A Theory of Monte Carlo A Theory of Monte Carlo Visibility SamplingVisibility Sampling
Ravi Ramamoorthi UC BerkeleyJohn Anderson Pixar (now at Google)Mark Meyer PixarDerek Nowrouzezahrai Disney Research Zurich University of Montreal
MotivationMotivation
Soft shadows critical in high quality rendering
Monte Carlo sampling of visibility most common
Which (non-adaptive) sampling patterns are better?
Theory of Monte Carlo visibility sampling Focus on (binary) visibility only; not general rendering
Monte Carlo Soft ShadowsMonte Carlo Soft Shadows
Light
Pixel
Experiment: Sampling StrategiesExperiment: Sampling Strategies
circle light uniform jitter RMS 6.6%
Experiment: Sampling StrategiesExperiment: Sampling Strategies
circle light stratified RMS 8.3%
Circle
Experiment: Sampling StrategiesExperiment: Sampling Strategies
square light uniform jitter RMS 13.4%
Experiment: Sampling StrategiesExperiment: Sampling Strategies
square light stratified RMS 10.4%
ContributionsContributions
Theory of Visibility Sampling for Linear Lights
Uniform Jitter Sampling Lowest error at center of stratum (uniform sampling) Can avoid banding, keep low error with uniform jitter
2D Pixel-Light Fourier Analysis
Planar Area Lights
Previous WorkPrevious Work
Early MC Sampling [Cook 86, Mitchell 87,96, HK 94]
Empirical Study Linear Lights [Ouellette & Fiume 01]
Adaptive Sampling [Mitchell 91, Guo98] Adaptive Filtering [Hachisuka 08, …, this session] Shadow Coherence [Agrawala et al. 00, Egan et al. 11] Can leverage our approach
Signal Processing and Frequency Analysis Space-Angle [Durand et al. 05] Sheared Visibility Spectrum [Egan et al. 11] Fourier Analysis of MC [Ouellete & Fiume 01, Durand 11]
Linear Lights: DiscontinuitiesLinear Lights: Discontinuities
Visibility0
1
PixelBlocker
Light
Discontinuity
Single Discontinuity:Heaviside Function
Two Discontinuities:Boxcar Function
Many Discontinuities:Many Box Functions
Single Discontinuity: Uniform SamplingSingle Discontinuity: Uniform Sampling Error of Uniform Sampling
Error depends only on stratum of discontinuity
Error depends only on discontinuity location
Worst-Case, Variance (N = samples) Depends only on sample placement in stratum
Absolute Error
Uniform Sampling
Visibility at Pixel
Single Discontinuity: ErrorsSingle Discontinuity: Errors
Sample Location Worst-Case RMS Error Variance
End-pts of stratume.g., QMC Halton
Random (Jittered)Stratified Sampling
Center (Uniform)
Uniform Sampling is optimal (but bands, bias)
Reduces variance by factor of 2 vs. stratified
Benefits of uniform without banding? Stratified best unbiased method for 1 discontinuity, but…
Two Discontinuities: Boxcar Two Discontinuities: Boxcar
Assume discontinuities in different strata For very complex visibility, stratification not useful anyway
Visibility0
1
Strata
V0
1
Ground truth net visibility = 0.75 + 0.75 = 1.5
Uniform Sampling: net visibility = 1+1 = 2, biasedStratified Sampling: net vis 00 / 01 / 10 / 11 = 0,1,2, high variance
Two Discontinuities: ErrorsTwo Discontinuities: Errors
Sampling Method Worst-Case RMS Error Variance
Stratified
Uniform Jitter
Uniform
Uniform Jitter exploits correlation of discontinuities Error stays the same as in the single discontinuity case While other methods (stratified) double the variance Multiple discontinuities: separate into individual box functions
Uniform Jitter has same variance as uniform Optimal with no bias or banding
Linear Light: UniformLinear Light: Uniform
Error Image (scaled up)Uniform 16 samples
3.97% RMS Error (best)Uniform causes banding
Linear Light: Uniform JitterLinear Light: Uniform Jitter
Error Image (scaled up)Uniform Jitter 16 samples
4.21% RMS ErrorNo banding, low error
Linear Light: StratifiedLinear Light: Stratified
