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A Novel Hemispherical Basis for Accurate and Efficient Rendering

Eurographics Symposium on Rendering 2004. 15th Eurographics Workshop on Rendering - 21-23 June, Norrköping, Sweden. A Novel Hemispherical Basis for Accurate and Efficient Rendering. P. Gautron J. K ř iv á nek S. Pattanaik K. Bouatouch. Problem Statement. BRDF. Incoming/Outgoing Radiance.

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A Novel Hemispherical Basis for Accurate and Efficient Rendering

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  1. Eurographics Symposium on Rendering 2004 15th Eurographics Workshop on Rendering - 21-23 June, Norrköping, Sweden A Novel Hemispherical Basis for Accurate and Efficient Rendering P. Gautron J. Křivánek S. Pattanaik K. Bouatouch

  2. Problem Statement BRDF Incoming/Outgoing Radiance F(, )  Sample set EGSR 2004 – Norrköping, Sweden

  3. Problem Statement  Original Function Piecewise linear approximation Need a more compact and smoothed representation Fast computation of integrals Better fitting EGSR 2004 – Norrköping, Sweden

  4. Contribution New set of basis functions Formula similar to Spherical Harmonics Designed for representing hemispherical functions Several rotation methods for projected functions Applications in lighting simulation EGSR 2004 – Norrköping, Sweden

  5. Previous work Basis functions Representation of hemispherical functions The new basis Definition Three approaches to hemispherical rotation Applications BRDF representation Environment mapping Directional radiance caching Outline EGSR 2004 – Norrköping, Sweden

  6. Outline Previous work Basis functions Representation of hemispherical functions The new basis Definition Three approaches to hemispherical rotation Applications BRDF representation Environment mapping Directional radiance caching EGSR 2004 – Norrköping, Sweden

  7. fi bi(x)  f(x) = fi = f(x)bi(x)dx fi gi gi bi(x)  g(x) = f(x)g(x)dx = Basis Functions EGSR 2004 – Norrköping, Sweden

  8. m m m (,) ()  Y K l l l (0,0) m (cos ) P l (1,-1) (1,0) (1,1) (2,-2) (2,-1) (2,0) (2,1) (2,2) Spherical Harmonics = EGSR 2004 – Norrköping, Sweden

  9. Spherical Harmonics Main Properties Simple projection and reconstruction Analytical rotations EGSR 2004 – Norrköping, Sweden

  10. SH For Hemispherical Functions Zero Hemisphere Original SH Equator discontinuity Artifacts EGSR 2004 – Norrköping, Sweden

  11. Optimization matrix Reflected Original SH SH SH SH For Hemispherical Functions Even Reflection [Westin92] Least-Squares Approximation [Sloan03] Original Avoid equator discontinuity Improve accuracy EGSR 2004 – Norrköping, Sweden

  12. R Above equator SH For Hemispherical Functions No rotation No dot product EGSR 2004 – Norrköping, Sweden

  13. No rotations No dot product SH For Hemispherical Functions Conclusion Do not fit the hemisphere Specific improvements EGSR 2004 – Norrköping, Sweden

  14. [Koenderink96] : Zernike Polynomials Accurate representation Used in CUReT BRDF Database No rotations [Makhotkin96] : Shifted Jacobi Polynomials Accurate representation Not used previously in computer graphics No rotations Hemispherical Basis Functions EGSR 2004 – Norrköping, Sweden

  15. Outline Previous work Basis functions Representation of hemispherical functions The new basis Definition Three approaches to hemispherical rotation Applications BRDF representation Environment mapping Directional radiance caching EGSR 2004 – Norrköping, Sweden

  16. m m m (,) ()  Y K = l l l (0,0) m (cos ) P l (1,-1) (1,0) (1,1) (2,-2) (2,-1) (2,0) (2,1) (2,2) Our Novel Basis Spherical Harmonics EGSR 2004 – Norrköping, Sweden

  17. Our Novel Basis Shifting EGSR 2004 – Norrköping, Sweden

  18. m m m (,) ()  H K l l l = ~ m (2cos -1) P l (0,0) (1,-1) (1,0) (1,1) (2,-2) (2,-1) (2,0) (2,1) (2,2) Our Novel Basis Hemispherical Harmonics EGSR 2004 – Norrköping, Sweden

  19. HSH Rotation 3 Methods Intuitive: conversion of HSH coefficients to SH Analytic: Comparison of SH and HSH basis functions Brute Force: Precomputation of rotation matrices EGSR 2004 – Norrköping, Sweden

  20. RSH C-1 C HSH Rotation Intuitive HSH SH R(SH) R(HSH) EGSR 2004 – Norrköping, Sweden

  21. HSH Rotation Intuitive RSH C-1 C HSH SH R(SH) R(HSH) Sparse Computed Numerically EGSR 2004 – Norrköping, Sweden

  22. HSH Rotation 3 Methods Intuitive: conversion of HSH coefficients to SH Analytic: Comparison of SH and HSH basis functions Brute Force: Precomputation of rotation matrices Reminders: Euler rotation angles Hemispherical data rotation EGSR 2004 – Norrköping, Sweden

