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

2. Presentation Topic Goal
Rotate a spherical function represented by Spherical 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 spherical function
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 spherical function
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. 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. Interpolation on Curved Surfaces
Align coordinate frames in interpolation p1 R p
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, Joseph Ivanic and Klaus Ruedenberg. Additions and corrections : Rotation matrices for 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 and stable determination of rotation matrices between spherical harmonics by direct recursion. 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