Abstract
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.
Supplemental Material
Available for Download
Supplemental movie and image files for, A theory of monte carlo visibility sampling
- 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
Index Terms
- A theory of monte carlo visibility sampling
Recommendations
Screen Space Ambient Occlusion Based Multiple Importance Sampling for Real-Time Rendering
We propose a new approximation technique for accelerating the Global Illumination algorithm for real-time rendering. The proposed approach is based on the Screen-Space Ambient Occlusion (SSAO) method, which approximates the global illumination for large,...
Accurate Direct Illumination Using Iterative Adaptive Sampling
This paper introduces a new multipass algorithm for efficiently computing direct illumination in scenes with many lights and complex occlusion. Images are first divided into 8\times 8 pixel blocks and for each point to be shaded within a block, a ...
Antithetic sampling for Monte Carlo differentiable rendering
Stochastic sampling of light transport paths is key to Monte Carlo forward rendering, and previous studies have led to mature techniques capable of drawing high-contribution light paths in complex scenes. These sampling techniques have also been applied ...
Comments