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High-Order Similarity Relations in Radiative Transfer

High-Order Similarity Relations in Radiative Transfer. Shuang Zhao 1 , Ravi Ramamoorthi 2 , and Kavita Bala 1 1 Cornell University 2 University of California, San Diego. Translucency is everywhere. Food. Skin. Architecture. Jewelry. Slide courtesy of Ioannis Gkioulekas.

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High-Order Similarity Relations in Radiative Transfer

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  1. High-Order Similarity Relations in Radiative Transfer Shuang Zhao1, Ravi Ramamoorthi2, and Kavita Bala1 1Cornell University2University of California, San Diego

  2. Translucency is everywhere Food Skin Architecture Jewelry Slide courtesy of IoannisGkioulekas

  3. Rendering translucency Appearance Radiativetransfer Scatteringparam.

  4. Rendering translucency Appearance 1 Appearance 1 ≈ ≠ Radiativetransfer Radiativetransfer Radiativetransfer Radiativetransfer Appearance 2 Appearance 2 Scatteringparam. 2 Scatteringparam. 1 Scatteringparam. 1 Scatteringparam. 2

  5. First-order methods Approx. identical appearance First-order approx. Limitedaccuracy Cheaper to render [Wang et al. 2009] Scatteringparam. 1 Scatteringparam. 2 Scatteringparam. 1 Scatteringparam. 2 [Arbree et al. 2011] [Frisvad et al. 2007] [Jensen et al. 2001]

  6. Similarity theory First-order approx. First-order approx. First-ordermethods Similarityrelations Similaritytheory [Wyman et al. 1989] Scatteringparam. 1 Scatteringparam. 2 Scatteringparam. 1 Scatteringparam. 2 Scatteringparam. 1 Scatteringparam. 2

  7. Similarity theory Provide fundamental insights into thestructure of material parameter space Similarityrelations Similaritytheory [Wyman et al. 1989] Scatteringparam. 1 Scatteringparam. 2

  8. Similarity theory Originates in applied optics[Wyman et al. 1989] Similar ideas explored in neutron transfer (Condensed History Monte Carlo)[Prinja & Franke 2005], [Bhan& Spanier 2007], … Similarityrelations Similaritytheory [Wyman et al. 1989] Scatteringparam. 1 Scatteringparam. 2

  9. Our contribution Introducing high-order similarity theory tocomputer graphics Novel algorithmsbenefiting forward &inverse rendering

  10. Our contribution: forward rendering User-specified (balancing performance and accuracy) Betteraccuracy Approx. identical appearance Our approach Cheaper to render Scatteringparam. 2 Scatteringparam. 1 100 ~ 200 lines of MATLAB code Up to 10X speedup

  11. Our contribution: inverse rendering Gradient descent methods perform much better Reparameterize Parameter space 2 Parameter space 1

  12. Background

  13. Material scattering parameters Extinction coefficient Light particle Absorption coefficient Scattering coefficient Interaction Scattered Absorbed Phase function

  14. Phase function Scattered Probability density for , parameterized as Forward Forward Isotropic scattering Forward scattering

  15. Similarity Theory

  16. Phase function moments nth Legendremoment Legendrepolynomial For a phase function “Average cosine”

  17. Similarity relations Low-frequency radiance Band-limited up to order-N in spherical harmonics domain … [Wyman et al. 1989]

  18. Similarity relations Radiance low-frequencyeverywhere identical appearance Order-N similarity relation Derivationin the paper [Wyman et al. 1989] … …

  19. Similarity relations Approximately Radiance low-frequencyeverywhere identical appearance Higher order,Better accuracy Order-N similarity relation …

  20. Challenge ? ? … Order-N similarity relation Order-N similarity relation Original(given) Altered(unknown) …

  21. Solving forAltered Parameters

  22. The problem Original parameters Constraints ? ? ? Altered parameters Forward Order-Nsimilarity relation …

  23. The problem Original parameters ? ? Altered parameters Forward Order-Nsimilarity relation Order-Nsimilarity relation … … …

