A multicomponent seismic record was transformed to the frequency-wavenumber domain using the quaternion Fourier transform. This transform was integrated into the projection onto convex sets algorithm to allow for the reconstruction of vector signals. The method took advantage of the spectral overlap of components in the frequency-wavenumber domain. The results of the method were compared with standard component-by-component reconstruction. Results were found for synthetic and real data including an example of 5D reconstruction of a converted-wave data set acquired over a heavy oil reservoir. We found an improvement in reconstruction quality producing fully sampled radial and transverse offset-azimuth gathers with a preserved vector relationship.