  Vector tomography methods intend to reconstruct and visualize vector fields
in restricted domains by measuring line integrals of projections of these
vector fields. Here, we deal with the reconstruction of irrotational vector
functions from boundary measurements. As the majority of inverse problems,
vector field recovery is an ill posed in the continuous domain and therefore
further assumptions, measurements and constraints should be imposed for the
full vector field estimation. The reconstruction idea in the discrete domain
relies on solving a numerical system of linear equations which derives from the
approximation of the line integrals along lines which trace the bounded domain.
This work presents an extensive description of a vector field recovery, the
fundamental assumptions and the ill conditioning of this inverse problem. More
importantly we show that this inverse problem is regularized via the domain
discretization, i.e. we show that the recovery of an irrotational vector field
within a discrete grid employing a finite set of longitudinal line integrals,
leads to a consistent linear system which has bounded solution errors. We
elaborate on the estimation of the solution's error and we prove that this
relative error is finite and therefore a stable vector field reconstruction is
ensured. Such theoretical aspects are critical for future implementations of
vector tomography in practical applications like the inverse bioelectric field
problem. We validate our theoretical results by performing simulations that
reconstruct smooth irrotational fields based solely on a finite number of
boundary measurements and without the need of any additional or prior
information (e.g. transversal line integrals or source free assumption).
