“Robust fitting of parallax-aware mixtures for path guiding” by Ruppert, Herholz and Lensch

  • ©Lukas Ruppert, Sebastian Herholz, and Hendrik P. A. Lensch

Conference:


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Title:

    Robust fitting of parallax-aware mixtures for path guiding

Session/Category Title:   Smart Sampling


Presenter(s)/Author(s):



Abstract:


    Effective local light transport guiding demands for high quality guiding information, i.e., a precise representation of the directional incident radiance distribution at every point inside the scene. We introduce a parallax-aware distribution model based on parametric mixtures. By parallax-aware warping of the distribution, the local approximation of the 5D radiance field remains valid and precise across large spatial regions, even for close-by contributors. Our robust optimization scheme fits parametric mixtures to radiance samples collected in previous rendering passes. Robustness is achieved by splitting and merging of components refining the mixture. These splitting and merging decisions minimize and bound the expected variance of the local radiance estimator. In addition, we extend the fitting scheme to a robust, iterative update method, which allows for incremental training of our model using smaller sample batches. This results in more frequent training updates and, at the same time, significantly reduces the required sample memory footprint. The parametric representation of our model allows for the application of advanced importance sampling methods such as radiance-based, cosine-aware, and even product importance sampling. Our method further smoothly integrates next-event estimation (NEE) into path guiding, avoiding importance sampling of contributions better covered by NEE. The proposed robust fitting and update scheme, in combination with the parallax-aware representation, results in faster learning and lower variance compared to state-of-the-art path guiding approaches.

References:


