  In vector tomography (VT), the aim is to reconstruct an unknown
multi-dimensional vector field using line integral data. In the case of a
2-dimensional VT, two types of line integral data are usually required. These
data correspond to integration of the parallel and perpendicular projection of
the vector field along integration lines. VT methods are non-invasive,
non-intrusive and offer more information on the field than classical point
measurements; they are typically used to reconstruct divergence-free (or
source-free) velocity and flow fields. In this paper, we show that VT can also
be used for the reconstruction of fields with non-zero divergence. In
particular, we study electric fields generated by dipole sources in bounded
domains which arise, for example, in electroencephalography (EEG) source
imaging. To the best of our knowledge, VT has not previously been used to
reconstruct such fields. We explain in detail the theoretical background, the
derivation of the electric field inverse problem and the numerical
approximation of the line integrals. We show that fields with non-zero
divergence can be reconstructed from the longitudinal measurements with the
help of two sparsity constraints that are constructed from the transverse
measurements and the vector Laplace operator. As a comparison to EEG source
imaging, we note that VT does not require mathematical modelling of the
sources. By numerical simulations, we show that the pattern of the electric
field can be correctly estimated using VT and the location of the source
activity can be determined accurately from the reconstructed magnitudes of the
field.
