Ambient dice

We present a family of basis functions designed to accurately and efficiently represent illumination signals on the unit sphere. The bases are built of locally supported functions, needing three to six basis functions in a given direction. This minimizes the number of memory transactions and bandwidth requirements needed for reconstruction. There are three variations of our basis. All are based on storing coefficients at the 12 vertices of an icosahedron. The first one stores the values directly, together with their directional derivatives and hybrid Bezier patches are used for interpolation. This allows one to achieve quality comparable to 3rd-5th order spherical harmonics while still requiring 27 coefficients for the reconstruction. The second variation encodes the signal in YCoCg space and uses a reduced quality, linear reconstruction for the chromaticity components - requiring only 15 coefficients while marginally reducing the quality. The third option exploits the partition of unity formed by cos2 and cos4 restricted to a hemisphere oriented along the directions of the icosahedron vertices. It uses 18 coefficients for the reconstruction, but trades the additional bandwidth requirements for simpler calculations. The quality of that version is still comparable to 3rd order spherical harmonics (SH). We name the basis Ambient Dice as a reference to both: the Ambient Cube basis - as ours is an extension of some of its properties - and the 20-sided dice commonly used in pen-and-paper role-playing games, which is an icosahedron.