    1. Hirotugu Akaike. 1974. A new look at the statistical model identification. In Selected Papers of Hirotugu Akaike. Springer, 215–222.Google Scholar
    2. James Arvo et al. 1986. Backward ray tracing. In Developments in Ray Tracing, Computer Graphics, Proc. of ACM SIGGRAPH 86 Course Notes. 259–263.Google Scholar
    3. Parthasarathy Bagchi and Irwin Guttman. 1988. Theoretical considerations of the multivariate von Mises-Fisher distribution. Journal of Applied Statistics 15, 2 (1988), 149–169.Google ScholarCross Ref
    4. Arindam Banerjee, Inderjit S Dhillon, Joydeep Ghosh, and Suvrit Sra. 2005. Clustering on the unit hypersphere using von Mises-Fisher distributions. Journal of Machine Learning Research 6, Sep (2005), 1345–1382.Google Scholar
    5. Mark Bangert, Philipp Hennig, and Uwe Oelfke. 2010. Using an infinite von Mises-Fisher mixture model to cluster treatment beam directions in external radiation therapy. In Machine Learning and Applications (ICMLA), 2010 Ninth International Conference on. IEEE, 746–751.Google ScholarDigital Library
    6. Thomas Bashford-Rogers, Kurt Debattista, and Alan Chalmers. 2012. A significance cache for accelerating global illumination. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 1837–1851.Google Scholar
    7. Christopher M. Bishop. 2006. Mixture Models and EM. In Pattern Recognition and Machine Learning. Springer Science+Business Media, LLC, New York.Google Scholar
    8. Benedikt Bitterli. 2016. Rendering resources. https://benedikt-bitterli.me/resources/.Google Scholar
    9. Norbert Bus and Tamy Boubekeur. 2017. Double Hierarchies for Directional Importance Sampling in Monte Carlo Rendering. Journal of Computer Graphics Techniques (JCGT) 6, 3 (28 August 2017), 25–37.Google Scholar
    10. Olivier Cappé and Eric Moulines. 2009. On-line expectation-maximization algorithm for latent data models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71, 3 (2009), 593–613.Google ScholarCross Ref
    11. David Cline, Justin Talbot, and Parris Egbert. 2005. Energy redistribution path tracing. In ACM Transactions on Graphics (TOG), Vol. 24. ACM, 1186–1195.Google ScholarDigital Library
    12. Ken Dahm and Alexander Keller. 2017. Learning light transport the reinforced way. In ACM SIGGRAPH 2017 Talks. ACM, 73.Google ScholarDigital Library
    13. Arthur P Dempster, Nan M Laird, and Donald B Rubin. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological) 39, 1 (1977), 1–22.Google ScholarCross Ref
    14. Philip Dutré, Eric P. Lafortune, and Yves D. Willems. 1993. Monte Carlo light tracing with direct computation of pixel intensities. In 3rd International Conference on Computational Graphics and Visualisation Techniques. Alvor, Portugal, 128–137.Google Scholar
    15. Luca Fascione, Johannes Hanika, Marcos Fajardo, Per Christensen, Brent Burley, and Brian Green. 2017. Path Tracing in Production – Part 1: Production Renderers. In ACM SIGGRAPH 2017 Courses (Los Angeles, California). Article 13, 39 pages.Google ScholarDigital Library
    16. Luca Fascione, Johannes Hanika, Daniel Heckenberg, Christopher Kulla, Marc Droske, and Jorge Schwarzhaupt. 2019. Path tracing in production: part 1: modern path tracing. In ACM SIGGRAPH 2019 Courses. ACM, 19.Google ScholarDigital Library
    17. Luca Fascione, Johannes Hanika, Rob Pieké, Ryusuke Villemin, Christophe Hery, Manuel Gamito, Luke Emrose, and André Mazzone. 2018. Path tracing in production. In ACM SIGGRAPH 2018 Courses. ACM, 15.Google ScholarDigital Library
    18. Nicholas I Fisher, Toby Lewis, and Brian JJ Embleton. 1987. Statistical analysis of spherical data. Cambridge university press.Google Scholar
    19. Iliyan Georgiev, Jaroslav Krivánek, Tomas Davidovic, and Philipp Slusallek. 2012. Light transport simulation with vertex connection and merging. ACM Trans. Graph. 31, 6 (2012), 192–1.Google ScholarDigital Library
    20. Pascal Grittmann, Arsène Pérard-Gayot, Philipp Slusallek, and Jaroslav Křivánek. 2018. Efficient Caustic Rendering with Lightweight Photon Mapping. In Computer Graphics Forum, Vol. 37. Wiley Online Library, 133–142.Google Scholar
    21. Jerry Guo, Pablo Bauszat, Jacco Bikker, and Elmar Eisemann. 2018. Primary Sample Space Path Guiding. In Eurographics Symposium on Rendering – EI & I, Wenzel Jakob and Toshiya Hachisuka (Eds.). Eurographics, The Eurographics Association, 73–82.Google Scholar
    22. Toshiya Hachisuka and Henrik Wann Jensen. 2009. Stochastic progressive photon mapping. In ACM Transactions on Graphics (TOG), Vol. 28. ACM, 141.Google ScholarDigital Library
    23. Toshiya Hachisuka, Jacopo Pantaleoni, and Henrik Wann Jensen. 2012. A Path Space Extension for Robust Light Transport Simulation. ACM Trans. Graph. 31, 6, Article 191 (Nov. 2012), 10 pages.Google ScholarDigital Library
