A method is presented to obtain tomographic inversions of line-integrated measurements in magnetized plasmas, both for the cases of scalar and vector fields. For the Hamiltonian flow of the magnetic field in tokamaks the vector tomographic problem can be reduced to a scalar one. In many fusion experiments the amount of measured data is insufficient to allow an inversion without further assumptions. We present a variational method, where such assumptions can easily be implemented by the use of constraints. We outline the procedures using the example of obtaining the function on tokamaks from Faraday rotation measurements.