Error Image (scaled up)Stratified 16 samples
5.36% RMS Error (worst)
2D Pixel Light Fourier Analysis2D Pixel Light Fourier Analysis
Fourier analysis for fixed depth blocker
Shadow spectrum line in pixel-light space
Wedge if blockers at multiple depths [Egan 2011]
Builds on [Egan 2011] and Fourier analysis of Monte Carlo [Durand 2011] but full pixel-light theory
Blocker
Receiver
Light
x
y
x
yOcclusion
Visibility Spatial Domain Fourier DomainΩx
Ωy
Fourier Analysis: Uniform SamplingFourier Analysis: Uniform Sampling
Sampling pattern same for all x in uniform sampling Fourier spectrum on vertical line, spacing depends on N
Product of visibility and sampling: Fourier convolution
Errors (only) when aliases touch spatial axis Banding since error concentrated in specific frequencies
Ωx
Ωy
Visibility Spectrum
Ωx
Ωy
Ωx
Ωy
Uniform Sampling Sampled Visibility
Fourier Analysis: Uniform JitterFourier Analysis: Uniform Jitter
Sampling pattern uniform spaced but jittered at each x Fourier spectrum dots for replicas become horizontal lines
Fourier convolution: Central spectrum plus noise
Errors diffused to entire spatial axis No banding: error not concentrated in specific frequencies
Ωx
Ωy
Visibility Spectrum
Ωx
Ωy
Ωx
Ωy
Uniform Jitter Sampling Sampled Visibility
Planar Area LightsPlanar Area Lights Apply linear 1D analysis to each “scanline” of 2D light
But possible bad correlation between scanlines
Uniform Jitter StratifiedBlocked Visible
Samples Correlated Samples De-Correlated
Frequency Analysis: Shape of LightFrequency Analysis: Shape of Light
For one pixel, Fourier spectrum of visibility, light
Uniform Jitter is a regular comb pattern
Simple visibility lies on a line (assume horizontal)
Ωx
Ωy
Frequency Analysis: Shape of LightFrequency Analysis: Shape of Light
Integral of product of visibility and sampling pattern True in spatial or Fourier domain Ground Truth is constant (0 frequency) term only
Errors when significant spectral overlap
Ωx
Ωy
Frequency Analysis: Shape of LightFrequency Analysis: Shape of Light
Smooth lights (e.g., Gaussian, circular) Multiply light by Gaussian same as multiply sampling pattern Fourier: Convolve sampling pattern by Gaussian
Overlap now only along horizontal line, not full pattern
Ωx
Ωy
ImplementationImplementation
Simple shaders in RenderMan (general RIB files)
NVIDIA Optix for real-time applications Sampling patterns in closest hit kernel
Practical Result: Uniform Jitter best published method to our knowledge for circle, linear lights
Grids: Circle Light: Stratified 20 Grids: Circle Light: Stratified 20
Grids: Circle Light: Uniform Jitter 20Grids: Circle Light: Uniform Jitter 20
Grids: Circle Light: Halton, Warp 25Grids: Circle Light: Halton, Warp 25
Grids: Circle and Gaussian LightsGrids: Circle and Gaussian Lights
Comparable gain across all sample counts
ConclusionConclusion
Comprehensive Theory of MC Visibility Sampling Linear (1D) Lights: Uniform Jitter is optimal Circular or Gaussian Light: Uniform Jitter is better Square Light: Uniform Jitter worse than Stratified Halton, blue noise do not perform better
Introduce new Statistical and Fourier approaches
Best sampling pattern depends on shape of light Can choose linear or circular instead of square lights
Practical gains of 20%-40% for almost no effort
Future analyses of Monte Carlo patterns Optimal pattern for planar lights still an open question
AcknowledgmentsAcknowledgments
Rob Cook for inspiration, discussing 2D pixel-light
Fredo Durand for notes on Fourier Monte Carlo
Kevin Egan, Florian Hecht, Christophe Hery, Juan Buhler for scenes in the paper
Li-Yi Wei and Christophe Hery for discussions of blue noise and sampling methods respectively
Anonymous reviewers for many helpful suggestions
To Jim Arvo, 1956-2011To Jim Arvo, 1956-2011