    24. JH Hannay and JF Nye. 2004. Fibonacci numerical integration on a sphere. Journal of Physics A: Mathematical and General 37, 48 (2004), 11591.Google ScholarCross Ref
    25. Sebastian Herholz, Oskar Elek, Jiří Vorba, Hendrik Lensch, and Jaroslav Křivánek. 2016. Product importance sampling for light transport path guiding. In Computer Graphics Forum, Vol. 35. Wiley Online Library, 67–77.Google Scholar
    26. Sebastian Herholz, Yangyang Zhao, Oskar Elek, Derek Nowrouzezahrai, Hendrik P. A. Lensch, and Jaroslav Křivánek. 2019. Volume Path Guiding Based on Zero-Variance Random Walk Theory. ACM Trans. Graph. 38, 3, Article 25 (June 2019), 19 pages.Google ScholarDigital Library
    27. Tim Hesterberg. 1995. Weighted Average Importance Sampling and Defensive Mixture Distributions. Technometrics 37, 2 (1995), 185–194.Google ScholarCross Ref
    28. Heinrich Hey and Werner Purgathofer. 2002. Importance sampling with hemispherical particle footprints. In Proceedings of the 18th spring conference on Computer graphics. ACM, 107–114.Google ScholarDigital Library
    29. David S. Immel, Michael F. Cohen, and Donald P. Greenberg. 1986. A Radiosity Method for Non-Diffuse Environments. Computer Graphics (Proceedings of SIGGRAPH) 20, 4 (Aug. 1986), 133–142.Google ScholarDigital Library
    30. Wenzel Jakob. 2010. Mitsuba renderer. http://www.mitsuba-renderer.org.Google Scholar
    31. Wenzel Jakob. 2012. Numerically stable sampling of the von Mises-Fisher distribution on S^2 (and other tricks). Interactive Geometry Lab, ETH Zürich, Tech. Rep (2012).Google Scholar
    32. Wenzel Jakob, Christian Regg, and Wojciech Jarosz. 2011. Progressive Expectation-Maximization for Hierarchical Volumetric Photon Mapping. Computer Graphics Forum (Proceedings of Eurographics Symposium on Rendering) 3, 4 (July 2011), 1287–1297.Google Scholar
    33. Henrik Wann Jensen. 1995. Importance driven path tracing using the photon map. In Rendering Techniques’ 95. Springer, 326–335.Google Scholar
    34. Henrik Wann Jensen. 1996. Global Illumination Using Photon Maps. In Proceedings of the Eurographics Workshop on Rendering Techniques ’96 (Porto, Portugal). Springer-Verlag, Berlin, Heidelberg, 21–30.Google ScholarDigital Library
    35. Henrik Wann Jensen. 2001. Realistic image synthesis using photon mapping. AK Peters/CRC Press.Google Scholar
    36. G Jona-Lasinio, M Piccioni, and A Ramponi. 1999. Selection of importance weights for monte carlo estimation of normalizing constants. Communications in Statistics – Simulation and Computation 28, 2 (1999), 441–462.Google ScholarCross Ref
    37. James T Kajiya. 1986. The rendering equation. In ACM Siggraph Computer Graphics, Vol. 20. ACM, 143–150.Google ScholarDigital Library
    38. Anton S. Kaplanyan, Johannes Hanika, and Carsten Dachsbacher. 2014. The Natural-Constraint Representation of the Path Space for Efficient Light Transport Simulation. ACM Trans. Graph. 33, 4, Article 102 (July 2014), 13 pages.Google ScholarDigital Library
    39. Jaroslav Křivánek, Kadi Bouatouch, Sumanta Pattanaik, and Jiří Žára. 2008. Making Radiance and Irradiance Caching Practical: Adaptive Caching and Neighbor Clamping. In ACM SIGGRAPH 2008 Classes (Los Angeles, California) (SIGGRAPH ’08). Association for Computing Machinery, New York, NY, USA, Article 77, 12 pages.Google Scholar
    40. Jaroslav Křivánek, Pascal Gautron, Kadi Bouatouch, and Sumanta Pattanaik. 2005a. Improved Radiance Gradient Computation. In Proceedings of the 21st Spring Conference on Computer Graphics (Budmerice, Slovakia) (SCCG ’05). Association for Computing Machinery, New York, NY, USA, 155–159.Google ScholarDigital Library
    41. Jaroslav Křivánek, Pascal Gautron, Sumanta Pattanaik, and Kadi Bouatouch. 2005b. Radiance caching for efficient global illumination computation. IEEE Transactions on Visualization and Computer Graphics 11, 5 (2005), 550–561.Google ScholarDigital Library
    42. Eric P. Lafortune and Yves D. Willems. 1993. Bi-Directional Path Tracing. In Proceedings of the International Conference on Computational Graphics and Visualization Techniques (Compugraphics), Vol. 93. Alvor, Portugal, 145–153.Google Scholar
    43. Eric P Lafortune and Yves D Willems. 1995. A 5D tree to reduce the variance of Monte Carlo ray tracing. In Rendering Techniques’ 95. Springer, 11–20.Google Scholar
    44. Julio Marco, Adrian Jarabo, Wojciech Jarosz, and Diego Gutierrez. 2018. Second-Order Occlusion-Aware Volumetric Radiance Caching. ACM Trans. Graph. 37, 2, Article 20 (July 2018), 14 pages.Google ScholarDigital Library