  24. The problem Original parameters ? ? Altered parameters Forward Order-Nsimilarity relation … …

  25. Altered phase function Original parameters ? ? Altered parameters Altered parameters Forward Remainingunknown … …

  26. Altered phase function Original parameters Legendre moments of ? Altered parameters Forward Remainingunknown Legendre moments of …

  27. Altered phase function Normalizationconstraint ? ? Altered parameters Altered parameters Order-1 Order-2 Order-3 Order-4 … Finding highestsatisfiableorder N

  28. Finding order N Existence condition • (Truncated Hausdorff moment problem) • [Curto and Fialkow 1991] Does phase function exist? Given desired Legendre moments Hankel matrices built using arepositive semi-definite Phase function exists

  29. Finding order N ? Altered parameters Order-1 Order-2 Order-3 Order-4 … Finding highestsatisfiableorder N

  30. Altered phase function ? Altered parameters Order-3 Order-3 Problem: not uniquely specified Invalid Valid Valid

  31. Constructing altered phase function Represent as a tabulatedfunction with pieces ? … … has Legendre moments 0 0 -1 -1 1 1 Need: non-negative

  32. Constructing altered phase function Represent as a tabulatedfunction with pieces ? Const. Need:

  33. Constructing altered phase function subject to Solve Smoothness term (favoring “uniform” solutions) Good Bad 0 0 -1 -1 1 1

  34. Constructing altered phase function subject to Solve Quadratic programming • Standard problem • Solvable with many tools/libraries • MATLAB, Gurobi, CVXOPT, … • Our MATLAB code is available online

  35. Constructing altered phase function ? Altered parameters Our approach Order-3 Invalid Valid Valid

  36. Constructing altered phase function Altered parameters Forward

  37. Summary Original parameters Compute order N Solve optimization Forward Forward Forward Altered parameters Altered parameters

  38. Application:Forward Rendering

  39. Our contribution: forward rendering User-specified (balancing performance and accuracy) Effort-freespeedups! Betteraccuracy Approx. identical appearance Our approach Cheaper to render Scatteringparam. 2 Scatteringparam. 1

  40. Application: forward rendering 0 1 large small No changein parameters Worse accuracy Greater speedup Better accuracy Lower speedup Perform test renderings to find optimal Reuse for high-resolution renderings or videos is a good start

  41. Experimental Results

  42. Performance vs. accuracy Reference (350 min) α = 0.05 (44 min, 8.0X) 30% Relative error 0%

  43. Performance vs. accuracy Reference (350 min) α = 0.05 (44 min, 8.0X) α = 0.10 (63 min, 5.6X) 30% 30% Relative error Relative error 0% 0%

  44. Performance vs. accuracy Visually identical α = 0.20 (103 min, 3.4X) α = 0.30 (126 min, 2.8X) α = 0.10 (63 min, 5.6X) α = 0.10 (63 min, 5.6X) 30% Relative error Relative error Relative error Relative error 0%

  45. Power of high-order relations Original parameters Altered parameters(Order-1) Forward Forward Used by first-order methods: Reduced scatteringcoefficient Satisfies order-1similarity relation

  46. Power of high-order relations Altered parameters(Order-3) Original parameters Altered parameters(Order-1) Forward Forward Forward

  47. Power of high-order relations Altered parameters(Order-1) Altered parameters(Order-3) Original parameters Original parameters Altered parameters(Order-1) Altered parameters(Order-3) 119 min (3.6X) 115 min (3.7X) 426 min (reference)

  48. More renderings Equal-time Equal-sample Reference23 min Ours20 min Ours178 min (2.7X) Reference473 min

  49. Conclusion ? … Order-N similarity relation Original Altered Introducing high-ordersimilarity relations to graphics Proposing a practical algorithmto solve for altered parameters

  50. Future work • Picking automatically and adaptively • Alternative versions of similarity theory

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