fast approximation to spherical harmonics rotation sumanta pattanaik university of central florida...

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 – Spherical Harmonics Rotation2/40

Presentation Topic

What? Rotate a spherical function represented by

spherical harmonics How?

Approximation by a truncated Taylor expansion

Why? Applications in real-time rendering and global

illumination


Unfortunate Finding

Our technique is only MARGINALLY FASTER than a previous technique.


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?

I’ll explain later.


Talk Overview

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


Talk Overview



Spherical Harmonics Basis functions on the sphere


Spherical Harmonics

)(L 00c +

22c 1

2c 02c+ + +

12c 22c+

01c 11c+ + 11c +


Spherical Harmonics

Function represented by a vector of coefficients:

Tnn

ml cccc 1

111

00][

n … order


Talk Overview



SH Rotation – Problem Definition

Given coefficients , representing a spherical

function

find coefficients for directly from coefficients .


SH Rotation Matrix

Rotation = linear transformation by R: R … spherical harmonics rotation matrix

Given the desired 3D rotation, find the matrix R

R


Talk Overview



Previous Work – Molecular Chemistry

[Ivanic and Ruedenberg 1996] [Choi et al. 1999]

Slow Bottleneck in rendering applications


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


zxzxz-decomposition [Kautz et al. 02]

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



R = RZ() RY() RZ()

Rotation around Z is simple and fast

Rotation around Y still a problem



Rotation around Y Decomposition of RY() into

RX(+90˚)

RZ()

RX(-90˚)

R = RZ() RX(+90˚) RZ() RX(-90˚) RZ() Rotation around X is fixed-angle

pre-computed matrix



Optimized implementation – unrolled code

4-6 x faster


Talk Overview



Our Rotation

1. zyz decomposition: R = RZ() RY() RZ()

2. RY() approximated by a truncated Taylor

expansion


Taylor Expansion of RY()

)0(2

)0()(2

22

d

d

d

d yyy

RRIR


Taylor Expansion of RY()

)0(2

)0()(2

22

d

d

d

d yyy

RRIR

“1.5-th order Taylor exp.” Fixed, sparse matrices


SH Rotation – Results

L2 error for a unit length input vector


Talk Overview



Application 1 – Radiance Caching Global illumination: smooth indirect term Sparse computation Interpolation


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


Radiance Caching Results


Radiance Caching Results

Direct illumination

Direct + indirect


More radiance caching: Temporal radiance caching, 3:45, room 210


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 with

the modulated normal Rotation implemented in fragment shader


Normal Mapping Results

Rotation Ignored Our Rotation


Comparison – Time per rotation (CPU)

DirectX October 2004: 1.8 x slower than Ivanic

0,00

5,00

10,00

15,00

20,00

25,00

30,00

35,00

40,00

0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

8,00

Order 6 Order 10

T [s

]

T [s

]


Talk Overview



Conclusion

Proposed approximate SH rotation Slightly faster than previous technique

SH Rotation Speed

1. Our approximation

2. DX 9.0c (up to order 6)

3. zxzxz-decomposition with unrolled code Lesson learned

Micro-optimization important for fair comparisons


Future Work

Fast approximate rotation for wavelets


Questions

?

??

?

? ??

?

Code on-line (SH rotation, radiance caching)http://moon.felk.cvut.cz/~xkrivanj/projects/rcaching


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.

fast approximation to spherical harmonics rotation sumanta pattanaik university of central florida...

Documents

d rotation

spherical harmonicshow

rza rx

rza ryb rzgrotation

vector of coefficients

ydecomposition of ryb

rzb rx

previous methodthe method