Doppler tomography for vector fields

The problem considered is to reconstruct a vector field in the case when, for each line, the distribution of the vector field components along the line is known. Such data are obtained from spectral analysis of signals, affected by Doppler shifts, caused by reflections on moving particles ('velocity spectra'). In the general setting, the problem is unsolved. Using only integrated data along lines, one encounters the same problem as when reconstructing vector fields by means of time-of-flight measurements. Reconstruction formulae are given in this case, establishing the known fact that only the curl of the vector field can be reconstructed without additional information. Inspired by the application to which the paper is most devoted, i.e. ultrasound Doppler measurements on blood flows, the case where the flow takes place in narrow channels is investigated in simulations and experiments.