More Web Proxy on the site http://driver.im/

research-article

Primary-space Adaptive Control Variates Using Piecewise-polynomial Approximations

Authors:

Adolfo MuñozAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 40, Issue 3

Article No.: 25, Pages 1 - 15

https://doi.org/10.1145/3450627

Published: 15 July 2021 Publication History

Abstract

We present an unbiased numerical integration algorithm that handles both low-frequency regions and high-frequency details of multidimensional integrals. It combines quadrature and Monte Carlo integration by using a quadrature-based approximation as a control variate of the signal. We adaptively build the control variate constructed as a piecewise polynomial, which can be analytically integrated, and accurately reconstructs the low-frequency regions of the integrand. We then recover the high-frequency details missed by the control variate by using Monte Carlo integration of the residual. Our work leverages importance sampling techniques by working in primary space, allowing the combination of multiple mappings; this enables multiple importance sampling in quadrature-based integration. Our algorithm is generic and can be applied to any complex multidimensional integral. We demonstrate its effectiveness with four applications with low dimensionality: transmittance estimation in heterogeneous participating media, low-order scattering in homogeneous media, direct illumination computation, and rendering of distribution effects. Finally, we show how our technique is extensible to integrands of higher dimensionality by computing the control variate on Monte Carlo estimates of the high-dimensional signal, and accounting for such additional dimensionality on the residual as well. In all cases, we show accurate results and faster convergence compared to previous approaches.

References

[1]

Steve Bako, Thijs Vogels, Brian McWilliams, Mark Meyer, Jan Novák, Alex Harvill, Pradeep Sen, Tony Derose, and Fabrice Rousselle. 2017. Kernel-predicting convolutional networks for denoising Monte Carlo renderings. ACM Trans. Graph. 36, 4 (2017), 97.

Digital Library

[2]

Laurent Belcour, Cyril Soler, Kartic Subr, Nicolas Holzschuch, and Fredo Durand. 2013. 5D covariance tracing for efficient defocus and motion blur. ACM Trans. Graph. 32, 3 (2013), 31.

Digital Library

[3]

Laurent Belcour, Guofu Xie, Christophe Hery, Mark Meyer, Wojciech Jarosz, and Derek Nowrouzezahrai. 2018. Integrating clipped spherical harmonics expansions. ACM Trans. Graph. 37, 2 (2018).

Digital Library

[4]

Jarle Berntsen, Terje O. Espelid, and Alan Genz. 1991. An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw. 17, 4 (1991), 437–451.

Digital Library

[5]

Benedikt Bitterli, Fabrice Rousselle, Bochang Moon, José A. Iglesias-Guitián, David Adler, Kenny Mitchell, Wojciech Jarosz, and Jan Novák. 2016. Nonlinearly weighted first-order regression for denoising Monte Carlo renderings. In Computer Graphics Forum, Vol. 35. Wiley Online Library, 107–117.

Digital Library

[6]

Jonathan Brouillat, Christian Bouville, Brad Loos, Charles Hansen, and Kadi Bouatouch. 2009. A bayesian Monte Carlo approach to global illumination. In Computer Graphics Forum, Vol. 28. Wiley Online Library, 2315–2329.

[7]

R. L. Burden and J. Douglas Faires. 2005. Numerical Analysis, 8th ed. Thomson Brooks/Cole.

[8]

Petrik Clarberg and Tomas Akenine-Möller. 2008. Exploiting visibility correlation in direct illumination. In Computer Graphics Forum, Vol. 27. Wiley Online Library, 1125–1136.

[9]

Robert L. Cook, Thomas Porter, and Loren Carpenter. 1984. Distributed ray tracing. In Proceedings of SIGGRAPH. 137–145.

Digital Library

[10]

Frédo Durand, Nicolas Holzschuch, Cyril Soler, Eric Chan, and François X. Sillion. 2005. A frequency analysis of light transport. ACM Trans. Graph. 24, 3 (2005), 1115–1126.

Digital Library

[11]

Shaohua Fan, Stephen Chenney, Bo Hu, Kam-Wah Tsui, and Yu-chi Lai. 2006. Optimizing control variate estimators for rendering. In Computer Graphics Forum, Vol. 25. Wiley Online Library, 351–357.

[12]

Alan Genz and Aftab Ahmad Malik. 1980. Remarks on Algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region. J. Comput. Appl. Math. 6, 4 (1980), 295–302.

[13]

Michaël Gharbi, Tzu-Mao Li, Miika Aittala, Jaakko Lehtinen, and Frédo Durand. 2019. Sample-based Monte Carlo denoising using a kernel-splatting network. ACM Trans. Graph. 38, 4 (2019), 1–12.

