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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 Presentation
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Deterministic Importance Sampling with Error Diffusion
László Szirmay-Kalos, László Szécsi
Budapest University of Technology
Eurographics Symposium on Rendering, 2009
Numerical integration
0 1
f: integrand
g: target density
samples
Quadrature error
fg
f/g
M2
1best:
Role of
Random sampling
undersampling
oversampling
Role of
Wanted
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?
Proposed method
• Simultaneously targets– Importance sampling
• Importance function– Point samples– Cheap
– Stratification• Minimize discrepancy in the target domain
• Simple!
Sample generation: Phase 1
f
I: importance function
I
Tentative samples
Normalizationconstant: b
Sample generation: Phase 2
f
g=I/b
M2
1
G
Frequency modulator
Comparator (quantizer)
-+ Integrator
Tentative samples
Realsamples
g(i) y(i)
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:
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:
Application in higher dimensions
Importancemap
pixels
Importancemap
Application in higher dimensions
sequence neighborhood
Application in higher dimensions
Importancemap
Equivalence• Deterministic importance sampling allowing
arbitrary importance functions and minimizing the error of distribution
• Delta-Sigma modulation• Error diffusion halftoning (e.g. Floyd-Steinberg)
Environment mapping with light source sampling
v=1
v=1
lighting reflection visibility
Light source sampling = Error diffusion halftoning of the Environment Map
Error diffusion
Random sampling
Similar complexity and running times!
Light source sampling results
Random Error diffusion Reference
Light source sampling results for diffuse objects
Random Error diffusion Reference
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
Product sampling: Diffuse objectsBRDF sampling Importance resampling Error diffusion
11 sec 13 sec 13 sec
Product sampling: Specular objectsBRDF sampling Importance resampling Error diffusion
11 sec 13 sec 13 sec
Product sampling with occlusions
BRDF sampling
Importanceresampling
Error diffusion
Even higher dimensions• Regular grid: Curse of dimensionality!
• Solution: Low-discrepancy seriescurrentsample
Errordistribution
sequence ofvisiting samples
Elemental interval property
8
12
7
3
4
5
6
9
10
11
1
2
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
Virtual point light source method
6D primary sample space
paths
VPLs ofa path
power
Geometryfactor
visibility
BRDF
VPL with error diffusion
6D primary sample space
Approximatevisibility
VPL with error diffusion results (4D, 16 real from 420 tentative)
Classical VPL Importance resampling Error diffusion
8D integration (equal time test)
Error diffusionClassical VPL
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.
Open question: Optimal error shaping filter
Higher weight for faster changing coordinate