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Fast Approximation to Spherical Harmonics Rotation

Computer. Graphics. Group. Fast Approximation to Spherical Harmonics Rotation. Jaroslav K řivánek Czech Technical University. Ja akko Konttinen University of Central Florida. Jiří Žára Czech Technical University. Sumanta Pattanaik University of Central Florida. Kadi Bouatouch

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Fast Approximation to Spherical Harmonics Rotation

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  1. Computer Graphics Group Fast Approximation to Spherical Harmonics Rotation Jaroslav Křivánek Czech Technical University Jaakko Konttinen University of Central Florida Jiří Žára Czech Technical University Sumanta Pattanaik University of Central Florida Kadi Bouatouch IRISA / INRIA Rennes

  2. Presentation Topic • Goal Rotate a spherical function represented bySpherical Harmonics • Proposed method Approximation through a truncated Taylor expansion Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  3. Spherical Harmonics • Basis functions on the sphere Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  4. + + + + + + + Spherical Harmonics + Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  5. Spherical Harmonics Image Robin Green, Sony computer Entertainment Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  6. Spherical Harmonics represented by a vector of coefficients: Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  7. Spherical Harmonics • Basis functions on the sphere l = 0 l = 1 l = 2 Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  8. SH Rotation – Problem Definition • Given coefficients , representing a sphericalfunction • find coefficients  for directly from coefficients . Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  9. Our Contribution • Novel, fast, approximate rotation • Based on a truncated Taylor Expansion of the SH rotation matrix • 4-6 times faster than [Kautz et al. 2002] • O(n2) complexity instead of O(n3) • Two applications • Global illumination (radiance interpolation) • Real-time shading (normal mapping) Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  10. Talk Overview • SH rotation • Previous Work • Our Rotation • Application in global illumination • Application in real-time shading • Conclusions Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  11. Talk Overview • SH rotation • Previous Work • Our Rotation • Application in global illumination • Application in real-time shading • Conclusions Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  12. SH Rotation – Problem Definition • Given coefficients , representing a sphericalfunction • find coefficients  for directly from coefficients . Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  13. SH Rotation Matrix • Rotation = linear transformation: Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  14. SH Rotation • Given the desired 3D rotation, find the matrix R Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  15. Talk Overview • SH rotation • Previous Work • Our Rotation • Application in global illumination • Application in real-time shading • Conclusions Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  16. Previous Work – Molecular Chemistry • [Ivanic and Ruedenberg 1996] • Recurrent relations: Rl = f(R1,Rl-1) • [Choi et al. 1999] • Through complex spherical harmonics • Fast for complex harmonics • Slow conversion to the real form Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  17. Previous Work – Computer Graphics • [Kautz et al. 2002] • zxzxz-decomposition • By far the fastest previous method Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  18. Previous Work – Summary • O(n3) complexity • Slow • Bottleneck in rendering applications Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  19. Talk Overview • SH rotation • Previous Work • Our Rotation • Application in global illumination • Application in real-time shading • Conclusions Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  20. Our Rotation • Fast, approximate rotation • Based on replacing the SH rotation matrix by its Taylor expansion • 4-6 times faster than [Kautz et al. 2002] Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  21. Rotation Decomposition • Decompose the 3D rotation into ZYZ Euler angles: R = RZ(a) RY(b) RZ(g) Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  22. Rotation Decomposition • R = RZ(a) RY(b) RZ(g) • Rotation around Z is simple and fast • Rotation around Y still a problem Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  23. Rotation Around Y • [Kautz et al. 2002] • Decomposition of Y into X(+90˚), Z, and X(-90˚) • R = RZ(a) RX(+90˚) RZ(b) RX(-90˚) RZ(g) • Rotation around Z is simple and fast • Rotation around X is fixed-angle • can be tabulated • The RXRZRX-part can still be improved… Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  24. Rotation Around Y – Our Approach • Second order truncated Taylor expansion of RY(b) Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  25. Taylor Expansion of RY(b) Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  26. Rotation Procedure – Taylor Expansion Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  27. Rotation Procedure – Taylor Expansion • “1.5-th order Taylor expansion” • Very sparse matrix Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  28. Full Rotation Procedure • Decompose the 3D rotation into ZYZ Euler angles: R = RZ(a) RY(b) RZ(g) • Rotate around Z by a • Use the “1.5-th order” Taylor expansion to rotate around Y by b • Rotate around Z by g Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  29. SH Rotation – Results • L2 error for a unit length input vector Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  30. Talk Overview • SH rotation • Previous Work • Our Rotation • Application in global illumination • Application in real-time shading • Conclusion Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  31. Application in GI - Radiance Caching • Sparse computation of indirect illumination • Interpolation • Enhanced with gradients Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  32. p1 p2 p Incoming Radiance Interpolation • Interpolate coefficient vectors 1 and 2 Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  33. Interpolation on Curved Surfaces Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  34. p p1 Interpolation on Curved Surfaces • Align coordinate frames in interpolation R Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  35. Results in Radiance Caching Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  36. Results in Radiance Caching Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  37. Talk Overview • SH rotation • Previous Work • Our Rotation • Application in global illumination • Application in real-time shading • Conclusion Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  38. GPU-based Real-time Shading • Original method by [Kautz et al. 2002] • Arbitrary BRDFs • represented by SH in the local coordinate frame • Environment Lighting • represented by SH in the global coordinate frame  ( ) Lout = Incident Radiance BRDF = coeff. dot product Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  39. GPU-based Real-time Shading (contd.) • must be rotated from global to local frame • zxzxz - rotation too complicated  on CPU Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  40. Our Extension – Normal Mapping • Normal modulated by a texture • Our rotation approximation • Rotation from the un-modulated to the modulated coordinate frame • Small rotation angle  good accuracy Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  41. Normal Mapping Results Rotation Ignored Our Rotation Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  42. Normal Mapping Results Rotation Ignored Our Rotation Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  43. Normal Mapping Results Rotation Ignored Our Rotation Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  44. Talk Overview • SH rotation • Previous Work • Our Rotation • Application in global illumination • Application in real-time shading • Conclusion Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  45. Conclusion and Future Work • Summary • Fast, approximate rotation • Truncated Taylor Expansion of the SH rotation matrix • 4-6 times faster than [Kautz et al. 2002] • O(n2) complexity instead of O(n3) • Applications in global illumination and real-time shading • Future Work • Rotation for Wavelets • Normal mapping for pre-computed radiance transfer Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  46. Thank You for your Attention ? ? Jaroslav Křivánek – Fast Spherical Harmonics Rotation

