“A Theory of Monte Carlo Visibility Sampling” by Ramamoorthi, Anderson, Meyer and Nowrouzezahrai

  • ©Ravi Ramamoorthi, John Anderson, Mark Meyer, and Derek Nowrouzezahrai




    A Theory of Monte Carlo Visibility Sampling



    Soft shadows from area lights are one of the most crucial effects in high-quality and production rendering, but Monte-Carlo sampling of visibility is often the main source of noise in rendered images. Indeed, it is common to use deterministic uniform sampling for the smoother shading effects in direct lighting, so that all of the Monte Carlo noise arises from visibility sampling alone. In this article, we analyze theoretically and empirically, using both statistical and Fourier methods, the effectiveness of different nonadaptive Monte Carlo sampling patterns for rendering soft shadows. We start with a single image scanline and a linear light source, and gradually consider more complex visibility functions at a pixel. We show analytically that the lowest expected variance is in fact achieved by uniform sampling (albeit at the cost of visual banding artifacts). Surprisingly, we show that for two or more discontinuities in the visibility function, a comparable error to uniform sampling is obtained by “uniform jitter” sampling, where a constant jitter is applied to all samples in a uniform pattern (as opposed to jittering each stratum as in standard stratified sampling). The variance can be reduced by up to a factor of two, compared to stratified or quasi-Monte Carlo techniques, without the banding in uniform sampling.
    We augment our statistical analysis with a novel 2D Fourier analysis across the pixel-light space. This allows us to characterize the banding frequencies in uniform sampling, and gives insights into the behavior of uniform jitter and stratified sampling. We next extend these results to planar area light sources. We show that the best sampling method can vary, depending on the type of light source (circular, Gaussian, or square/rectangular). The correlation of adjacent “light scanlines” in square light sources can reduce the effectiveness of uniform jitter sampling, while the smoother shape of circular and Gaussian-modulated sources preserves its benefits—these findings are also exposed through our frequency analysis. In practical terms, the theory in this article provides guidelines for selecting visibility sampling strategies, which can reduce the number of shadow samples by 20–40%, with simple modifications to existing rendering code.


