A 4D Hyperspherical Interpretation of q-Space

3D q-space can be viewed as the surface of a 4D hypersphere. In this paper, we seek to develop a 4D hyperspherical interpretation of q-space by projecting it onto a hypersphere and subsequently modeling the q-space signal via 4D hyperspherical harmonics (HSH). Using this orthonormal basis, we analytically derive familiar q-space metrics and introduce a novel hyperspherical diffusivity measure. Importantly, we derive the integral transform describing the relationship between the diffusion signal and propagator on a hypersphere. We also show that the HSH basis expends less fitting parameters than other well-established methods to achieve comparable signal reconstruction. All in all, this work provides a new way of looking at q-space.