    45. Geoffrey McLachlan and Thriyambakam Krishnan. 2007. The EM algorithm and extensions. Vol. 382. John Wiley & Sons.Google Scholar
    46. Geoffrey J McLachlan, Sharon X Lee, and Suren I Rathnayake. 2019. Finite mixture models. Annual review of statistics and its application 6 (2019), 355–378.Google Scholar
    47. Thomas Müller, Markus Gross, and Jan Novák. 2017. Practical Path Guiding for Efficient Light-Transport Simulation. In Computer Graphics Forum, Vol. 36. Wiley Online Library, 91–100.Google Scholar
    48. Thomas Müller, Brian Mcwilliams, Fabrice Rousselle, Markus Gross, and Jan Novák. 2019. Neural Importance Sampling. ACM Trans. Graph. 38, 5, Article 145 (Oct. 2019), 19 pages.Google ScholarDigital Library
    49. Richard F Murray and Yaniv Morgenstern. 2010. Cue combination on the circle and the sphere. Journal of vision 10, 11 (2010), 15–15.Google ScholarCross Ref
    50. Radford M Neal and Geoffrey E Hinton. 1998. A view of the EM algorithm that justifies incremental, sparse, and other variants. In Learning in graphical models. Springer, 355–368.Google Scholar
    51. Jerzy Neyman and Egon Sharpe Pearson. 1933. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 231, 694-706 (1933), 289–337.Google Scholar
    52. Vincent Pegoraro, Ingo Wald, and Steven G. Parker. 2008. Sequential Monte Carlo integration in low-anisotropy participating media. In Proceedings of the Eurographics Conference on Rendering. 1097–1104.Google Scholar
    53. Florian Reibold, Johannes Hanika, Alisa Jung, and Carsten Dachsbacher. 2018. Selective Guided Sampling with Complete Light Transport Paths. ACM Trans. Graph. 37, 6, Article 223 (Dec. 2018), 14 pages.Google ScholarDigital Library
    54. Gideon Schwarz. 1978. Estimating the dimension of a model. The annals of statistics 6, 2 (1978), 461–464.Google Scholar
    55. Jorge Schwarzhaupt, Henrik Wann Jensen, and Wojciech Jarosz. 2012. Practical Hessian-Based Error Control for Irradiance Caching. ACM Trans. Graph. 31, 6, Article 193 (Nov. 2012), 10 pages.Google ScholarDigital Library
    56. Martin Šik and Jaroslav Křivánek. 2019. Implementing One-Click Caustics in Corona Renderer. In Eurographics Symposium on Rendering – DL-only and Industry Track, Tamy Boubekeur and Pradeep Sen (Eds.). The Eurographics Association, 61–67.Google Scholar
    57. Naonori Ueda, Ryohei Nakano, Zoubin Ghahramani, and Geoffrey E Hinton. 2000a. SMEM algorithm for mixture models. Neural computation 12, 9 (2000), 2109–2128.Google Scholar
    58. Naonori Ueda, Ryohei Nakano, Zoubin Ghahramani, and Geoffrey E Hinton. 2000b. Split and merge EM algorithm for improving Gaussian mixture density estimates. Journal of VLSI signal processing systems for signal, image and video technology 26, 1-2 (2000), 133–140.Google ScholarCross Ref
    59. Eric Veach and Leonidas Guibas. 1995a. Bidirectional estimators for light transport. In Photorealistic Rendering Techniques. Springer, 145–167.Google Scholar
    60. Eric Veach and Leonidas J. Guibas. 1995b. Optimally Combining Sampling Techniques for Monte Carlo Rendering. In Annual Conference Series (Proceedings of SIGGRAPH), Vol. 29. ACM Press, 419–428.Google Scholar
    61. Jakob J. Verbeek, Jan R. Nunnink, and Nikos Vlassis. 2006. Accelerated EM-Based Clustering of Large Data Sets. Data Min. Knowl. Discov. 13, 3 (Nov. 2006), 291–307.Google ScholarDigital Library
    62. Jiří Vorba, Johannes Hanika, Sebastian Herholz, Thomas Müller, Jaroslav Křivánek, and Alexander Keller. 2019. Path guiding in production. In ACM SIGGRAPH 2019 Courses. ACM, 18.Google ScholarDigital Library
    63. Jiří 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 (TOG) 33, 4 (2014), 101.Google ScholarDigital Library
    64. Jiří Vorba and Jaroslav Křivánek. 2016. Adjoint-driven Russian roulette and splitting in light transport simulation. ACM Transactions on Graphics (TOG) 35, 4 (2016), 42.Google ScholarDigital Library
    65. Hai xian Wang, Bin Luo, Quan bing Zhang, and Sui Wei. 2004. Estimation for the number of components in a mixture model using stepwise split-and-merge EM algorithm. Pattern Recognition Letters 25, 16 (2004), 1799–1809.Google ScholarDigital Library
    66. G.J. Ward and P.S. Heckbert. 1992. Irradiance gradients. Technical Report.Google Scholar
    67. Gregory J. Ward, Francis M. Rubinstein, and Robert D. Clear. 1988. A Ray Tracing Solution for Diffuse Interreflection. In Proceedings of the 15th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’88). Association for Computing Machinery, New York, NY, USA, 85–92.Google Scholar
    68. Quan Zheng and Matthias Zwicker. 2019. Learning to Importance Sample in Primary Sample Space. Computer Graphics Forum (2019).Google Scholar


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