Fast, exact (but unstable) spin spherical harmonic transforms

Abstract

In many applications data are measured or defined on a spherical
manifold; spherical harmonic transforms are then required to access
the frequency content of the data.  We derive algorithms to perform
forward and inverse spin spherical harmonic transforms for functions
of arbitrary spin number.  These algorithms involve recasting the spin
transform on the two-sphere S^2 as a Fourier transform on the
two-torus T^2.  Fast Fourier transforms are then used to compute
Fourier coefficients, which are related to spherical harmonic
coefficients through a linear transform.  By recasting the problem as
a Fourier transform on the torus we appeal to the usual Shannon
sampling theorem to develop spherical harmonic transforms that are
theoretically exact for band-limited functions, thereby providing an
alternative sampling theorem on the sphere.  The computational
complexity of our forward and inverse spin spherical harmonic
transforms scale as O(L^3) for any arbitrary spin number, where L is
the harmonic band-limit of the spin function on the sphere.  Numerical
experiments are performed and unfortunately the forward transform is
found to be unstable for band-limits above L~32.  The instability is
due to the poorly conditioned linear system relating Fourier and
spherical harmonic coefficients.  The inverse transform is expected to
be stable, although it is not possible to verify this hypothesis.