Digital Library

[14]

Ibón Guillén, Carlos Ureña, Alan King, Marcos Fajardo, Iliyan Georgiev, Jorge López-Moreno, and Adrian Jarabo. 2017. Area-preserving parameterizations for spherical ellipses. In Computer Graphics Forum, Vol. 36. Wiley Online Library, 179–187.

[15]

Toshiya Hachisuka, Wojciech Jarosz, Richard Peter Weistroffer, Kevin Dale, Greg Humphreys, Matthias Zwicker, and Henrik Wann Jensen. 2008. Multidimensional adaptive sampling and reconstruction for ray tracing. In ACM Transactions on Graphics, Vol. 27. ACM, 33.

[16]

Stefan Heinrich. 2001. Multilevel Monte Carlo methods. In Proceedings of the International Conference on Large-Scale Scientific Computing. Springer, Berlin, 58–67.

[17]

Binh-Son Hua, Adrien Gruson, Victor Petitjean, Matthias Zwicker, Derek Nowrouzezahrai, Elmar Eisemann, and Toshiya Hachisuka. 2019. A survey on gradient-domain rendering. In Computer Graphics Forum, Vol. 38. Wiley Online Library, 455–472.

[18]

Wenzel Jakob. 2010. Mitsuba Renderer. Retrieved from http://www.mitsuba-renderer.org.

[19]

Adrian Jarabo, Belen Masia, Adrien Bousseau, Fabio Pellacini, and Diego Gutierrez. 2014. How do people edit light fields? ACM Trans. Graph. 33, 4 (2014).

Digital Library

[20]

Wojciech Jarosz, Craig Donner, Matthias Zwicker, and Henrik Wann Jensen. 2008. Radiance caching for participating media. ACM Trans. Graph. 27, 1 (2008), 1–11.

Digital Library

[21]

Henrik Wann Jensen and Per H. Christensen. 1998. Efficient simulation of light transport in scenes with participating media using photon maps. In Proceedings of SIGGRAPH. 311–320.

[22]

Jared M. Johnson, Dylan Lacewell, Andrew Selle, and Wojciech Jarosz. 2011. Gaussian quadrature for photon beams in tangled. In Proceedings of SIGGRAPH 2011 Talks. ACM, 54.

Digital Library

[23]

James T. Kajiya. 1986. The rendering equation. In Computer Graphics (Proceedings of SIGGRAPH), Vol. 20. ACM, 143–150.

Digital Library

[24]

Csaba Kelemen, László Szirmay-Kalos, György Antal, and Ferenc Csonka. 2002. A simple and robust mutation strategy for the metropolis light transport algorithm. In Computer Graphics Forum, Vol. 21. Wiley Online Library, 531–540.

[25]

Alexander Keller. 2001. Hierarchical Monte Carlo image synthesis. Math. Comput. Simul. 55, 1–3 (2001), 79–92.

Digital Library

[26]

Markus Kettunen, Marco Manzi, Miika Aittala, Jaakko Lehtinen, Frédo Durand, and Matthias Zwicker. 2015. Gradient-domain path tracing. ACM Trans. Graph. 34, 4 (2015).

Digital Library

[27]

Ivo Kondapaneni, Petr Vévoda, Pascal Grittmann, Tomaš Skřivan, Philipp Slusallek, and Jaroslav Křivánek. 2019. Optimal multiple importance sampling. ACM Trans. Graph. 38, 4 (2019).

Digital Library

[28]

Christopher Kulla and Marcos Fajardo. 2012. Importance sampling techniques for path tracing in participating media. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 1519–1528.

[29]

Peter Kutz, Ralf Habel, Yining Karl Li, and Jan Novák. 2017. Spectral and decomposition tracking for rendering heterogeneous volumes. ACM Trans. Graph. 36, 4 (2017), 1–16.

Digital Library

[30]

Eric P. Lafortune and Yves D. Willems. 1994. The ambient term as a variance reducing technique for Monte Carlo ray tracing. In Photorealistic Rendering Techniques. Springer, Berlin, 168–176.

[31]

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, Berlin, 11–20.

[32]

Julio Marco, Adrian Jarabo, Wojciech Jarosz, and Diego Gutierrez. 2018. Second-order occlusion-aware volumetric radiance caching. ACM Trans. Graph. 37, 2 (2018), 1–14.

Digital Library

[33]

Ricardo Marques, Christian Bouville, Mickaël Ribardière, Luís Paulo Santos, and Kadi Bouatouch. 2013. A spherical gaussian framework for Bayesian Monte Carlo rendering of glossy surfaces. IEEE Trans. Visual. Comput. Graph. 19, 10 (2013), 1619–1632.

[34]

Soham Uday Mehta, Ravi Ramamoorthi, Mark Meyer, and Christophe Hery. 2012. Analytic tangent irradiance environment maps for anisotropic surfaces. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 1501–1508.

