A null-scattering path integral formulation of light transport

Abstract

Unbiased rendering of general, heterogeneous participating media currently requires using null-collision approaches for estimating transmittance and generating free-flight distances. A long-standing limitation of these approaches, however, is that the corresponding path pdfs cannot be computed due to the black-box nature of the null-collision rejection sampling process. These techniques therefore cannot be combined with other sampling techniques via multiple importance sampling (MIS), which significantly limits their robustness and generality. Recently, Galtier et al. [2013] showed how to derive these algorithms directly from the radiative transfer equation (RTE). We build off this generalized RTE to derive a path integral formulation of null scattering, which reveals the sampling pdfs and allows us to devise new, express existing, and combine complementary unbiased techniques via MIS. We demonstrate the practicality of our theory by combining, for the first time, several path sampling techniques in spatially and spectrally varying media, generalizing and outperforming the prior state of the art.

References

1. James Arvo. 1993. Transfer Functions in Global Illumination. In ACM SIGGRAPH '93 Course Notes - Global Illumination.
2. James Richard Arvo. 1995. Analytic methods for simulated light transport. Ph.D. Dissertation. Yale University.
3. Hugo W. Bertini. 1963. Monte Carlo simulations on intranuclear cascades. Technical Report ORNL-3383. Oak Ridge National Laboratory, Oak Ridge, TN, USA.
4. Benedikt Bitterli and Wojciech Jarosz. 2017. Beyond Points and Beams: Higher-Dimensional Photon Samples for Volumetric Light Transport. ACM Transactions on Graphics (Proc. SIGGRAPH) 36, 4 (July 2017).
5. Benedikt Bitterli, Srinath Ravichandran, Thomas Müller, Magnus Wrenninge, Jan Novák, Steve Marschner, and Wojciech Jarosz. 2018. A radiative transfer framework for non-exponential media. ACM Transactions on Graphics (Proc. SIGGRAPH Asia) 37, 6 (Nov. 2018).
6. Brent Burley, David Adler, Matt Jen-Yuan Chiang, Hank Driskill, Ralf Habel, Patrick Kelly, Peter Kutz, Yining Karl Li, and Daniel Teece. 2018. The Design and Evolution of Disney's Hyperion Renderer. ACM Transactions on Graphics 37, 3, Article 33 (July 2018).
7. John C. Butcher and Harry Messel. 1958. Electron Number Distribution in Electron-Photon Showers. Phys. Rev. 112 (Dec. 1958). Issue 6.
8. John C. Butcher and Harry Messel. 1960. Electron number distribution in electron-photon showers in air and aluminium absorbers. Nuclear Physics 20 (1960).
9. Subrahmanyan Chandrasekhar. 1960. Radiative Transfer. Dover Publications.
10. Per Christensen, Julian Fong, Jonathan Shade, Wayne Wooten, Brenden Schubert, Andrew Kensler, Stephen Friedman, Charlie Kilpatrick, Cliff Ramshaw, Marc Bannister, Brenton Rayner, Jonathan Brouillat, and Max Liani. 2018. RenderMan: An Advanced Path-Tracing Architecture for Movie Rendering. ACM Transactions on Graphics 37, 3, Article 30 (Aug. 2018).
11. Per H. Christensen and Wojciech Jarosz. 2016. The Path to Path-Traced Movies. Foundations and Trends in Computer Graphics and Vision 10, 2 (Oct. 2016).
12. W. A. Coleman. 1968. Mathematical verification of a certain Monte Carlo sampling technique and applications of the technique to radiation transport problems. Nuclear Science and Engineering 32, 1 (April 1968).
13. Luca Fascione, Johannes Hanika, Marcos Fajardo, Per Christensen, Brent Burley, Brian Green, Rob Pieké, Christopher Kulla, Christophe Hery, Ryusuke Villemin, Daniel Heckenberg, and André Mazzone. 2017. SIGGRAPH 2017 Course Notes: Path tracing in production (Parts 1 and 2). In SIGGRAPH Courses.
14. Luca Fascione, Johannes Hanika, Mark Leone, Marc Droske, Jorge Schwarzhaupt, Tomáš Davidovič, Andrea Weidlich, and Johannes Meng. 2018. Manuka: A Batch-Shading Architecture for Spectral Path Tracing in Movie Production. ACM Transactions on Graphics (Proc. SIGGRAPH) 37, 3, Article 31 (Aug. 2018).
