New linear transforms for data on a Fourier 2-sphere with application to diffusion MRI

This paper describes a new family of linear transforms for data restricted to the surface of a 2-sphere in three-dimensional Fourier space. These transforms generalize the existing Funk-Radon Transform, which has previously been used with great success to extract microstructural tissue orientation information from high angular resolution magnetic resonance diffusion imaging data. Several properties of the new transforms are described, and computationally efficient implementations are derived using spherical harmonic basis functions. A special case from this family, called the FunkRadon and Cosine Transform, is introduced and evaluated. The method is illustrated with simulated and real diffusion weighted MRI data.