Error analysis of estimators that use combinations of stochastic sampling strategies for direct illumination

1Disney Research Zürich 2University of Montreal 3University College London

In Computer Graphics Forum (Proceedings of EGSR), 2014

The "optimal" sampling strategy varies as a function of the sample count as well as spatially over pixels. A formal study of the variance of combinations of strategies is necessary to understand such behaviour. We compare combinations of four different sampling strategies with multiple importance sampling to render the image (left). The rates at which numerical integration errors decrease with increasing sample counts (per pixel; spp) are shown at 7 pixels (a-g). Our theoretical analysis provides insight into such behaviour, and motivates a new, jittered antithetic importance sampling, estimator (black) for rendering.


We present a theoretical analysis of error of combinations of Monte Carlo estimators used in image synthesis. Importance sampling and multiple importance sampling are popular variance-reduction strategies. Unfortunately, neither strategy improves the rate of convergence of Monte Carlo integration. Jittered sampling (a type of stratified sampling), on the other hand is known to improve the convergence rate. Most rendering software optimistically combine importance sampling with jittered sampling, hoping to achieve both. We derive the exact error of the combination of multiple importance sampling with jittered sampling. In addition, we demonstrate a further benefit of introducing negative correlations (antithetic sampling) between estimates to the convergence rate. As with importance sampling, antithetic sampling is known to reduce error for certain classes of integrands without affecting the convergence rate. In this paper, our analysis and experiments reveal that importance and antithetic sampling, if used judiciously and in conjunction with jittered sampling, may improve convergence rates. We show the impact of such combinations of strategies on the convergence rate of estimators for direct illumination.



Kartic Subr, Derek Nowrouzezahrai, Wojciech Jarosz, Jan Kautz, Kenny Mitchell. Error analysis of estimators that use combinations of stochastic sampling strategies for direct illumination. Computer Graphics Forum (Proceedings of EGSR), 33(4):93–102, June 2014.
    author = "Subr, Kartic and Nowrouzezahrai, Derek and Jarosz, Wojciech and Kautz, Jan and Mitchell, Kenny",
    title = "Error analysis of estimators that use combinations of stochastic sampling strategies for direct illumination",
    journal = "Computer Graphics Forum (Proceedings of EGSR)",
    volume = "33",
    number = "4",
    month = "jun",
    year = "2014",
    pages = "93–102",
    DOI = "10.1111/cgf.12416",
    keywords = "variance analysis, Gaussian copula, convergence rate"
© The Author(s). This is the author's version of the work. It is posted here by permission of The Eurographics Association for your personal use. Not for redistribution. The definitive version is available at