    Agrawala, M., Ramamoorthi, R., Heirich, A., and Moll, L. 2000. Efficient image-based methods for rendering soft shadows. In Proceedings of the ACM SIGGRAPH 00 Conferrence. 375–384. Google ScholarDigital Library
    Ben-Artzi, A., Ramamoorthi, R., and Agrawala, M. 2006. Efficient shadows for sampled environment maps. J. Graph. Tools 11, 1, 13–36.Google ScholarCross Ref
    Candes, E. 2006. Compressive sampling. In Proceedings of the International Congress of Mathematics. Number 3, 1433–1452.Google Scholar
    Candes, E., Romberg, J., and Tao, T. 2006. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59, 8, 1207–1223.Google ScholarCross Ref
    Candes, E. and Tao, T. 2006. Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inf. Theory 52, 12, 5406–5425. Google ScholarDigital Library
    Cook, R. 1986. Stochastic sampling in computer graphics. ACM Trans. Graph. 5, 1, 51–72. Google ScholarDigital Library
    Dippe, M. and Wold, E. 1985. Antialiasing through stochastic sampling. In Proceedings of the ACM SIGGRAPH 85 Conference. 69–78. Google ScholarDigital Library
    Dunbar, D. and Humphreys, G. 2006. A spatial data structure for fast poisson-disk sample generation. ACM Trans. Graph. 25, 3, 503–508. Google ScholarDigital Library
    Durand, F. 2011. A frequency analysis of monte-carlo and other numerical integration schemes. Tech. rep. MIT-CSAIL-TR-2011-052 http://hdl.handle.net/1721.1/67677, MIT CSAIL.Google Scholar
    Durand, F., Drettakis, G., and Puech, C. 1997. The visibility skeleton: A powerful and efficient multi-purpose global visibility tool. In Proceedings of the ACM SIGGRAPH 97 Conference. 89–100. Google ScholarDigital Library
    Durand, F., Holzschuch, N., Soler, C., Chan, E., and Sillion, F. 2005. A frequency analysis of light transport. ACM Trans. Graph. 25, 3, 1115–1126. Google ScholarDigital Library
    Egan, K., Hecht, F., Durand, F., and Ramamoorthi, R. 2011. Frequency analysis and sheared filtering for shadow light fields of complex occluders. ACM Trans. Graph. 30, 2. Google ScholarDigital Library
    Guo, B. 1998. Progressive radiance evaluation using directional coherence maps. In Proceedings of the ACM SIGGRAPH 98 Conference. 255–266. Google ScholarDigital Library
    Hachisuka, T., Jarosz, W., Weistroffer, R., Dale, K., Humphreys, G., Zwicker, M., and Jensen, H. 2008. Multidimensional adaptive sampling and reconstruction for ray tracing. ACM Trans. Graph. 27, 3. Google ScholarDigital Library
    Heinrich, S. and Keller, A. 1994. Quasi-Monte carlo methods in computer graphics. Tech. rep. 242/3, University of Kaiserslautern.Google Scholar
    Keller, A. 1997. Instant radiosity. In Proceedings of the ACM SIGGRAPH 97 Conference. 49–56. Google ScholarDigital Library
    Lagae, A. and Dutre, P. 2008. A comparison of methods for generating poisson disk patterns. Comput. Graph. Forum 27, 1, 114–129.Google ScholarCross Ref
    Lanman, D., Raskar, R., Agrawal, A., and Taubin, G. 2008. Shield fields: modeling and capturing 3D occluders. ACM Trans. Graph. 27, 5. Google ScholarDigital Library
    Lee, M., Redner, A., and Uselton, S. 1985. Statistically optimized sampling for distributed ray tracing. In Proceedings of the ACM SIGGRAPH 85 Conference. 61–68. Google ScholarDigital Library
    Mitchell, D. 1987. Generating antialiased images at low sampling densities. In Proceedings of the ACM SIGGRAPH 87 Conference. 65–72. Google ScholarDigital Library
    Mitchell, D. 1991. Spectrally optimal sampling for distribution ray tracing. In Proceedings of the ACM SIGGRAPH 91 Conference. 157–164. Google ScholarDigital Library
    Mitchell, D. 1996. Consequences of stratified sampling in graphics. In Proceedings of the ACM SIGGRAPH 96 Conference. 277–280. Google ScholarDigital Library
    Ng, R., Ramamoorthi, R., and Hanrahan, P. 2004. Triple product wavelet integrals for all-frequency relighting. ACM Trans. Graph. 23, 3, 475–485. Google ScholarDigital Library
    Niederreiter, H. 1992. Random Number Generation and Quasi-Monte Carlo Methods. SIAM. Google ScholarDigital Library
    Ouellette, M. and Fiume, E. 2001. On numerical solutions to one-dimensional integration problems with application to linear light sources. ACM Trans. Graph. 20, 4, 232–279. Google ScholarDigital Library
    Overbeck, R., Donner, C., and Ramamoorthi, R. 2009. Adaptive Wavelet Rendering. ACM Trans. Graph. 28, 5. Google ScholarDigital Library
    Purgathofer, W. 1986. A statistical model for adaptive stochastic sampling. In Proceedings of the Eurographics Conference. 145–152.Google Scholar
    Ramamoorthi, R., Koudelka, M., and Belhumeur, P. 2005. A Fourier theory for cast shadows. IEEE Trans. Pattern Anal. Mach. Intell. 27, 2, 288–295. Google ScholarDigital Library
    Ramamoorthi, R., Mahajan, D., and Belhumeur, P. 2007. A first order analysis of lighting, shading, and shadows. ACM Trans. Graph. 26, 1. Google ScholarDigital Library
    Sen, P. and Darabi, S. 2010. Compressive estimation for signal integration in rendering. Comput. Graph. Forum 29, 4, 1355–1363. Google ScholarDigital Library
    Shirley, P. and Chiu, K. 1997. A low distortion map between disk and square. J. Graph. Tools 2, 3, 45–52. Google ScholarDigital Library
    Soler, C. and Sillion, F. 1998. Fast calculation of soft shadow textures using convolution. In Proceedings of the ACM SIGGRAPH 98 Conference. 321–332. Google ScholarDigital Library
    Wei, L. 2008. Parallel poisson disk sampling. ACM Trans. Graph. 27, 3. Google ScholarDigital Library
    Wei, L. 2010. Multi-Class blue noise sampling. ACM Trans. Graph. 29, 4. Google ScholarDigital Library
    Yellot, J. 1983. Spectral consequences of photoreceptor sampling in the rhesus retina. Science 221, 382–385.Google ScholarCross Ref

ACM Digital Library Publication:

Overview Page: