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 ára Czech Technical University
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Presentation Topic

• Goal
• Rotate a spherical function represented
  bySpherical Harmonics
• Proposed method
• Approximation through a truncated Taylor
  expansion

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Spherical Harmonics

• Basis functions on the sphere

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Spherical Harmonics

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Spherical Harmonics
Image Robin Green, Sony computer Entertainment
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Spherical Harmonics
represented by a vector of coefficients
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Spherical Harmonics

• Basis functions on the sphere

l 0
l 1
l 2
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SH Rotation Problem Definition

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

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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)

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Talk Overview

• SH rotation
• Previous Work
• Our Rotation
• Application in global illumination
• Application in real-time shading
• Conclusions

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Talk Overview

• SH rotation
• Previous Work
• Our Rotation
• Application in global illumination
• Application in real-time shading
• 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

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SH Rotation

• Given the desired 3D rotation, find the matrix R

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Talk Overview

• SH rotation
• Previous Work
• Our Rotation
• Application in global illumination
• Application in real-time shading
• Conclusions

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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

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Previous Work Computer Graphics

• Kautz et al. 2002
• zxzxz-decomposition
• By far the fastest previous method

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Previous Work Summary

• O(n3) complexity
• Slow
• Bottleneck in rendering applications

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Talk Overview

• SH rotation
• Previous Work
• Our Rotation
• Application in global illumination
• Application in real-time shading
• Conclusions

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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

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Rotation Decomposition

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

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Rotation Decomposition

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

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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

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Rotation Around Y Our Approach

• Second order truncated Taylor expansion of RY(b)

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Taylor Expansion of RY(b)
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Rotation Procedure Taylor Expansion
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Rotation Procedure Taylor Expansion

• 1.5-th order Taylor expansion
• Very sparse matrix

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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

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SH Rotation Results

• L2 error for a unit length input vector

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Talk Overview

• SH rotation
• Previous Work
• Our Rotation
• Application in global illumination
• Application in real-time shading
• Conclusion

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Application in GI - Radiance Caching

• Sparse computation of indirect illumination
• Interpolation
• Enhanced with gradients

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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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Results in Radiance Caching
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Results in Radiance Caching
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Talk Overview

• SH rotation
• Previous Work
• Our Rotation
• Application in global illumination
• Application in real-time shading
• Conclusion

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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
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GPU-based Real-time Shading (contd.)

• must be rotated from global to local
  frame
• zxzxz - rotation too complicated ? on CPU

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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

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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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Normal Mapping Results
Rotation Ignored
Our Rotation
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Talk Overview

• SH rotation
• Previous Work
• Our Rotation
• Application in global illumination
• Application in real-time shading
• Conclusion

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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

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Thank You for your Attention
?
?
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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.