deterministic importance sampling with error diffusion
DESCRIPTION
Deterministic Importance Sampling with Error Diffusion. L ászló Szirmay-Kalos, L ászló Szécsi Budapest University of Technology. Eurographics Symposium on Rendering, 2009. Numerical i ntegration. f : integrand. g : target density. 1. 0. samples. Quadrature error. f/g. f. best:. g. - PowerPoint PPT PresentationTRANSCRIPT
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Deterministic Importance Sampling with Error Diffusion
László Szirmay-Kalos, László Szécsi
Budapest University of Technology
Eurographics Symposium on Rendering, 2009
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Numerical integration
0 1
f: integrand
g: target density
samples
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Quadrature error
fg
f/g
M2
1best:
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Role of
Random sampling
undersampling
oversampling
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Role of
Wanted
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Previous work• Importance sampling:
– Transformation of uniform samples– Rejection sampling
• Metropolis (Veach97)• Population Monte Carlo (Lai07)• Importance re-sampling (Talbot05), thresholding (Burke05)
• Stratification:
• Low-discrepancy series (Shirley91,Keller95,Kollig02)• Poisson-disk/blue-noise (Cook86,Dunbar06,Kopf06)• Tiling (Ostromukhov05-07, Lagae06) • Sample relaxation (Agarwal03,Kollig03,Wan05,Spencer09)
2D only?
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Proposed method
• Simultaneously targets– Importance sampling
• Importance function– Point samples– Cheap
– Stratification• Minimize discrepancy in the target domain
• Simple!
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Sample generation: Phase 1
f
I: importance function
I
Tentative samples
Normalizationconstant: b
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Sample generation: Phase 2
f
g=I/b
M2
1
G
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Frequency modulator
Comparator (quantizer)
-+ Integrator
Tentative samples
Realsamples
g(i) y(i)
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Frequency domain analysis
)()()1()( 11 zyznzzzg
Delay Light-blue noise
White noise:
-+ Integrator
Tentative samples
Realsamples
g(i) y(i)
n(i)
Transfer function in the Z-transform domain:
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Delta-Sigma modulator:Noise-Shaping Feedback Coder
H(z)
quantizer
+
-
)()())(1()( zyznzHzg
No delayControllable blue noise
Tentative samples
g(i)+
g(i) y(i)+
Realsamples
Noise shaping
filter
Transfer function in the Z-transform domain:
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Application in higher dimensions
Importancemap
pixels
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Importancemap
Application in higher dimensions
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sequence neighborhood
Application in higher dimensions
Importancemap
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Equivalence• Deterministic importance sampling allowing
arbitrary importance functions and minimizing the error of distribution
• Delta-Sigma modulation• Error diffusion halftoning (e.g. Floyd-Steinberg)
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Environment mapping with light source sampling
v=1
v=1
lighting reflection visibility
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Light source sampling = Error diffusion halftoning of the Environment Map
Error diffusion
Random sampling
Similar complexity and running times!
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Light source sampling results
Random Error diffusion Reference
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Light source sampling results for diffuse objects
Random Error diffusion Reference
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Environment mapping with product sampling
lighting reflection visibility
• Separate importance map for every shaded point• Computational cost ???:
– Similar to importance re-sampling– Negligible overhead more complex scenes
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Product sampling: Diffuse objectsBRDF sampling Importance resampling Error diffusion
11 sec 13 sec 13 sec
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Product sampling: Specular objectsBRDF sampling Importance resampling Error diffusion
11 sec 13 sec 13 sec
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Product sampling with occlusions
BRDF sampling
Importanceresampling
Error diffusion
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Even higher dimensions• Regular grid: Curse of dimensionality!
• Solution: Low-discrepancy seriescurrentsample
Errordistribution
sequence ofvisiting samples
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Elemental interval property
8
12
7
3
4
5
6
9
10
11
1
2
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The algorithm in d-dimensions
8
12
7
3
4
5
6
9
10
11
1
2
8,I(u8)
2,I(u2)
5,I(u5)
11,I(u11)
4,I(u4)
10,I(u10)
1,I(u1)
7,I(u7)
12,I(u12)
6,I(u6)
9,I(u9)
3,I(u3)
d-dimensional arrayd-dimensional cube
+ normalization constant b
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Virtual point light source method
6D primary sample space
paths
VPLs ofa path
power
Geometryfactor
visibility
BRDF
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VPL with error diffusion
6D primary sample space
Approximatevisibility
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VPL with error diffusion results (4D, 16 real from 420 tentative)
Classical VPL Importance resampling Error diffusion
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8D integration (equal time test)
Error diffusionClassical VPL
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Conclusions
• Delta-sigma modulation is a powerful sampling algorithm.
• In lower dimensions sampling is equivalent to the error diffusion halftoning of the importance image.
• In higher dimensions, implicit cell structure of low-discrepancy series can help to fight the curse of dimensionality.
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Open question: Optimal error shaping filter
Higher weight for faster changing coordinate