Bistro: Unbiased rendering of a complex scene with global illumination (22 indirect bounces, resulting in a 48-dimensional integration domain). Traditional Monte Carlo-based rendering results in high variance even with importance sampling techniques. In contrast, our technique combines multiple importance sampling with an adaptive piecewise-polynomial control variate (4D in this example): Our control variate closely approximates the low-frequency regions of the signal, while leaving the high-frequency details on the residual, which is estimated using Monte Carlo integration. This results in lower variance with faster convergence. Except for the reference, the images were generated using 512 samples per pixel.

Bistro: Unbiased rendering of a complex scene with global illumination (22 indirect bounces, resulting in a 48-dimensional integration domain). Traditional Monte Carlo-based rendering results in high variance even with importance sampling techniques. In contrast, our technique combines multiple importance sampling with an adaptive piecewise-polynomial control variate (4D in this example): Our control variate closely approximates the low-frequency regions of the signal, while leaving the high-frequency details on the residual, which is estimated using Monte Carlo integration. This results in lower variance with faster convergence. Except for the reference, the images were generated using 512 samples per pixel.

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-base 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 distributed 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.

Video

Figures

Integration of two two-dimensional functions (a), its piecewise-polynomial approximation used as control variate (b, boundaries of each region in green), and the corresponding residual (c, where red and blue are the positive and negative residual, respectively).Renderings of two purely absorbing media, with high (first row, Hetvol) and low (second row, Smoke) densities, computed using delta tracking, residual ratio tracking, and our adaptive residual ratio tracking (full image). The three methods have approximately the same number of media queries.
Equal-samples (64 spp) comparison between Monte Carlo, Simpson-Trapezoid quadrature and our technique for computing single scattering from a point light source in isotropic homogenous media. Our technique yields more accurate results and recovers both the smooth global structure of light transport and the high frequency details of the scene, while remaining unbiased.Equal-samples (64 spp) comparison between Monte Carlo and our technique for computing two-bounce scattering from a collimated beam in isotropic homogenous media. While pure Monte Carlo generates high-frequency noise, our approach excels at the smooth regions, accurately handling the sharp details.Comparison of the different approaches of our technique against Monte Carlo integration for the same number of evaluations of direct illumination. In all cases, Monte Carlo produces noisier images even with MIS. In contrast, our technique leverages MIS adapting the control variate to the integrand, yielding better results both per pixel ('Ours 2D') and for the whole image space ('Ours 4D'). Furthermore, amortizing the control variate among the whole image space reduces noise in low frequency areas, removes structured noise, and serves as antialiasing. All results are calculated using 155 spp.
Comparison between our approach (left column), Monte Carlo, and previous related work in four different scenes (with increasing dimensionality), at equal number of samples (64 spp). The scenes feature several distributed effects including motion blur, depth-of-field and soft shadows. In all cases, Monte Carlo produces renders with high variance, while Hachisuka et al.'s approach achieves good results in smooth domains, but tends to overblur the sharp regions of the scene. In contrast, our unbiased method outperforms previous work keeping the high contrast areas sharp.

Bibtex

@article{crespo2021primary,
  title={Primary-Space Adaptive Control Variates using Piecewise-Polynomial Approximations},
  author={Crespo, Miguel and Jarabo, Adrian and Mu{\~n}oz, Adolfo},
  year={2021},
  journal = {ACM Transactions on Graphics},
  doi = {10.1145/3450627},
}

Acknowledgements

We thank Ibón Guillén for comments and discussion throughout the project; Manuel Lagunas for help with figures; Diego Gutierrez for advice on the reviews; Felix Bernal for helping with the Single Scattering implementation; all the members of the Graphics \& Imaging Lab that helped with proof-reading; and the reviewers for the in-depth reviews. The Pool and Chess are by Hachisuka et al.; Cornell Box, House, Classroom and MIS Test are by Benedikt Bitterli; Violin was modeled by Tahseen; Helicopter was modeled by Mond; Volley Balls models by Shri; Dragon and Budha are from the Stanford 3D Scanning Repository; Bistro was modelled by Amazon Lumberyard. Lightfields used in Figure 4 are courtesy of Jarabo et al. [2014]. This project has been funded by the European Research Council (ERC) under the EU's Horizon 2020 research and innovation programme (project CHAMELEON, grant No 682080), DARPA (project REVEAL, HR0011-16-C-0025) and the Spanish Ministry of Economy and Competitiveness (project TIN2016-78753-P and PID2019-105004GB-I00).