research-article

Public Access

Unbiased and consistent rendering using biased estimators

Authors:

Benedikt Bitterli,

Iliyan Georgiev,

Wojciech JaroszAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 41, Issue 4

Article No.: 48, Pages 1 - 13

https://doi.org/10.1145/3528223.3530160

Published: 22 July 2022 Publication History

Abstract

We introduce a general framework for transforming biased estimators into unbiased and consistent estimators for the same quantity. We show how several existing unbiased and consistent estimation strategies in rendering are special cases of this framework, and are part of a broader debiasing principle. We provide a recipe for constructing estimators using our generalized framework and demonstrate its applicability by developing novel unbiased forms of transmittance estimation, photon mapping, and finite differences.

Supplemental Material

MP4 File

presentation

Download
115.42 MB

SRT File

presentation

Download
26.83 KB

References

[1]

Sai Praveen Bangaru, Tzu-Mao Li, and Frédo Durand. 2020. Unbiased Warped-Area Sampling for Differentiable Rendering. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 39, 6 (Nov. 2020), 245:1--245:18.

Digital Library

[2]

Benedikt Bitterli. 2016. Rendering resources. https://benedikt-bitterli.me/resources/.

[3]

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 (Proceedings of SIGGRAPH Asia) 37, 6 (Nov. 2018), 225:1--225:17.

[4]

Jose H. Blanchet, Nan Chen, and Peter W. Glynn. 2015. Unbiased Monte Carlo Computation of Smooth Functions of Expectations via Taylor Expansions. In 2015 Winter Simulation Conference (WSC). 360--367.

[5]

Jose H. Blanchet and Peter W. Glynn. 2015. Unbiased Monte Carlo for Optimization and Functions of Expectations via Multi-Level Randomization. In 2015 Winter Simulation Conference (WSC). 3656--3667.

[6]

T. E. Booth. 2007. Unbiased Monte Carlo Estimation of the Reciprocal of an Integral. Nuclear Science and Engineering 156, 3 (July 2007), 403--407.

[7]

Jérémi Dauchet, Jean-Jacques Bezian, Stéphane Blanco, Cyril Caliot, Julien Charon, Christophe Coustet, Mouna El Hafi, Vincent Eymet, Olivier Farges, Vincent Forest, Richard Fournier, Mathieu Galtier, Jacques Gautrais, Anaïs Khuong, Lionel Pelissier, Benjamin Piaud, Maxime Roger, Guillaume Terrée, and Sebastian Weitz. 2018. Addressing Nonlinearities in Monte Carlo. Scientific Reports 8, 1 (Dec. 2018), 13302.

[8]

Anthony B. Davis and Mark B. Mineev-Weinstein. 2011. Radiation Propagation in Random Media: From Positive to Negative Correlations in High-Frequency Fluctuations. Journal of Quantitative Spectroscopy and Radiative Transfer 112, 4 (March 2011), 632--645.

[9]

Iliyan Georgiev, Jaroslav Křivánek, Tomáš Davidovič, and Philipp Slusallek. 2012. Light Transport Simulation with Vertex Connection and Merging. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 31, 6 (Nov. 2012), 192:1--192:10.

Digital Library

[10]

Iliyan Georgiev, Zackary Misso, Toshiya Hachisuka, Derek Nowrouzezahrai, Jaroslav Křivánek, and Wojciech Jarosz. 2019. Integral Formulations of Volumetric Transmittance. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 38, 6 (Nov. 2019), 154:1--154:17.

[11]

Michael B. Giles. 2008. Multilevel Monte Carlo Path Simulation. Operations Research 56, 3 (June 2008), 607--617.

Digital Library

[12]

Toshiya Hachisuka, Wojciech Jarosz, and Henrik Wann Jensen. 2010. A Progressive Error Estimation Framework for Photon Density Estimation. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 29, 6 (Dec. 2010), 144:1--144:12.

[13]

Toshiya Hachisuka and Henrik Wann Jensen. 2009. Stochastic Progressive Photon Mapping. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 28, 5 (Dec. 2009), 130:1--130:8.

Digital Library

[14]

S. Heinrich. 1998. A Multilevel Version of the Method of Dependent Tests. In Proc. of the 3rd St. Petersburg Workshop on Simulation. St. Petersburg University Press, 31--35.

[15]

Adrian Jarabo, Carlos Aliaga, and Diego Gutierrez. 2018. A Radiative Transfer Framework for Spatially-Correlated Materials. ACM Transactions on Graphics (Proceedings of SIGGRAPH) 37, 4 (July 2018), 83:1--83:13.

