Unifying spherical convolutions through lifting
We develop a unified description of convolution on the sphere by lifting to the rotation group. A number of definitions of spherical convolution have been proposed, differing in whether the kernel is axisymmetric or directional, whether the output is on the sphere or the rotation group, whether the signals carry spin, and in their computational cost. No unified treatment exists. Regarding the sphere as the homogeneous space $S^2 \cong \text{SO(3)}/\text{SO(2)}$, we lift a spin $s$ field to the rotation group, convolve on $\text{SO(3)}$, and project back to the sphere through the adjoint of the lift for a potentially different spin $s'$. We recover every rotation-equivariant convolution considered previously through constraints on the kernel, spin choices and inclusion or omission of the lift and projection. For maps between fields of fixed input and output spin, we prove that directional kernels, output on the sphere and equivariance cannot be achieved simultaneously. We make explicit that spin fields are sections of associated line bundles, linking the physics and bundle literatures, and express the spin rotation phase as the character of the twist of the section. We present harmonic-space algorithms for the exact and efficient computation of lifted spherical convolutions of band-limited fields by appealing to sampling theorems on the sphere and rotation group. All operators are implemented in the open-source s2conv code, and numerical experiments confirm the reductions and the equivariance of the operators to machine precision. Finally, we illustrate the operators on fields measured on the Earth and on the celestial sky.