Sparse zonal harmonic factorization for efficient SH rotation

Abstract

We present a sparse analytic representation for spherical functions, including those expressed in a Spherical Harmonic (SH) expansion, that is amenable to fast and accurate rotation on the GPU. Exploiting the fact that each band-l SH basis function can be expressed as a weighted sum of 2l + 1 rotated band-l Zonal Harmonic (ZH) lobes, we develop a factorization that significantly reduces this number. We investigate approaches for promoting sparsity in the change-of-basis matrix, and also introduce lobe sharing to reduce the total number of unique lobe directions used for an order-N expansion from N² to 2N-1. Our representation does not introduce approximation error, is suitable for any type of spherical function (e.g., lighting or transfer), and requires no offline fitting procedure; only a (sparse) matrix multiplication is required to map to/from SH. We provide code for our rotation algorithms, and apply them to several real-time rendering applications.

References

1. Edmonds, A. 1960. Angular Momentum in Quantum Mechanics. Princeton University Press.
2. Habel, R., Mustata, B., and Wimmer, M. 2008. Efficient spherical harmonics lighting with the preetham skylight model. In Proceedings of Eurographics Short Papers.
3. Kajiya, J. T. and Von Herzen, B. P. 1984. Ray tracing volume densities. In Proceedings of the 11th Annual Conference on Computer Graphics and Interactive Techniques. ACM, New York.
4. Kautz, J., Sloan, P.-P., and Snyder, J. 2002. Fast, arbitrary BRDF shading for low-frequency lighting using spherical harmonics. In Proceedings of the 13th Eurographics Workshop on Rendering (EGRW'02). Eurographics Association.
5. Kazhdan, M. 2007. An approximate and efficient method for optimal rotation alignment of 3d models. IEEE Trans. Pattern Anal. Mach. Intell. 29.
6. Křivánek, J., Konttinen, J., Bouatouch, K., Pattanaik, S., and Žára, J. 2006. Fast approximation to spherical harmonics rotation. In ACM SIGGRAPH Sketches. ACM, New York.
7. Lacroix, N. H. J. 1984. On common zeros of Legendre's associated functions. Math. Comput. 43, 167.
8. Lessig, C., deWitt, T., and Fiume, E. 2010. Efficient and stable rotation of finite spherical harmonics expansions. Tech. rep., University of Toronto, http://www.dgp.toronto.edu/~lessig/shrk/.
9. Lisle, I. G. and Huang, S.-L. T. 2007. Algorithms for spherical harmonic lighting. In Proceedings of the GRAPHITE'07 Conference. ACM, New York.
10. Mayaux, B. 2010. Saint Jean Cathedral - O3D web demo. http://www. patapom.com/O3D/Cathedral.html.
11. Michalewicz, Z. 1998. Genetic Algorithms + Data Structures = Evolution Programs, 3rd Ed. Springer.
12. Nowrouzezahrai, D. and Snyder, J. 2009. Fast global illumination on dynamic height fields. In Proceedings of the Eurographics Symposium on Rendering.
13. Ramamoorthi, R. and Hanrahan, P. 2002. Frequency space environment map rendering. In Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques. ACM, New York.
14. Ren, Z., Wang, R., Snyder, J., Zhou, K., Liu, X., Sun, B., Sloan, P.-P., Bao, H., Peng, Q., and Guo, B. 2006. Real-Time soft shadows in dynamic scenes using spherical harmonic exponentiation. In Proceedings of the SIGGRAPH'06 Conference. ACM, New York.
15. Sloan, P.-P., Govindaraju, N. K., Nowrouzezahrai, D., and Snyder, J. 2007. Image-Based proxy accumulation for real-time soft global illumination. In Proceedings of the 15th Pacific Conference on Computer Graphics and Applications. IEEE, Los Alamitos, CA.
16. Sloan, P.-P., Kautz, J., and Snyder, J. 2002. Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments. In Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques. ACM, New York.
17. Sloan, P.-P., Luna, B., and Snyder, J. 2005. Local, deformable precomputed radiance transfer. In ACM SIGGRAPH Papers. ACM, New York.
18. Tsai, Y.-T. and Shih, Z.-C. 2006. All-Frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation. In Proceedings of the SIGGRAPH'06 Conference. ACM, New York.
19. Wang, R., Ng, R., Luebke, D., and Humphreys, G. 2006. Efficient wavelet rotation for environment map rendering. In Proceedings of the Eurographics Symposium on Rendering. Springer.
20. Westin, S. H., Arvo, J. R., and Torrance, K. E. 1992. Predicting reflectance functions from complex surfaces. In Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques. ACM, New York.

Index Terms

• Computing methodologies

  • Computer graphics

    • Image manipulation

      • Texturing