We develop a novel sampling theorem for functions defined on the
three-dimensional rotation group SO(3) by connecting the rotation group to the
three-torus through a periodic extension. Our sampling theorem requires 4L3
samples to capture all of the information content of a signal band-limited at
L, reducing the number of required samples by a factor of two compared to
other equiangular sampling theorems. We present fast algorithms to compute the
associated Fourier transform on the rotation group, the so-called Wigner
transform, which scale as O(L4), compared to the naive scaling of O(L6).
For the common case of a low directional band-limit N, complexity is reduced
to O(NL3). Our fast algorithms will be of direct use in speeding up the
computation of directional wavelet transforms on the sphere. We make our SO3
code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for
publication. Code available at http://www.sothree.or