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

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

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

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

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

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

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

Path integral pixel value all paths contribution function 9 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 11 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“: 12 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 13 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek - Path Integral Formulation of Light Transport

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

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

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

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

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 BRDF lobe sampling

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

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

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

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

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

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

BIDIRECTIONAL PATH TRACING

Bidirectional path tracing Path tracingLight tracing Bidirectional path sampling 27 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 28 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 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

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

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

Naive BPT implementation 32 Jaroslav Křivánek – Bidirectional Path Sampling Techniques

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

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

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

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

NEARLY THERE…

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

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

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. P S Images  Eric Tabellion  Marcos Fajardo 42 Course: Recent Advances in Light Transport Simulation Jaroslav Křivánek – Bidirectional Path Sampling Techniques