3-D Vector tomography of Doppler-transformed fields by filtered-backprojection

Filtered-backprojection formulae for the tomographic reconstruction of three-dimensional, divergenceless vector fields from their line integrals (Doppler transforms) are obtained, for both the cases of longitudinal and transverse measurement. The longitudinal formula is numerically simulated, and shown to be robust in the presence of noise. If, in the longitudinal case, there are non-zero normal components of the field on the boundary of the reconstruction region, there exists a harmonic component in the field that requires direct boundary measurements to determine. The derivation and implementation of the formulae are simple, and they are therefore easily adapted to specific experimental situations.