[35]

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.

[36]

Thomas Müller, Brian Mcwilliams, Fabrice Rousselle, Markus Gross, and Jan Novák. 2019. Neural importance sampling. ACM Trans. Graph. 38, 5 (2019).

Digital Library

[37]

Adolfo Muñoz. 2014. Higher-order ray marching. In Computer Graphics Forum, Vol. 33. Wiley Online Library, 167–176.

[38]

Jan Novák, Iliyan Georgiev, Johannes Hanika, and Wojciech Jarosz. 2018. Monte Carlo methods for volumetric light transport simulation. In Computer Graphics Forum, Vol. 37. Wiley Online Library, 551–576.

[39]

Jan Novák, Derek Nowrouzezahrai, Carsten Dachsbacher, and Wojciech Jarosz. 2012. Virtual ray lights for rendering scenes with participating media. ACM Trans. Graph. 31, 4 (2012), 60:1–60:11.

Digital Library

[40]

Jan Novák, Andrew Selle, and Wojciech Jarosz. 2014. Residual ratio tracking for estimating attenuation in participating media. ACM Trans. Graph. 33, 6 (2014).

Digital Library

[41]

Art Owen and Yi Zhou. 2000. Safe and effective importance sampling. J. Amer. Statist. Assoc. 95, 449 (2000), 135–143.

[42]

Art B. Owen. 2013. Monte Carlo Theory, Methods and Examples.

[43]

Ken Perlin and Eric M. Hoffert. 1989. Hypertexture. In Computer Graphics (Proceedings of SIGGRAPH), Vol. 23. ACM, 253–262.

[44]

William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. 2007. Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press, Cambridge, UK.

Digital Library

[45]

Ravi Ramamoorthi and Pat Hanrahan. 2001. An efficient representation for irradiance environment maps. In Proceedings of SIGGRAPH. 497–500.

Digital Library

[46]

Ravi Ramamoorthi and Pat Hanrahan. 2002. Frequency space environment map rendering. ACM Trans. Graph. 21, 3 (2002), 517–526.

Digital Library

[47]

Ravi Ramamoorthi, Dhruv Mahajan, and Peter Belhumeur. 2007. A first-order analysis of lighting, shading, and shadows. ACM Trans. Graph. 26, 1 (2007), 2.

Digital Library

[48]

Christian P. Robert and George Casella. 2004. Monte Carlo statistical methods. Springer New York.

[49]

Fabrice Rousselle, Wojciech Jarosz, and Jan Novák. 2016. Image-space control variates for rendering. ACM Trans. Graph. 35, 6 (2016), 169.

Digital Library

[50]

Fabrice Rousselle, Claude Knaus, and Matthias Zwicker. 2012. Adaptive rendering with non-local means filtering. ACM Trans. Graph. 31, 6 (2012), 195.

Digital Library

[51]

Martin Šik and Jaroslav Krivanek. 2018. Survey of Markov chain Monte Carlo methods in light transport simulation. IEEE Trans. Visual. Comput. Graph. 26, 4 (2018), 1821--1840.

[52]

Arthur H. Stroud and Don Secrest. 1966. Gaussian quadrature formulas. Prentice-Hall.

[53]

Carlos Ureña, Marcos Fajardo, and Alan King. 2013. An area-preserving parametrization for spherical rectangles. In Computer Graphics Forum, Vol. 32. Wiley Online Library, 59–66.

[54]

Eric Veach. 1997. Robust Monte Carlo Methods for Light Transport Simulation. Vol. 1610. PhD thesis, Stanford University.

Digital Library

[55]

Eric Veach and Leonidas J. Guibas. 1995. Optimally combining sampling techniques for Monte Carlo rendering. In Proceedings of SIGGRAPH’95. ACM, 419–428.

[56]

Petr Vévoda, Ivo Kondapaneni, and Jaroslav Křivánek. 2018. Bayesian online regression for adaptive direct illumination sampling. ACM Trans. Graph. 37, 4 (2018), 125.

Digital Library

[57]

Jiří Vorba, Ondřej Karlík, Martin Šik, Tobias Ritschel, and Jaroslav Křivánek. 2014. On-line learning of parametric mixture models for light transport simulation. ACM Trans. Graph. 33, 4 (2014), 101.

Digital Library

[58]

Gregory J. Ward, Francis M. Rubinstein, and Robert D. Clear. 1988. A ray tracing solution for diffuse interreflection. In Proceedings of SIGGRAPH.

[59]

Rex West, Iliyan Georgiev, Adrien Gruson, and Toshiya Hachisuka. 2020. Continuous multiple importance sampling. ACM Trans. Graph. 39, 4 (2020).