[16]

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 (Jan. 2011), 5:1--5:19.

Digital Library

[17]

Wojciech Jarosz, Derek Nowrouzezahrai, Robert Thomas, Peter-Pike Sloan, and Matthias Zwicker. 2011b. Progressive Photon Beams. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 30, 6 (Dec. 2011), 181:1--181:12.

[18]

Henrik Wann Jensen. 2001. Realistic Image Synthesis Using Photon Mapping. AK Peters, Ltd., Natick, MA, USA.

Digital Library

[19]

Jiantao Jiao and Yanjun Han. 2020. Bias Correction with Jackknife, Bootstrap, and Taylor Series. IEEE Transactions on Information Theory 66, 7 (July 2020), 4392--4418.

Digital Library

[20]

Daniel Jonsson, Joel Kronander, Jonas Unger, Thomas B. Schon, and Magnus Wrenninge. 2020. Direct Transmittance Estimation in Heterogeneous Participating Media Using Approximated Taylor Expansions. IEEE Transactions on Visualization and Computer Graphics (2020), 1--1.

[21]

A. Keller. 2001. Hierarchical Monte Carlo Image Synthesis. Mathematics and Computers in Simulation 55, 1--3 (2001), 79--92.

Digital Library

[22]

Markus Kettunen, Eugene d'Eon, Jacopo Pantaleoni, and Jan Novak. 2021. An Unbiased Ray-Marching Transmittance Estimator. (Feb. 2021). arXiv:2102.10294 [cs.GR]

[23]

Claude Knaus and Matthias Zwicker. 2011. Progressive Photon Mapping: A Probabilistic Approach. ACM Transactions on Graphics 30, 3 (May 2011), 25:1--25:13.

Digital Library

[24]

Peter Kutz, Ralf Habel, Yining Karl Li, and Jan Novák. 2017. Spectral and Decomposition Tracking for Rendering Heterogeneous Volumes. ACM Transactions on Graphics (Proceedings of SIGGRAPH) 36, 4 (July 2017), 111:1--111:16.

[25]

Don McLeish. 2011. A General Method for Debiasing a Monte Carlo Estimator. Monte Carlo Methods and Applications 17, 4 (Jan. 2011).

[26]

Zackary Misso, Benedikt Bitterli, Iliyan Georgiev, and Wojciech Jarosz. 2022. Unbiased and consistent rendering using biased estimators supplemental code and data.

[27]

Sarat Babu Moka, Dirk P. Kroese, and Sandeep Juneja. 2019. Unbiased Estimation of the Reciprocal Mean for Non-Negative Random Variables. In 2019 Winter Simulation Conference (WSC). 404--415.

[28]

Jan Novák, Andrew Selle, and Wojciech Jarosz. 2014. Residual Ratio Tracking for Estimating Attenuation in Participating Media. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 33, 6 (Nov. 2014), 179:1--179:11.

Digital Library

[29]

Adithya Pediredla, Yasin Karimi Chalmiani, Matteo Giuseppe Scopelliti, Maysamreza Chamanzar, Srinivasa Narasimhan, and Ioannis Gkioulekas. 2020. Path Tracing Estimators for Refractive Radiative Transfer. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia) 39, 6 (Nov. 2020), 241:1--241:15.

Digital Library

[30]

Matt Pharr, Wenzel Jakob, and Greg Humphreys. 2016. Physically Based Rendering: From Theory to Implementation (3rd ed.). Morgan Kaufmann, Cambridge, MA.

Digital Library

[31]

Hao Qin, Xin Sun, Qiming Hou, Baining Guo, and Kun Zhou. 2015. Unbiased Photon Gathering for Light Transport Simulation. ACM Transactions on Graphics 34, 6 (Oct. 2015).

Digital Library

[32]

Chang-han Rhee. 2013. Unbiased Estimation with Biased Samplers. Ph. D. Dissertation. Stanford University.

[33]

Chang-han Rhee and Peter W. Glynn. 2012. A New Approach to Unbiased Estimation for SDE's. (July 2012). arXiv:1207.2452 [q-fin.CP]

[34]

Chang-Han Rhee and Peter W. Glynn. 2015. Unbiased Estimation with Square Root Convergence for SDE Models. Operations Research 63, 5 (Oct. 2015), 1026--1043.

[35]

Delio Vicini, Wenzel Jakob, and Anton Kaplanyan. 2021. A Non-Exponential Transmittance Model for Volumetric Scene Representations. ACM Transactions on Graphics (Proceedings of SIGGRAPH) 40, 4 (Aug. 2021), 136:1--136:16.

