Fast Approximation to Spherical Harmonics Rotation

Fast Approximation to Spherical Harmonics Rotation

Sumanta Pattanaik

University of Central Florida

Kadi Bouatouch

IRISA / INRIA Rennes

Jaroslav Křivánek

Czech Technical University

Jaakko Konttinen

University of Central Florida

Jiří Žára

Czech Technical University

ComputerGraphicsGroup

Jaroslav Křivánek – Fast Spherical Harmonics Rotation2/45

Presentation Topic Goal

Rotate a spherical function represented bySpherical Harmonics

Proposed method

Approximation through a truncated Taylor expansion


Spherical Harmonics Basis functions on the sphere


Spherical Harmonics

)(L 00c +

22c 1

2c 02c+ + +

12c 22c+

01c 11c+ + 11c +


Spherical Harmonics

Image Robin Green,Sony computer Entertainment


Spherical Harmonics

represented by a vector of coefficients:

Tnn

ml cccc 1

111

00][

1

0

)()(n

l

lm

lm

ml

ml YcL


Spherical Harmonics Basis functions on the sphere

l = 0

l = 1

l = 2


SH Rotation – Problem Definition

Given coefficients , representing a spherical

function

find coefficients for directly from coefficients .


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)


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




SH Rotation – Problem Definition

Given coefficients , representing a spherical

function

find coefficients for directly from coefficients .


SH Rotation Matrix

Rotation = linear transformation: R0R

1R

2R

0l 1l 2l 3l


SH Rotation

Given the desired 3D rotation, find the matrix R

R




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


Previous Work – Computer Graphics

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


Previous Work – Summary

O(n3) complexity Slow Bottleneck in rendering applications




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]


Rotation Decomposition

Decompose the 3D rotation into ZYZ Euler angles: R = RZ() RY() RZ()


Rotation Decomposition

R = RZ() RY() RZ()

Rotation around Z is simple and fast

Rotation around Y still a problem


Rotation Around Y

[Kautz et al. 2002] Decomposition of Y into X(+90˚), Z, and X(-

90˚) R = RZ() RX(+90˚) RZ() RX(-90˚) RZ()

Rotation around Z is simple and fast Rotation around X is fixed-angle

can be tabulated

The RXRZRX-part can still be improved…


Rotation Around Y – Our Approach

Second order truncated Taylor expansion of RY()


Taylor Expansion of RY()

)0(2

)0()(2

22

d

d

d

d yyy

RRIR


Rotation Procedure – Taylor Expansion

)0(2

)0()(2

22

d

d

d

d yyy

RRIR


Rotation Procedure – Taylor Expansion

)0(2

)0()(2

22

d

d

d

d yyy

RRIR

“1.5-th order Taylor expansion”

Very sparse matrix


Full Rotation Procedure

1. Decompose the 3D rotation into ZYZ Euler angles: R = RZ() RY() RZ()

2. Rotate around Z by

3. Use the “1.5-th order” Taylor expansion to rotate around Y by

4. Rotate around Z by


SH Rotation – Results L2 error for a

unit length input vector


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


Application in GI - Radiance Caching Sparse computation of indirect illumination Interpolation Enhanced with gradients


Incoming Radiance Interpolation

Interpolate coefficient vectors 1 and 2

p1

p2p


Interpolation on Curved Surfaces


Interpolation on Curved Surfaces Align coordinate frames in interpolation

p

p1

R


Results in Radiance Caching


Results in Radiance Caching




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

Incident Radiance BRDF

= coeff. dot product( )Lout =


GPU-based Real-time Shading (contd.)

must be rotated from global to local frame

zxzxz - rotation too complicated on CPU


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


Normal Mapping Results

Rotation Ignored Our Rotation










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


Thank You for your Attention

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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 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.

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Fast Approximation to Spherical Harmonics Rotation