15. Mathieu Galtier, Stephane Blanco, Cyril Caliot, Christophe Coustet, Jérémi Dauchet, Mouna El Hafi, Vincent Eymet, Richard Fournier, Jacques Gautrais, Anais Khuong, Benjamin Piaud, and Guillaume Terrée. 2013. Integral formulation of null-collision Monte Carlo algorithms. Journal of Quantitative Spectroscopy and Radiative Transfer 125 (2013).
16. Manuel Gamito. 2018. Path Tracing in Production: Path Tracing the Framestorian Way. In ACM SIGGRAPH 2018 Courses. ACM, New York, NY, USA, Article 15.
17. Iliyan Georgiev, Thiago Ize, Mike Farnsworth, Ramón Montoya-Vozmediano, Alan King, Brecht Van Lommel, Angel Jimenez, Oscar Anson, Shinji Ogaki, Eric Johnston, Adrien Herubel, Declan Russell, Frédéric Servant, and Marcos Fajardo. 2018. Arnold: A Brute-Force Production Path Tracer. ACM Transactions on Graphics (Proc. SIGGRAPH) 37, 3, Article 32 (Aug. 2018).
18. Iliyan Georgiev, Jaroslav Kfivánek, Toshiya Hachisuka, Derek Nowrouzezahrai, and Wojciech Jarosz. 2013. Joint Importance Sampling of Low-Order Volumetric Scattering. ACM Transactions on Graphics (Proc. SIGGRAPH Asia) 32, 6 (Nov. 2013).
19. Johannes Hanika, Marc Droske, and Luca Fascione. 2015. Manifold Next Event Estimation. Computer Graphics Forum 34, 4 (July 2015).
20. Sebastian Herholz, Oskar Elek, Jiří Vorba, Hendrik Lensch, and Jaroslav Křivánek. 2016. Product Importance Sampling for Light Transport Path Guiding. Computer Graphics Forum 35, 4 (2016).
21. David Immel, Michael Cohen, and Donald Greenberg. 1986. A radiosity method for non-diffuse environments. Computer Graphics (Proc. SIGGRAPH) 20, 4 (1986).
22. Wenzel Jakob. 2013. Light Transport on Path-Space Manifolds. Ph.D. Dissertation. Cornell University.
23. Wenzel Jakob and Steve Marschner. 2012. Manifold Exploration: A Markov Chain Monte Carlo Technique for Rendering Scenes with Difficult Specular Transport. ACM Transactions on Graphics (Proc. SIGGRAPH) 31, 4, Article 58 (July 2012).
24. Adrian Jarabo, Carlos Aliaga, and Diego Gutierrez. 2018. A Radiative Transfer Framework for Spatially-Correlated Materials. ACM Transactions on Graphics (Proc. SIGGRAPH) 37, 4 (2018).
25. Wojciech Jarosz, Derek Nowrouzezahrai, Iman Sadeghi, and Henrik Wann Jensen. 2011a. A Comprehensive Theory of Volumetric Radiance Estimation Using Photon Points and Beams. ACM Transactions on Graphics 30, 1 (Feb. 2011).
26. Wojciech Jarosz, Derek Nowrouzezahrai, Robert Thomas, Peter-Pike Sloan, and Matthias Zwicker. 2011b. Progressive Photon Beams. ACM Transactions on Graphics (Proc. SIGGRAPH Asia) 30, 6 (Dec. 2011).
27. Wojciech Jarosz, Matthias Zwicker, and Henrik Wann Jensen. 2008. The Beam Radiance Estimate for Volumetric Photon Mapping. Computer Graphics Forum (Proc. Eurographics) 27, 2 (April 2008).
28. Henrik Wann Jensen and Per H. Christensen. 1998. Efficient Simulation of Light Transport in Scenes With Participating Media Using Photon Maps. In Annual Conference Series (Proc. SIGGRAPH).
29. Henrik Wann Jensen, Stephen R. Marschner, Marc Levoy, and Pat Hanrahan. 2001. A Practical Model for Subsurface Light Transport. Annual Conference Series (Proc. SIGGRAPH) 35 (2001).
30. James T. Kajiya. 1986. The Rendering Equation. Computer Graphics (Proc. SIGGRAPH) 20, 4 (Aug. 1986).
31. Markus Kettunen, Marco Manzi, Miika Aittala, Jaakko Lehtinen, Frédo Durand, and Matthias Zwicker. 2015. Gradient-domain Path Tracing. ACM Transactions on Graphics (Proc. SIGGRAPH) 34, 4, Article 123 (July 2015).
32. Christopher Kulla and Marcos Fajardo. 2012. Importance Sampling Techniques for Path Tracing in Participating Media. Computer Graphics Forum (Proc. Eurographics Symposium on Rendering) 31, 4 (2012).
33. Peter Kutz, Ralf Habel, Yining Karl Li, and Jan Novák. 2017. Spectral and Decomposition Tracking for Rendering Heterogeneous Volumes. ACM Transactions on Graphics (Proc. SIGGRAPH) 36, 4 (July 2017).
34. Jaroslav Křivánek, Iliyan Georgiev, Toshiya Hachisuka, Petr Vévoda, Martin Šik, Derek Nowrouzezahrai, and Wojciech Jarosz. 2014. Unifying Points, Beams, and Paths in Volumetric Light Transport Simulation. ACM Transactions on Graphics (Proc. SIGGRAPH) 33, 4 (July 2014).
35. Eric Lafortune and Yves Willems. 1993. Bi-directional path tracing. In Proc. Compugraphics.