Digital Library

[36]

Jiří Vorba and Jaroslav Křivánek. 2016. Adjoint-Driven Russian Roulette and Splitting in Light Transport Simulation. ACM Transactions on Graphics (Proceedings of SIGGRAPH) 35, 4 (July 2016), 42:1--42:11.

Digital Library

[37]

Christopher S. Withers. 1987. Bias Reduction by Taylor Series. Communications in Statistics - Theory and Methods 16, 8 (Jan. 1987), 2369--2383.

[38]

Tizian Zeltner, Iliyan Georgiev, and Wenzel Jakob. 2020. Specular Manifold Sampling for Rendering High-Frequency Caustics and Glints. ACM Transactions on Graphics (Proceedings of SIGGRAPH) 39, 4 (July 2020).

Digital Library

Cited By

Lee FJhang JChang C(2024)Photon-Driven Manifold SamplingProceedings of the ACM on Computer Graphics and Interactive Techniques10.1145/36753757:3(1-16)Online publication date: 9-Aug-2024
https://dl.acm.org/doi/10.1145/3675375
West RMukherjee S(2024)Stylized Rendering as a Function of ExpectationACM Transactions on Graphics10.1145/365816143:4(1-19)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658161
Fan ZGuo JWang YXiao TZhang HZhou CChen ZHong PGuo YYan L(2024)Specular PolynomialsACM Transactions on Graphics10.1145/365813243:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658132
Show More Cited By

Index Terms

Unbiased and consistent rendering using biased estimators
1. Computing methodologies
  1. Computer graphics
    1. Rendering
      1. Ray tracing

Recommendations

On the adaptive linear estimators, using biased Cramér-Rao bound

The idea of minimizing the variance in biased estimation along with controlling the gradient of bias is well established for the case of singular Fisher information matrix (FIM) in order to find the biased estimators. In this paper, the biased Cramer-...
Uniformly Best Biased Estimators in Non-Bayesian Parameter Estimation

In this paper, a new structured approach for obtaining uniformly best non-Bayesian biased estimators, which attain minimum-mean-square-error performance at any point in the parameter space, is established. We show that if a uniformly best biased (UBB) ...
Rethinking Biased Estimation: Improving Maximum Likelihood and the Cramér–Rao Bound

One of the prime goals of statistical estimation theory is the development of performance bounds when estimating parameters of interest in a given model, as well as constructing estimators that achieve these limits. When the parameters to be estimated ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 41, Issue 4

July 2022

1978 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3528223

Issue’s Table of Contents

Copyright © 2022 ACM.

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 the author(s) 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: 22 July 2022

Published in TOG Volume 41, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

National Science Foundation

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
605
Total Downloads

Downloads (Last 12 months)298
Downloads (Last 6 weeks)37

Reflects downloads up to 25 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Lee FJhang JChang C(2024)Photon-Driven Manifold SamplingProceedings of the ACM on Computer Graphics and Interactive Techniques10.1145/36753757:3(1-16)Online publication date: 9-Aug-2024
https://dl.acm.org/doi/10.1145/3675375
West RMukherjee S(2024)Stylized Rendering as a Function of ExpectationACM Transactions on Graphics10.1145/365816143:4(1-19)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658161
Fan ZGuo JWang YXiao TZhang HZhou CChen ZHong PGuo YYan L(2024)Specular PolynomialsACM Transactions on Graphics10.1145/365813243:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658132
Sawhney RMiller BGkioulekas ICrane K(2023)Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary ConditionsACM Transactions on Graphics10.1145/359239842:4(1-20)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592398
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
Sugimoto RChen TJiang YBatty CHachisuka T(2023)A Practical Walk-on-Boundary Method for Boundary Value ProblemsACM Transactions on Graphics10.1145/359210942:4(1-16)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592109
West RGeorgiev IHachisuka T(2022)Marginal Multiple Importance SamplingSIGGRAPH Asia 2022 Conference Papers10.1145/3550469.3555388(1-8)Online publication date: 29-Nov-2022
https://dl.acm.org/doi/10.1145/3550469.3555388
Rioux-Lavoie DSugimoto RÖzdemir TShimada NBatty CNowrouzezahrai DHachisuka T(2022)A Monte Carlo Method for Fluid SimulationACM Transactions on Graphics10.1145/3550454.355545041:6(1-16)Online publication date: 30-Nov-2022
https://dl.acm.org/doi/10.1145/3550454.3555450

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

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

Figures

Tables

Media

View Issue’s Table of Contents