We compute reconstructions of 4D and 5D fast-ion phase-space distribution functions in fusion plasmas from synthetic projections of these functions. The fast-ion phase-space distribution functions originating from neutral beam injection (NBI) at TCV and Wendelstein 7-X (W7-X) at full, half, and one-third injection energies can be distinguished and particle densities of each component inferred based on 20 synthetic spectra of projected velocities at TCV and 680 at W7-X. Further, we demonstrate that an expansion into a basis of slowing-down distribution functions is equivalent to regularization using slowing-down physics as prior information. Using this technique in a Tikhonov formulation, we infer the particle density fractions for each NBI energy for each NBI beam from synthetic measurements, resulting in six unknowns at TCV and 24 unknowns at W7-X. Additionally, we show that installing 40 LOS in each of 17 ports at W7-X, providing full beam coverage and almost full angle coverage, produces the highest quality reconstructions.