Stochastic sampling in computer graphics
ACM Transactions on Graphics (TOG)
Generating antialiased images at low sampling densities
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Spectrally optimal sampling for distribution ray tracing
Proceedings of the 18th annual conference on Computer graphics and interactive techniques
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Consequences of stratified sampling in graphics
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
The visibility skeleton: a powerful and efficient multi-purpose global visibility tool
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
Progressive radiance evaluation using directional coherence maps
Proceedings of the 25th annual conference on Computer graphics and interactive techniques
Fast calculation of soft shadow textures using convolution
Proceedings of the 25th annual conference on Computer graphics and interactive techniques
A low distortion map between disk and square
Journal of Graphics Tools
Statistically optimized sampling for distributed ray tracing
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Antialiasing through stochastic sampling
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Efficient image-based methods for rendering soft shadows
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
ACM Transactions on Graphics (TOG)
Triple product wavelet integrals for all-frequency relighting
ACM SIGGRAPH 2004 Papers
A Fourier Theory for Cast Shadows
IEEE Transactions on Pattern Analysis and Machine Intelligence
A frequency analysis of light transport
ACM SIGGRAPH 2005 Papers
A spatial data structure for fast Poisson-disk sample generation
ACM SIGGRAPH 2006 Papers
A first-order analysis of lighting, shading, and shadows
ACM Transactions on Graphics (TOG)
Parallel Poisson disk sampling
ACM SIGGRAPH 2008 papers
Multidimensional adaptive sampling and reconstruction for ray tracing
ACM SIGGRAPH 2008 papers
Shield fields: modeling and capturing 3D occluders
ACM SIGGRAPH Asia 2008 papers
ACM SIGGRAPH Asia 2009 papers
Multi-class blue noise sampling
ACM SIGGRAPH 2010 papers
Frequency analysis and sheared filtering for shadow light fields of complex occluders
ACM Transactions on Graphics (TOG)
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
Compressive estimation for signal integration in rendering
EGSR'10 Proceedings of the 21st Eurographics conference on Rendering
Fourier analysis of stochastic sampling strategies for assessing bias and variance in integration
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
Probabilistic visibility evaluation for direct illumination
EGSR '13 Proceedings of the Eurographics Symposium on Rendering
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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.