  23. Euler’s Rotation Theorem « An arbitrary rotation may be described by only three parameters » ZYZ Angles EGSR 2004 – Norrköping, Sweden

  24. m m m m m m (,) (,) () ()   H K Y K = l l l l l l = ~ m (2cos -1) P l m (cos ) P l HSH Rotation Rotation Around Vertical Axis EGSR 2004 – Norrköping, Sweden

  25. m m m m m m (,) (,) () ()   H K Y K = l l l l l l = ~ m (2cos -1) P l m (cos ) P l HSH Rotation Rotation Around Other Axes EGSR 2004 – Norrköping, Sweden

  26. C1 x β (0,0) C2 x C3 x C4 x (1,-1) (1,0) (1,1) Deletion Matrix: projection of « cut » basis functions high frequency dense matrix computed numerically Partial Deletion Deleting vanishing part EGSR 2004 – Norrköping, Sweden

  27. βSH βHSH HSH Rotation Analytic Idea: Use SH rotation matrices HSH-projected function SH-projected function using same coefficients SH rotation Impact of SH rotation on HSH projected function βSH = arccos(2cos(βHSH)-1) EGSR 2004 – Norrköping, Sweden

  28. ≈50° x 0.5 x 0.5 40° 20° 60° 80° HSH Rotation Brute Force 50° Rotation around Y Axis ? Precomputed Rotation Matrices EGSR 2004 – Norrköping, Sweden

  29. Outline Previous work Basis functions Representation of hemispherical functions The new basis Definition Three approaches to hemispherical rotation Applications BRDF representation Environment mapping Directional radiance caching EGSR 2004 – Norrköping, Sweden

  30. Application: BRDF Representation Principle BRDF = 4D Function Parabolic Parameterization EGSR 2004 – Norrköping, Sweden

  31. Application: BRDF Representation EGSR 2004 – Norrköping, Sweden

  32. Application: BRDF Representation Accuracy Less Ringing Higher Frequency SH HSH EGSR 2004 – Norrköping, Sweden

  33. CPU CPU Rotation Conversion GPU Environment BRDF Application: Environment Mapping Principle For each vertex Additional Step EGSR 2004 – Norrköping, Sweden

  34. Application: Environment Mapping Performance Rotation on CPU for SH and HSH Added conversion (sparse matrix) Accuracy overcomes computational overhead EGSR 2004 – Norrköping, Sweden

  35. Application : Radiance Caching Irradiance Caching Scheme Goal : computation of indirect diffuse lighting   EGSR 2004 – Norrköping, Sweden

  36. Application : Radiance Caching Irradiance Caching Scheme Goal : computation of indirect diffuse lighting EGSR 2004 – Norrköping, Sweden

  37. Application : Radiance Caching Irradiance Caching Scheme Goal : computation of indirect diffuse lighting Interpolation EGSR 2004 – Norrköping, Sweden

  38. Application : Radiance Caching Goal : computation of indirect glossy lighting HSH HSH EGSR 2004 – Norrköping, Sweden

  39. Application : Radiance Caching Goal : computation of indirectglossylighting EGSR 2004 – Norrköping, Sweden

  40. Application : Radiance Caching Goal : computation of indirectglossylighting Interpolation EGSR 2004 – Norrköping, Sweden

  41. Application : Radiance Caching Goal : computation of indirectglossylighting  Incident Radiance BRDF  dot product EGSR 2004 – Norrköping, Sweden

  42. Application : Radiance Caching Results Low frequency BRDFs Rotational gradient replaced by rotation New translational gradients formulas EGSR 2004 – Norrköping, Sweden

  43. Conclusion New basis more accurate than SH 3 methods for computing rotations Easy to use in SH applications : BRDF Representation, Environment Mapping, Global Illumination More details on Radiance Caching in « Radiance Caching for Efficient Global Illumination Computation » (J. Krivanek, P. Gautron, S. Pattanaik, K. Bouatouch) IRISA Technical Report #1623 EGSR 2004 – Norrköping, Sweden

  44. Analytic formulas for SH HSH Conversion Matrix HSH Rotation Matrices Improve Radiance Caching Hardware Interactive Global Illumination Perspectives EGSR 2004 – Norrköping, Sweden

  45. Any Questions ? Rendered using Radiance Caching EGSR 2004 – Norrköping, Sweden

  46. Papers Download A Novel Hemispherical Basis for Accurate and Efficient Rendering Radiance Caching for Efficient Global Illumination Computation http://www.cgg.cvut.cz/~xkrivanj/papers/index.htm EGSR 2004 – Norrköping, Sweden

  47. BRDF Representation Accuracy Phong BRDF EGSR 2004 – Norrköping, Sweden

  48. BRDF Representation Accuracy Anisotropic Ward BRDF EGSR 2004 – Norrköping, Sweden

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