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Deterministic Importance Sampling with Error Diffusion

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.

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Deterministic Importance Sampling with Error Diffusion

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  1. Deterministic Importance Sampling with Error Diffusion László Szirmay-Kalos, László Szécsi Budapest University of Technology Eurographics Symposium on Rendering, 2009

  2. Numerical integration f: integrand g: target density 1 0 samples

  3. Quadrature error f/g f best: g

  4. Role of undersampling oversampling Random sampling

  5. Role of Wanted

  6. 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?

  7. Proposed method • Simultaneously targets • Importance sampling • Importance function • Point samples • Cheap • Stratification • Minimize discrepancy in the target domain • Simple!

  8. Sample generation: Phase 1 f I I: importance function Normalization constant: b Tentative samples

  9. Sample generation: Phase 2 G f g=I/b

  10. Frequency modulator Comparator (quantizer) Tentative samples Real samples Integrator  y(i) g(i) + -

  11. Frequency domain analysis White noise: n(i) Tentative samples Real samples Integrator  y(i) g(i) + - Transfer function in the Z-transform domain: Delay Light-blue noise

  12. Delta-Sigma modulator:Noise-Shaping Feedback Coder Tentative samples Real samples quantizer y(i) g(i) g(i) + + + Noise shaping filter H(z) - Transfer function in the Z-transform domain: Controllable blue noise No delay

  13. Application in higher dimensions pixels Importance map

  14. Application in higher dimensions Importance map

  15. Application in higher dimensions Importance map neighborhood sequence

  16. Equivalence • Deterministic importance sampling allowing arbitrary importance functions and minimizing the error of distribution • Delta-Sigma modulation • Error diffusion halftoning (e.g. Floyd-Steinberg)

  17. Environment mapping with light source sampling v=1 v=1 lighting reflection visibility

  18. Light source sampling = Error diffusion halftoning of the Environment Map Error diffusion Similar complexity and running times! Random sampling

  19. Light source sampling results Random Error diffusion Reference

  20. Light source sampling results for diffuse objects Random Error diffusion Reference

  21. Environment mapping with product sampling visibility lighting  reflection • Separate importance map for every shaded point • Computational cost ???: • Similar to importance re-sampling • Negligible overhead more complex scenes

  22. Product sampling: Diffuse objects BRDF sampling Error diffusion Importance resampling 11 sec 13 sec 13 sec

  23. Product sampling: Specular objects BRDF sampling Error diffusion Importance resampling 11 sec 13 sec 13 sec

  24. Product sampling with occlusions BRDF sampling Error diffusion Importance resampling

  25. Even higher dimensions  • Regular grid: Curse of dimensionality! • Solution: Low-discrepancy series current sample sequence of visiting samples Error distribution

  26. 8 5 11 2 7 4 10 1 6 12 3 9 Elemental interval property

  27. 8 5 11 2 7 4 10 1 6 12 3 9 The algorithm in d-dimensions + normalization constant b 11, I(u11) 5, I(u5) 2, I(u2) 8, I(u8) 7, I(u7) 1, I(u1) 10, I(u10) 4, I(u4) 3, I(u3) 9, I(u9) 6, I(u6) 12, I(u12) d-dimensional cube d-dimensional array

  28. Virtual point light source method paths power visibility VPLs of a path BRDF Geometry factor 6D primary sample space

  29. VPL with error diffusion Approximate visibility 6D primary sample space

  30. VPL with error diffusion results (4D, 16 real from 420 tentative) Classical VPL Error diffusion Importance resampling

  31. 8D integration (equal time test) Error diffusion Classical VPL

  32. 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.

  33. Open question: Optimal error shaping filter Higher weight for faster changing coordinate

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