“Fast approximation to spherical harmonics rotation” by Křivánek, Konttinen, Pattanaik, Bouatouch and Zara

We present a fast and simple approximation of spherical harmonic rotation which decreases the asymptotic complexity and achieves a speedup of four to six in practical applications. The rotation procedure’s simplicity allows implementation in a pixel shader of standard graphics hardware. The main idea is to replace the general spherical harmonic rotation matrix by its truncated Taylor expansion, resulting in a sparse matrix. We show applications in global illumination (radiance interpolation) and GPU-based real-time shading (normal mapping on surfaces with arbitrary BRDFs illuminated by low-frequency environment lighting). Although the rotation approximation is accurate only for small rotation angles, we show this is not a serious limitation in our applications.

1. 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, Eurographics Association, 291–296.
2. Křivánek, J., Gautron, P., Pattanaik, S., and Bouatouch, K. 2005. Radiance caching for efficient global illumination computation. IEEE TVCG 11, 5.
3. Sloan, P.-P., Luna, B., and Snyder, J. 2005. Local, deformable precomputed radiance transfer. ACM Trans. Graph. 24, 3, 1216–1223.