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

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

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

4. 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

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

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

7. EGSR 2004 – Norrköping, Sweden 7 Basis Functions f i = f(x)b i (x)dx  f(x) = fifi b i (x) g(x) = gigi b i (x)  f(x)g(x)dx = fifi gigi

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

26. EGSR 2004 – Norrköping, Sweden 26 Partial Deletion β Deleting vanishing part (0,0) C 1 x (1,-1) C 2 x (1,0) C 3 x (1,1) C 4 x Deletion Matrix : projection of « cut » basis functions computed numerically high frequency dense matrix

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

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

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

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

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

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

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

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

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

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

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

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

39. EGSR 2004 – Norrköping, Sweden 39 Application : Radiance Caching Goal : computation of indirect glossy lighting

40. EGSR 2004 – Norrköping, Sweden 40 Application : Radiance Caching Interpolation Goal : computation of indirect glossy lighting 

41. EGSR 2004 – Norrköping, Sweden 41 Application : Radiance Caching Incident RadianceBRDF  dot product  Goal : computation of indirect glossy lighting

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

43. 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

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

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

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

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

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