36. Eric Lafortune and Yves Willems. 1996. Rendering participating media with bidirectional path tracing. Photorealistic Rendering Techniques (Proc. Eurographics Workshop on Rendering (1996).
37. Laurence B. Miller. 1967. Monte Carlo analysis of reactivity coefficients in fast reactors; general theory and applications. Technical Report ANL-7307. Argonne National Laboratory, Argonne, IL, USA.
38. Thomas Müller, Markus Gross, and Jan Novák. 2017. Practical Path Guiding for Efficient Light-Transport Simulation. Computer Graphics Forum (Proc. Eurographics Symposium on Rendering) 36, 4 (June 2017).
39. Ken Museth. 2013. VDB: High-resolution Sparse Volumes with Dynamic Topology. ACM Transactions on Graphics 32, 3, Article 27 (July 2013).
40. Jan Novák, Iliyan Georgiev, Johannes Hanika, and Wojciech Jarosz. 2018. Monte Carlo Methods for Volumetric Light Transport Simulation. Computer Graphics Forum 37, 2 (May 2018).
41. Jan Novák, Andrew Selle, and Wojciech Jarosz. 2014. Residual Ratio Tracking for Estimating Attenuation in Participating Media. ACM Transactions on Graphics (Proc. SIGGRAPH Asia) 33, 6 (Nov. 2014).
42. Mark Pauly, Thomas Kollig, and Alexander Keller. 2000. Metropolis Light Transport for Participating media. In Rendering Techniques (Proc. Eurographics Workshop on Rendering).
43. Ken Perlin. 1985. An Image Synthesizer. Computer Graphics (Proc. SIGGRAPH) 19, 3 (July 1985).
44. Ken Perlin and Eric Hoffert. 1989. Hypertexture. Computer Graphics (Proc. SIGGRAPH) 23, 3 (July 1989).
45. Matt Pharr, Wenzel Jakob, and Greg Humphreys. 2016. Physically Based Rendering: From Theory To Implementation (3rd ed.).
46. Matthias Raab, Daniel Seibert, and Alexander Keller. 2008. Unbiased global illumination with participating media. In Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer.
47. T. M. Sutton, F. B. Brown, F. G. Bischoff, D. B. MacMillan, C. L. Ellis, J. T. Ward, C. T. Ballinger, D. J. Kelly, and L. Schindler. 1999. The physical models and statistical procedures used in the RACER Monte Carlo code. Technical Report KAPL-4840. Knolls Atomic Power Laboratory, Niskayuna, NY, USA.
48. László Szirmay-Kalos, Iliyan Georgiev, Milán Magdics, Balázs Molnár, and Dávid Légrády. 2017. Unbiased Estimators to Render Procedurally Generated Inhomogeneous Participating Media. Computer Graphics Forum (Proc. Eurographics) 36, 2 (2017).
49. László Szirmay-Kalos, Milán Magdics, and Mateu Sbert. 2018. Multiple Scattering in Inhomogeneous Participating Media Using Rao-Blackwellization and Control Variates. Computer Graphics Forum (Proc. Eurographics) 37, 2 (May 2018).
50. László Szirmay-Kalos, Balázs Tóth, and Milán Magdics. 2011. Free path sampling in high resolution inhomogeneous participating media. Computer Graphics Forum 30, 1 (2011).
51. Eric Veach. 1997. Robust Monte Carlo methods for light transport simulation. Ph.D. Dissertation. Stanford, CA, USA.
52. Eric Veach and Leonidas Guibas. 1995. Optimally combining sampling techniques for Monte Carlo rendering. Annual Conference Series (Proc. SIGGRAPH) 29 (1995).
53. Jiři Vorba, Ondřej Karlík, Martin Šik, Tobias Ritschel, and Jaroslav Křivánek. 2014. On-line Learning of Parametric Mixture Models for Light Transport Simulation. ACM Transactions on Graphics (Proc. SIGGRAPH) 33, 4 (Aug. 2014).
54. Bruce Walter, Shuang Zhao, Nicolas Holzschuch, and Kavita Bala. 2009. Single Scattering in Refractive Media with Triangle Mesh Boundaries. ACM Transactions on Graphics 28, 3, Article 92 (July 2009).
55. Alexander Wilkie, Sehera Nawaz, Mark Droske, Andrea Weidlich, and Johannes Hanika. 2014. Hero wavelength spectral sampling. Computer Graphics Forum (Proc. Eurographics Symposium on Rendering) 33, 4 (June 2014).
56. E. R. Woodcock, T. Murphy, P. J. Hemmings, and T. C. Longworth. 1965. Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry. In Applications of Computing Methods to Reactor Problems. Argonne National Laboratory.
57. C. D. Zerby, R. B. Curtis, and Hugo W. Bertini. 1961. The relativistic doppler problem. Technical Report ORNL-61-7-20. Oak Ridge National Laboratory, TN, USA.

Index Terms

• Computing methodologies

  • Computer graphics

    • Rendering

      • Ray tracing