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A Lie Theoretic Proposal on Algorithms for the Spherical Harmonic Lighting

Chapter
Part of the Mathematics for Industry book series (MFI, volume 4)

Abstract

The spherical harmonics are the angular portion of the solution to the Laplace equation in spherical coordinates and provide a frequency-basis for representing functions on the sphere. The spherical harmonic lighting, as defined by Robin Green at Sony Computer Entertainment in 2003, is a family of real-time rendering techniques that may produce certain realistic shading and shadowing with relatively small overhead lighting. All such spherical harmonic lighting techniques involve replacing parts of standard lighting equations with spherical functions that have been projected into a frequency space using the spherical harmonics as a basis (or a weight space of irreducible finite dimensional representation of the rotation group). In this chapter, using a group theoretical background of spherical harmonics and rather simple realization of the space of functions on the two dimensional sphere in the frame work of representation theory, we propose a possible geometry preserving algebraic/efficient computing, which might accelerate the (numerical and exact) computations slightly for spherical harmonic lighting.

Keywords

Spherical harmonic lighting Global illumination  Spherical harmonics Irreducible representation Intertwiner   Casimir element Legendre polynomials 

References

  1. 1.
    Basri R, Jacobs DW (2003) Lambertian reflectance and linear subspaces. IEEE Trans Pattern Anal Mach Intell 25:218–233Google Scholar
  2. 2.
    Dobashi Y, Kaneda K, Nakashima E, Yamashita H, Nishita T (1995–1999) A quick rendering method using basis functions for interactive lighting design. Comput Graph Forum (Proc EUROGRAPHICS’95) 14(3):229–240Google Scholar
  3. 3.
    Faraut J (2008) Cambridge Studies in Advanced Mathematics. In: Analysis on lie groups, vol 110. Cambridge University Press, CambridgeGoogle Scholar
  4. 4.
    Green R (2003) Spherical harmonic lighting: gritty details, game developers conference 2003Google Scholar
  5. 5.
    Howe R, Chye TE (1992) Non-abelian harmonic analysis. Applications of \(\mathit{SL}(2,\mathbb{R})\). Springer, New YorkGoogle Scholar
  6. 6.
    Schönefeld V (2005) Spherical harmonics. Seminal paper. http://videoarch1.s-inf.de/volker/prosem_paper.pdf
  7. 7.
    Seymour M (2013) The science of spherical harmonics at weta digital. http://www.fxguide.com/featured/the-science-of-spherical-harmonics-at-weta-digital/
  8. 8.
    Shirley P (2001) Realistic ray tracing. A K Peters, NatickGoogle Scholar
  9. 9.
    Sloan P-P (2008) Stupid spherical harmonics (SH) tricks. Game developers conference 2008Google Scholar
  10. 10.
    Sloan P-P, Kautz J, Snyder J (2002) Precomputed radiance transfer for real-time rendering in dynamic. Low-frequency lighting environments. Microsoft research and SIGGRAPHGoogle Scholar
  11. 11.
    Sugiura M (1975) Unitary representations and harmonic analysis. North-Holland/Kodansha, New YorkGoogle Scholar
  12. 12.
    Wakayama M (2013) Representation theory for digital image expression via spherical harmonics (in Japanese). In: Nishii R et al (eds) Mathematical approach to research problems of science and technology—theoretical basis and developments in mathematical modelling. MI lecture notes, vol 46. Kyushu University, FukuokaGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu University/JST CRESTFukuokaJapan

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