Digital Library

[60]

E. Woodcock, T. Murphi, P. Hemmings, and S. Longworth. 1965. Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry. In Proceedings of the Conference on Applications of Computing Methods to Reactors.

[61]

Quan Zheng and Matthias Zwicker. 2019. Learning to importance sample in primary sample space. In Computer Graphics Forum, Vol. 38. Wiley Online Library, 169–179.

[62]

Eric Ziegel. 1987. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, UK.

[63]

Matthias Zwicker, Wojciech Jarosz, Jaakko Lehtinen, Bochang Moon, Ravi Ramamoorthi, Fabrice Rousselle, Pradeep Sen, Cyril Soler, and S.-E. Yoon. 2015. Recent advances in adaptive sampling and reconstruction for Monte Carlo rendering. In Computer Graphics Forum, Vol. 34. Wiley Online Library, 667–681.

Digital Library

Cited By

Zhu JHery CBode LAliaga CJarabo AYan LChiang M(2024)A Realistic Multi-scale Surface-based Cloth Appearance ModelACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657426(1-10)Online publication date: 13-Jul-2024
https://dl.acm.org/doi/10.1145/3641519.3657426
Hua QGrittmann PSlusallek P(2023)Revisiting controlled mixture sampling for rendering applicationsACM Transactions on Graphics10.1145/359243542:4(1-13)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592435
Nicolet BRousselle FNovak JKeller AJakob WMüller T(2023)Recursive Control Variates for Inverse RenderingACM Transactions on Graphics10.1145/359213942:4(1-13)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592139
Show More Cited By

Index Terms

Primary-space Adaptive Control Variates Using Piecewise-polynomial Approximations
1. Computing methodologies
  1. Computer graphics
    1. Rendering

Recommendations

Image-space control variates for rendering

We explore the theory of integration with control variates in the context of rendering. Our goal is to optimally combine multiple estimators using their covariances. We focus on two applications, re-rendering and gradient-domain rendering, where we ...
Coupling control variates for Markov chain Monte Carlo

We show that Markov couplings can be used to improve the accuracy of Markov chain Monte Carlo calculations in some situations where the steady-state probability distribution is not explicitly known. The technique generalizes the notion of control ...
Combining antithetic variates and control variates in simulation experiments

Antithetic variates and control variates are two well-known variance reduction techniques. We consider combining antithetic variates and control variates to estimate the mean response in a stochastic simulation experiment. When applying antithetic ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 40, Issue 3

June 2021

264 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3463476

Editor:
Marc Alexa
TU Berlin, Germany

Issue’s Table of Contents

Copyright © 2021 Association for Computing Machinery.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 15 July 2021

Accepted: 01 February 2021

Revised: 01 January 2021

Received: 01 August 2020

Published in TOG Volume 40, Issue 3

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Refereed

Funding Sources

European Research Council (ERC) under the EU’s Horizon 2020 research and innovation programme (project CHAMELEON)
DARPA (Project REVEAL)
Spanish Ministry of Economy and Competitiveness
Spanish Ministry of Science and Innovation

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

6
Total Citations
View Citations
196
Total Downloads

Downloads (Last 12 months)50
Downloads (Last 6 weeks)7

Reflects downloads up to 24 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Zhu JHery CBode LAliaga CJarabo AYan LChiang M(2024)A Realistic Multi-scale Surface-based Cloth Appearance ModelACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657426(1-10)Online publication date: 13-Jul-2024
https://dl.acm.org/doi/10.1145/3641519.3657426
Hua QGrittmann PSlusallek P(2023)Revisiting controlled mixture sampling for rendering applicationsACM Transactions on Graphics10.1145/359243542:4(1-13)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592435
Nicolet BRousselle FNovak JKeller AJakob WMüller T(2023)Recursive Control Variates for Inverse RenderingACM Transactions on Graphics10.1145/359213942:4(1-13)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592139
Salaün CGruson AHua BHachisuka TSingh G(2022)Regression-based Monte Carlo integrationACM Transactions on Graphics10.1145/3528223.353009541:4(1-14)Online publication date: 22-Jul-2022
https://dl.acm.org/doi/10.1145/3528223.3530095
Fraboni BWebanck ABonneel NIehl J(2022)Volumetric Multi‐View RenderingComputer Graphics Forum10.1111/cgf.1448141:2(379-392)Online publication date: 24-May-2022
https://doi.org/10.1111/cgf.14481
González García FMartin IPatow G(2022)Feature-based clustered geometry for interpolated Ray-castingComputers and Graphics10.1016/j.cag.2021.08.019102:C(175-186)Online publication date: 1-Feb-2022
https://dl.acm.org/doi/10.1016/j.cag.2021.08.019

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents