Fast Approximation to Spherical Harmonics Rotation

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Fast Approximation to Spherical Harmonics Rotation
Jaroslav Krivánek Czech Technical University
Jaakko Konttinen University of Central Florida
Sumanta Pattanaik University of Central Florida
Kadi Bouatouch IRISA / INRIA Rennes
Jirí Žára Czech Technical University
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Presentation Topic

• What?
• Rotate a spherical function represented
  byspherical harmonics
• How?
• Approximation by a truncated Taylor expansion
• Why?
• Applications in real-time rendering and global
  illumination

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Unfortunate Finding

• Our technique is only MARGINALLY FASTER than a
  previous technique.

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Questions You Might Want to Ask

• Q1 So why taking up a SIGGRAPH sketch slot?
• Found out only very recently.
• Q2 And before?
• Implementation of previous work was NOT
  OPTIMIZED.
• Q3 Does optimization change that much?
• In this case, it does (4-6 times speedup).
• Q4 How did you find out?
• Ill explain later.

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Talk Overview

• Spherical Harmonics
• Spherical Harmonics Rotation
• Previous Techniques
• Our Rotation Approximation
• Applications Results
• Conclusions

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Talk Overview

• Spherical Harmonics
• Spherical Harmonics Rotation
• Previous Techniques
• Our Rotation Approximation
• Applications Results
• Conclusions

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Spherical Harmonics

• Basis functions on the sphere

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Spherical Harmonics

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Spherical Harmonics
Function represented by a vector of coefficients
n order
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Talk Overview

• Spherical Harmonics
• Spherical Harmonics Rotation
• Previous Techniques
• Our Rotation Approximation
• Applications Results
• Conclusions

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SH Rotation Problem Definition

• Given coefficients ?, representing a
  sphericalfunction
• find coefficients ? for directly from
  coefficients ?.

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SH Rotation Matrix

• Rotation linear transformation by R
• R spherical harmonics rotation matrix
• Given the desired 3D rotation, find the matrix R

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Talk Overview

• Spherical Harmonics
• Spherical Harmonics Rotation
• Previous Techniques
• Our Rotation Approximation
• Applications Results
• Conclusions

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Previous Work Molecular Chemistry

• Ivanic and Ruedenberg 1996
• Choi et al. 1999
• Slow
• Bottleneck in rendering applications

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Previous Work Computer Graphics

• Kautz et al. 2002
• zxzxz-decomposition
• THE fastest previous method
• THE method we compare against
• THE method nearly as fast as ours
• if properly optimized

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zxzxz-decomposition Kautz et al. 02

• Decompose the 3D rotation into ZYZ Euler angles
  R RZ(a) RY(b) RZ(g)

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zxzxz-decomposition Kautz et al. 02

• R RZ(a) RY(b) RZ(g)
• Rotation around Z is simple and fast
• Rotation around Y still a problem

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zxzxz-decomposition Kautz et al. 02

• Rotation around Y
• Decomposition of RY(b) into
• RX(90)
• RZ(b)
• RX(-90)
• R RZ(a) RX(90) RZ(b) RX(-90) RZ(g)
• Rotation around X is fixed-angle
• pre-computed matrix

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zxzxz-decomposition Kautz et al. 02

• Optimized implementation unrolled code

• 4-6 x faster

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Talk Overview

• Spherical Harmonics
• Spherical Harmonics Rotation
• Previous Techniques
• Our Rotation Approximation
• Applications Results
• Conclusions

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Our Rotation

• zyz decomposition R RZ(a) RY(b) RZ(g)
• RY(b) approximated by a truncated Taylor
  expansion

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Taylor Expansion of RY(b)
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Taylor Expansion of RY(b)

• 1.5-th order Taylor exp.
• Fixed, sparse matrices

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SH Rotation Results

• L2 error for a unit length input vector

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Talk Overview

• Spherical Harmonics
• Spherical Harmonics Rotation
• Previous Techniques
• Our Rotation Approximation
• Applications Results
• Conclusions

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Application 1 Radiance Caching

• Global illumination smooth indirect term
• Sparse computation
• Interpolation

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Incoming Radiance Interpolation

• Interpolate coefficient vectors ?1 and ?2

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Interpolation on Curved Surfaces
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Interpolation on Curved Surfaces

• Align coordinate frames in interpolation

R
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Radiance Caching Results
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Radiance Caching Results
Direct illumination
Direct indirect
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More radiance caching Temporal radiance caching,
345, room 210
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Application 2 Normal Mapping

• Original method by Kautz et al. 2002
• Environment map
• Arbitrary BRDF
• Extended with normal mapping
• Needs per-pixel rotation to align withthe
  modulated normal
• Rotation implemented in fragment shader

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Normal Mapping Results
Rotation Ignored
Our Rotation
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Normal Mapping Results
Rotation Ignored
Our Rotation
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Comparison Time per rotation (CPU)
Order 6
Order 10
T ms
T ms

• DirectX October 2004 1.8 x slower than Ivanic

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Talk Overview

• Spherical Harmonics
• Spherical Harmonics Rotation
• Previous Techniques
• Our Rotation Approximation
• Applications Results
• Conclusions

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Conclusion

• Proposed approximate SH rotation
• Slightly faster than previous technique
• SH Rotation Speed
• Our approximation
• DX 9.0c (up to order 6)
• zxzxz-decomposition with unrolled code
• Lesson learned
• Micro-optimization important for fair comparisons

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Future Work

• Fast approximate rotation for wavelets

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Questions
Code on-line (SH rotation, radiance
caching) http//moon.felk.cvut.cz/xkrivanj/projec
ts/rcaching
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Appendix Bibliography

• Krivánek et al. 2005 Jaroslav Krivánek, 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)63426347,
  1996.Joseph Ivanic and Klaus Ruedenberg.
  Additions and corrections Rotation matrices for
  real spherical harmonics. J. Phys. Chem. A,
  102(45)90999100, 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)88258831, 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 291296. Eurographics
  Association, 2002.