  47. Appendix – Bibliography • [Křivánek et al. 2005] Jaroslav Křivánek, Pascal Gautron, Sumanta Pattanaik, and Kadi Bouatouch. Radiance caching for efficient global illumination computation. IEEE Transactions on Visualization and Computer Graphics, 11(5), September/October 2005. • [Ivanic and Ruedenberg 1996]Joseph Ivanic and Klaus Ruedenberg. Rotation matrices for real spherical harmonics.direct determination by recursion. J. Phys. Chem., 100(15):6342–6347, 1996.Joseph Ivanic and Klaus Ruedenberg. Additions and corrections : Rotation matricesfor real spherical harmonics. J. Phys. Chem. A, 102(45):9099–9100, 1998. • [Choi et al. 1999]Cheol Ho Choi, Joseph Ivanic, Mark S. Gordon, and Klaus Ruedenberg. Rapid andstable determination of rotation matrices between spherical harmonics by directrecursion. J. Chem. Phys., 111(19):8825–8831, 1999. • [Kautz et al. 2002] Jan Kautz, Peter-Pike Sloan, and John Snyder. Fast, arbitrary BRDF shading for low-frequency lighting using spherical harmonics. In Proceedings of the 13th Eurographics workshop on Rendering, pages 291–296. Eurographics Association, 2002. Jaroslav Křivánek – Fast Spherical Harmonics Rotation

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