A Novel Sampling Theorem on the Rotation Group

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 S03 code implementing these algorithms publicly available.