general form of estimator No importance transport, no adjoint equations!!! Path sampling 19 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

PATH SAMPLING & PATH PDF

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

Example – Path tracing 1. A priori distrib. 2. Direction sampling 3. Connect vertices Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Use of local path sampling Path tracingLight tracing Bidirectional path tracing 23 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

Probability density function (PDF) path PDF joint PDF of path vertices 25 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

Probability density function (PDF) path PDF joint PDF of path vertices product of (conditional) vertex PDFs Path tracing example: 27 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Vertex sampling Importance sampling principle 1. Sample from an a priori distrib. 2. Sample direction from an existing vertex 3. Connect sub-paths BRDF lobe sampling emissionsampling high thruput connections Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

BRDF lobe sampling Vertex sampling Sample direction from an existing vertex 29 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Measure conversion BRDF lobe sampling Sample direction from an existing vertex 30 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport w.r.t. area w.r.t. proj. solid angle

Summary Path integral pixel value all paths contribution function MC estimator path pdf sampled path 31

Summary Algorithms  different path sampling techniques  different path PDF 32 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

Time for questions… Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Path Integral Formulation of Light Transport